2.1 Microbundles and classifying spaces

Let $\operatorname{TOP}(k)=\{g\colon\mathbb{R}^k\xrightarrow{\approx}\mathbb{R}^k\vert f(0)=0\}$ and let $\operatorname{O}(k)$ be the group of orthogonal k-dimensional matrices. Then there are obvious inclusions $\operatorname{TOP}(k)\hookrightarrow \operatorname{TOP}(k+1)$ and $\operatorname{O}(k)\hookrightarrow \operatorname{O}(k+1)$ , and we denote the corresponding direct limits by $\operatorname{TOP}$ and $\operatorname{O}$ , respectively. We will use the notation $\operatorname{CAT}$ to stand in for $\operatorname{TOP}$ and $\operatorname{O}$ . The classifying spaces $\operatorname{\mathcal{B}\!\operatorname{TOP}}$ and $\operatorname{\mathcal{B}\!\operatorname{O}}$ then classify stable $\mathbb{R}^n$ fibre bundles and stable vector bundles, respectively. The universal stable vector bundle has an underlying stable topological $\mathbb{R}^n$ fibre bundle and its classifying map will be denoted by $\xi\colon \operatorname{\mathcal{B}\!\operatorname{O}}\to\operatorname{\mathcal{B}\!\operatorname{TOP}}$ . Similarly, let $\mathcal{B}_{\operatorname{DIFF}}$ and $\mathcal{B}_{\operatorname{TOP}}$ denote the classifying spaces of stable $\operatorname{DIFF}$ microbundles and stable $\operatorname{TOP}$ microbundles, respectively (see [Reference Kirby and SiebenmannKS77, Essay IV, § 10]), and let $\xi'$ denote the classifying map of the universal $\operatorname{DIFF}$ microbundle’s underlying $\operatorname{TOP}$ microbundle.

Proof. The Kister–Mazur theorem [Reference KisterKis64, Reference Kuiper and LashofKL66, Reference Siebenmann, Guillou and HählSGH73] gives that isomorphism classes of stable $\operatorname{CAT}$ bundles over a CW-complex X are in one-to-one correspondence with isomorphism classes of stable $\operatorname{CAT}$ microbundles over X. This means that there is a natural bijection $\kappa_{\operatorname{CAT}}\colon[X,\operatorname{\mathcal{B}\!\operatorname{CAT}}]\leftrightarrow [X,\mathcal{B}_{\operatorname{CAT}}]$ defined as the composition of the natural bijections

\[ [X,\operatorname{\mathcal{B}\!\operatorname{CAT}}]\to \operatorname{CAT}(X) \to \operatorname{Mic}_{\operatorname{CAT}}(X) \to [X,\mathcal{B}_{\operatorname{CAT}}]\]

where $\operatorname{CAT}(X)$ denotes the set of isomorphism classes of stable $\operatorname{CAT}$ bundles over X (i.e. stable $\mathbb{R}^n$ fibre bundles if $\operatorname{CAT}=\operatorname{TOP}$ and stable vector bundles if $\operatorname{CAT}=\operatorname{DIFF}$ ) and $\operatorname{Mic}_{\operatorname{CAT}}(X)$ denotes the set of isomorphism classes of stable $\operatorname{CAT}$ microbundles over X. In fact, by the definition we can conclude that $\kappa_{\operatorname{CAT}}=(u_{\operatorname{CAT}})_*$ . Then naturality gives the following commutative diagram.

By a simple diagram chase one can see that there exists an element $f\in[\mathcal{B}_{\operatorname{CAT}},\operatorname{\mathcal{B}\!\operatorname{CAT}}]$ such that $(u_{\operatorname{CAT}})^*(f)=\operatorname{Id}_{\operatorname{\mathcal{B}\!\operatorname{CAT}}}$ and such that $(u_{\operatorname{CAT}})_*(f)=\operatorname{Id}_{\mathcal{B}_{\operatorname{CAT}}}$ . Hence, f is the homotopy inverse of $u_{\operatorname{CAT}}$ , completing the proof.

We have the following square which is a homotopy pullback

and it follows that the homotopy fibres of $\xi$ and $\xi'$ are homotopy equivalent. We will denote this space by $\operatorname{TOP}\!/\!\operatorname{O}$ . Boardman and Vogt [Reference Boardman and VogtBV68] showed that we can ‘deloop’ this fibre to obtain a space $\mathcal{B}(\operatorname{TOP}\!/\!\operatorname{O})$ and that we can extend $\xi$ to the right to obtain the fibration sequence

\[\operatorname{TOP}\!/\!\operatorname{O} \to \mathcal{B}\!\operatorname{O}\xrightarrow{\xi} \mathcal{B}\!\operatorname{TOP} \xrightarrow{p} \mathcal{B}(\operatorname{TOP}\!/\!\operatorname{O}).\]

By the above exposition, we can dispense with considering $\mathcal{B}_{\operatorname{TOP}}$ and $\mathcal{B}_{\operatorname{DIFF}}$ and we will consider the classifying map of a $\operatorname{TOP}$ microbundle to be a map to $\operatorname{\mathcal{B}\!\operatorname{TOP}}$ and the classifying map of a $\operatorname{DIFF}$ microbundle to be a map to $\operatorname{\mathcal{B}\!\operatorname{O}}$ . This is helpful as much is known about the homotopy types of $\operatorname{\mathcal{B}\!\operatorname{O}}$ , $\operatorname{\mathcal{B}\!\operatorname{TOP}}$ and $\operatorname{TOP}\!/\!\operatorname{O}$ , whereas it is useful to work solely with $\operatorname{CAT}$ microbundles, rather than passing between microbundles and vector bundles depending on the category.

2.2 Definition of the Casson–Sullivan invariant

Let X be an n-dimensional topological manifold with (potentially empty) boundary $\partial X$ . Further assume that we already have a smooth structure on the boundary $\partial X$ . The manifold X admits a stable topological tangent microbundle $t_X\colon X\to\operatorname{\mathcal{B}\!\operatorname{TOP}}$ . The first obstruction for X to be smoothable is to be able to lift $t_X$ to a stable $\operatorname{DIFF}$ microbundle $\tau_X\colon X\to \mathcal{B}\!\operatorname{O}$ which extends the lift $\tau_{\partial X}$ which is already defined on $\partial X$ , which we have since we already supposed the existence of a smooth structure on the boundary. We have the following diagram.

It is known [Reference Kirby and SiebenmannKS77, Essay IV, § 10.12] that there is a 6-connected map $\operatorname{TOP}\!/\!\operatorname{O}\to K(\mathbb{Z}/2,3)$ , and this means we can consider the composite as $p\circ t_X$ as a map $X\to K(\mathbb{Z}/2,4)$ since X is four-dimensional. The lift $\tau_{\partial X}$ corresponds to a null-homotopy $h_t(\tau_{\partial X})$ of $p\circ (t_{X}\!\vert_{\partial X})$ and, since the inclusion $\partial X\to X$ is a cofibration, this homotopy extends to a homotopy

\[ \widetilde{h}_t(\tau_{\partial X})\colon X\times I \to \mathcal{B} (\operatorname{TOP}\!/\!\operatorname{O}) \]

such that $\widetilde{h}_0(\tau_{\partial X})=p\circ t_X$ and such that $\widetilde{h}_1(\tau_{\partial X})$ defines an element

\[ \widetilde{h}_1(\tau_{\partial X})\in[(X,\partial X),(K(\mathbb{Z}/2,4),\ast)]\cong H^4(X,\partial X;\mathbb{Z}/2). \]

Definition 2.2. Let $(X,\partial X)$ be as above. We define the Kirby–Siebenmann invariant

\[ \operatorname{ks}(X,\partial X):=\widetilde{h}_1(\tau_{\partial X})\in H^4(X,\partial X;\mathbb{Z}/2). \]

We wish to define the Casson–Sullivan invariant as the Kirby–Siebenmann invariant of the mapping cylinder of a homeomorphism, but there is a subtlety that must be addressed first. Let $f\colon X\to X'$ be a homeomorphism of (formally) smooth manifolds. Let $t_X$ , $t_{X'}$ denote the classifying maps of the $\operatorname{TOP}$ microbundles of X and X′ and let $\tau_X$ and $\tau_{X'}$ denote their corresponding lifts to $\operatorname{\mathcal{B}\!\operatorname{O}}$ given by their (formal) smooth structures. The pullback $f^*(\tau_{X'})=\tau_{X'}\circ f$ is not a lift of $t_X$ immediately (rather, it is a lift of $t_{X'}\circ f$ ), but there is a homotopy which is unique up to homotopy to make it a lift of $t_X$ .

To get uniqueness of this homotopy up to homotopy, one can again use Kirby and Siebenman [Reference Kirby and SiebenmannKS77, Essay IV, Proposition 8.1] on $(X\times I) \times I$ , and then use the same argument as above.

Since the map $\xi\colon \operatorname{\mathcal{B}\!\operatorname{O}}\to \operatorname{\mathcal{B}\!\operatorname{TOP}}$ is a fibration, we can lift the homotopy $h(f)_t$ from Lemma 2.8 to a homotopy $\widetilde{h}(f)_t$ such that $\widetilde{h}(f)_0=\tau_{X'}\circ f=f^*(\tau_{X'})$ and such that $\widetilde{h}(f)_1$ is a lift of $t_X$ . Since this homotopy is unique up to homotopy, we will cease to mention this and instead whenever we write $f^*(\tau_{X'})$ it should be taken to mean that we have homotoped this such that it is a lift of $t_X$ (in the unique way). In fact, one can extend the argument in the proof of Lemma 2.8 to see that all higher homotopies are unique up to homotopy, etc., and hence there is an essentially unique classifying map for the tangent microbundle. We will not need this stronger statement.

We now define the Casson–Sullivan invariant.

Definition 2.11. Let X and X′ be n-dimensional (formally) smooth manifolds with (potentially empty) boundaries $\partial X\cong\partial X'$ and let $f\colon X\to X'$ be a homeomorphism restricting to a diffeomorphism on the boundary. Let $M_f$ be the mapping cylinder

\[ M_f:= \frac{(X\times I)\sqcup X'}{(\{x\}\times\{1\})\sim f(x)}\]

and note that $\tau_X\cup_{\partial} f^*(\tau_X)'$ defines a lift of $t_{M_f}$ on $\partial M_f$ . The reduced suspension (of pairs) construction gives an isomorphism

(2.1) \begin{equation}\varpi\colon H^3(X,\partial X;\mathbb{Z}/2) \xrightarrow{\cong} H^4(M_f,\partial M_f;\mathbb{Z}/2).\end{equation}

We then define the Casson–Sullivan invariant $\operatorname{cs}(f)$ as

\[\operatorname{cs}(f):= \varpi^{-1}(\operatorname{ks} (M_f,\partial M_f))\in H^3(X,\partial X;\mathbb{Z}/2).\]

We establish an equivalent characterisation of $\varpi$ which will sometimes be useful.

Hence, $\varpi = \operatorname{PD}^{-1}\circ i_* \circ \operatorname{PD}$ , where $i\colon X\times\{0\}\to X\times I$ denotes the inclusion map.

where $\delta$ denotes the connecting homomorphism in the long exact sequence in cohomology of the triple $(X\times I, \partial(X\times I),(\partial X\times I)\cup X\times\{1\})$ and j denotes the inclusion map of pairs

\[ (X\times\{0\},\partial X)\to (\partial (X\times I),(\partial X\times I)\cup X\times\{1\}). \]

One may readily verify that the maps $\delta$ and $j_*$ are isomorphisms by considering the other two adjacent terms in the long exact sequence and excision, respectively. Commutativity can be obtained using the correct version of naturality of relative cap product. The proof is finished by noting that $\varpi=\delta\circ (j_*)^{-1}$ , since this is precisely the map that induces the (reduced) suspension isomorphism of pairs.

Remark 2.13. It will often be useful for us to think of $\operatorname{cs}(f)$ instead as the element

\[ \varpi(\operatorname{cs}(f))\in H^4(M_f,\partial M_f;\mathbb{Z}/2). \]

In the rest of this paper, we will reserve the symbol $\varpi$ to always mean the isomorphism given in (2.1) so that is clear when we are going between these two elements.

In this paper we will consider only homeomorphisms of 4-manifolds. In this case it is clear by Poincaré duality that the Casson–Sullivan invariant vanishes for all homeomorphisms if X is simply-connected.

2.3 Dependence of the Casson–Sullivan invariant on smooth structures

This subsection is devoted to describing the extent to which the Casson–Sullivan invariant depends on the choice of a (formal) smooth structure. We will begin by giving a different definition for the Casson–Sullivan invariant which more naturally exhibits it as an element of $H^3(X,\partial X;\mathbb{Z}/2)$ .

Let $f\colon X\to X'$ be a homeomorphism and let X be a (formally) smoothable 4-manifold with smooth boundary $\partial X$ (endow $\partial X'$ with a smooth structure using f). For any choice of lifts $\tau_X$ , $\tau_{X'}$ of the topological tangent bundles $t_X$ , $t_{X'}$ we have the following (very non-commutative) diagram, augmented from (2.2).

We will now work to define $\delta(f,\tau_X,\tau_{X'})$ in the diagram, but roughly one should think of it as the ‘difference’ between the lifts $\tau_X$ and $f^*(\tau_{X'})$ . First, note that we have an action of $[X,\partial X;\operatorname{TOP}\!/\!\operatorname{O}]$ on homotopy classes of lifts of the stable tangent microbundle to $\operatorname{\mathcal{B}\!\operatorname{O}}$ , defined using the H-space structure on $\operatorname{\mathcal{B}\!\operatorname{O}}$ which we will denote by the symbol $\oplus$ .

Lemma 2.15. Let $(X,\partial X)$ be a topological 4-manifold with boundary and stable tangent microbundle $t_X$ . There is a well-defined action of $[(X,\partial X), \operatorname{TOP}\!/\!\operatorname{O}]$ on lifts of the stable tangent microbundle to $\operatorname{\mathcal{B}\!\operatorname{O}}$ relative to $\tau_{\partial X}$ -1pt defined by

\[ \delta \cdot \tau_X:=\tau_X\oplus \iota\delta. \]

Furthermore, this action is free and transitive.

Proof. Let $\tau_X\colon X\to \operatorname{\mathcal{B}\!\operatorname{O}}$ be a lift of $t_X$ extending $\tau_{\partial X}$ and let $[X,\partial X;\operatorname{\mathcal{B}\!\operatorname{O}}]_{t_X}$ denote homotopy classes of lifts of $t_X$ extending $\tau_{\partial X}$ relative to the boundary (i.e. lifts of $t_X$ up to homotopy through lifts of $t_X$ restricting to $\tau_{\partial X}$ on $\partial X$ ). By [Reference BauesBau77, Theorem 1.3.8], and the surrounding discussion, there is a bijection

\[ [(X,\partial X), \operatorname{TOP}\!/\!\operatorname{O}] \to [X,\partial X;\operatorname{\mathcal{B}\!\operatorname{O}}]_{t_X}\]

given by $\delta \mapsto \tau_X\oplus\iota\delta$ . This induces an action via the H-space structure on $\operatorname{TOP}\!/\!\operatorname{O}$ , and it being free and transitive follows by the above map being a bijection.

We can now define $\delta(f,\tau_X,\tau_{X'})$ .

Proposition 2.17. Let X be a formally smoothable topological 4-manifold and $f\colon X\to X'$ a homeomorphism. Let $\tau_X$ and $\widetilde{\tau}_X$ be two lifts of the stable tangent microbundle of X, let $\tau_{X'}$ and $\widetilde{\tau}_{X'}$ be two lifts of the stable tangent microbundle of X′, let $a\in [(X,\partial X), \operatorname{TOP}\!/\!\operatorname{O}]$ be the unique element given by Lemma 2.15 such that $a\cdot \tau_X=\widetilde{\tau}_X$ , and similarly let b be such that $b\cdot \tau_{X'}=\widetilde{\tau}_{X'}$ . Then

\[ a+\delta(f,\widetilde{\tau}_X,\widetilde{\tau}_{X'}) = f^*(b) + \delta(f,\tau_X,\tau_{X'}).\]

Hence, it follows that if $X=X'$ , $\tau_X=\tau_X'$ , $\widetilde{\tau_{X}}=\widetilde{\tau}_{X'}$ , and f acts trivially on $H^3(X,\partial X;\mathbb{Z}/2)$ then $\delta(f)$ is defined and does not depend on the choice of lift of the stable tangent microbundle of X.

Proof. We have that

\[ (a+\delta(f,\widetilde{\tau}_X,\widetilde{\tau}_{X'}))\cdot\tau_X=\delta(f,\widetilde{\tau}_X,\widetilde{\tau}_{X'})\cdot(a\cdot\tau_X)=\delta(f,\widetilde{\tau}_X,\widetilde{\tau}_{X'})\cdot\widetilde{\tau}_X=f^*\widetilde{\tau}_{X'},\]

where the first equality is given by using the action, the second from the definition of a, and the third from the definition of $\delta(f,\widetilde{\tau}_X,\widetilde{\tau}_{X'})$ . We also have that

\[ (f^*(b) + \delta(f,\tau_X,\tau_{X'}))\cdot \tau_X = f^*(b)\cdot (\delta(f,\tau_X,\tau_{X'})\cdot\tau_X)= f^*(b)\cdot f^*\tau_{X'}=f^*(b\cdot \tau_{X'})=f^*\widetilde{\tau}_{X'},\]

where the first equality is again given by using the action, the second from the definition of $\delta(f,\tau_X,\tau_{X'})\cdot\tau_X)$ , the third by the fact that the H-space structure is compatible with taking precomposition by the homeomorphism f, and the final equality comes from the definition of b. Since the action of $[(X,\partial X);\operatorname{TOP}\!/\!\operatorname{O}]$ is free by Lemma 2.15, this completes the proof.

It remains to be seen that this definition for $\delta(f)$ matches up with the definition of the Casson–Sullivan invariant, given as the Kirby–Siebenmann invariant of the mapping cylinder.

If we put together the previous two propositions, we immediately obtain the following corollary.

Corollary 2.19. Let X and X′ be topological manifolds each with two (formal) smooth structures $\mathscr{S}^1_X$ and $\mathscr{S}^2_X$ , and $\mathscr{S}_{X'}^1$ and $\mathscr{S}_{X'}^2$ , respectively. Denote the corresponding lifts of the stable tangent microbundle by $\tau_X$ and $\widetilde{\tau}_X$ , and $\tau_{X'}$ and $\widetilde{\tau}_{X'}$ , respectively. Let $f\colon X\to X'$ be a homeomorphism, and let $\operatorname{cs}(f,\mathscr{S}^1_X,\mathscr{S}^1_{X'})$ and $\operatorname{cs}(f,\mathscr{S}^2_X,\mathscr{S}^2_{X'})$ denote the Casson–Sullivan invariants of f with respect to the corresponding (formal) smooth structures. Let $a\in [(X,\partial X), \operatorname{TOP}\!/\!\operatorname{O}]$ be the unique element given by Lemma 2.15 such that $a\cdot \tau_X =\widetilde{\tau}_X$ , and similarly let b be the unique element such that $b\cdot \tau_{X'}=\widetilde{\tau}_{X'}$ . Then we have that

\[ a + \operatorname{cs}(f,\mathscr{S}^1_X,\mathscr{S}^1_{X'})=f^*(b)+\operatorname{cs}(f,\mathscr{S}^2_X,\mathscr{S}^2_{X'}).\]

If we only consider self-homeomorphisms, then $\tau_X=\tau_{X'}$ , $\widetilde{\tau}_X=\widetilde{\tau}_{X'}$ and $a=b$ , giving the formula

\[ a+\operatorname{cs}(f,\mathscr{S}^1_X)=f^*(a)+\operatorname{cs}(f,\mathscr{S}^2_X). \]

This means, for f a self-homeomorphism, that we can define $\operatorname{cs}(f)$ not only for smooth 4-manifolds, but also for smoothable manifolds in the case where all self-homeomorphisms must act trivially on $H^3(-;\mathbb{Z}/2)$ . For example, this occurs if the fundamental group is cyclic. In fact, in these cases one could also define $\operatorname{cs}(f)$ for non-smoothable manifolds by first removing a point, and then using the fact that all topological 4-manifolds admit a smooth structure away from a point [Reference Freedman and QuinnFQ90, § 8.2]. We will not pursue this in the rest of the paper, and instead simplify matters by only considering our manifolds to be smooth.

2.4 Properties

We now establish properties of the Casson–Sullivan invariant. We begin by showing that it is a pseudo-isotopy invariant.

Definition 2.20. Let X and X′ be a pair of topological manifolds and let $f,g\colon X\to X'$ be a pair of homeomorphisms. If X has boundary then f and g restrict to a fixed homeomorphism $f_0\colon \partial X\to \partial X'$ . We say that f is pseudo-isotopic to g if there exists a homeomorphism

\[ F\colon X\times I\to X'\times I\]

such that:

1. (i) $F\vert_{\partial X \times I}=f_0\times \operatorname{Id}$ ;

2. (ii) $F\vert_{X\times \{0\}}=f\colon X\times\{0\}\to X\times\{0\}$ ;

3. (iii) $F\vert_{X\times\{1\}}=g\colon X\times\{1\}\to X\times\{1\}$ .

Such an F is called a pseudo-isotopy. We say that f and g are isotopic if they are pseudo-isotopic via a level-preserving pseudo-isotopy.

Now let X and X′ be smooth manifolds and $f\colon X\to X'$ be a homeomorphism such that f restricts to a diffeomorphism $f_0\colon \partial X\to \partial X'$ . We say that f is smoothable if f is isotopic to a diffeomorphism. We say that f is pseudo-smoothable if f is pseudo-isotopic to a diffeomorphism.

By definition isotopy implies pseudo-isotopy, so if we can obstruct a homeomorphism from being pseudo-smoothable, it cannot be smoothable either.

1. (i) If f is pseudo-smoothable then $\operatorname{cs}(f)=0$ .

2. (ii) If f is pseudo-isotopic to g then $\operatorname{cs}(f)=\operatorname{cs}(g)$ .

We now prove (ii). As in Remark 2.14, $\varpi\operatorname{cs}(f)$ is the homotopy class of $p\circ \tau_{X\times I}$ extended by the null-homotopies given by $\mathscr{S}_X$ and $f^*(\mathscr{S}_{X'})$ . Similarly, $\varpi\operatorname{cs}(g)$ is the homotopy class of $p\circ \tau_{X\times I}$ extended by the null-homotopies given by $\mathscr{S}_X$ and $g^*(\mathscr{S}_{X'})$ . Let F be the pseudo-isotopy from g to f. Then $F^*(\mathscr{S}_{X'}\times I)$ describes a null-homotopy of $p\circ \tau_{X\times I}$ extended by the null-homotopies given by $f^*(\mathscr{S}_{X'})$ and $g^*(\mathscr{S}_{X'})$ , that is, a homotopy between the null-homotopies given by $f^*(\mathscr{S}_X)$ and $g^*(\mathscr{S}_X)$ . This gives us a homotopy relative to the boundary between the relative homotopy classes defining $\varpi\operatorname{cs}(f)$ and $\varpi\operatorname{cs}(g)$ , and hence $\operatorname{cs}(f)=\operatorname{cs}(g)$ .

We can say more in the case of self-homeomorphisms. Let $\widetilde{\pi}_0\operatorname{Homeo}(X,\partial X)$ denote the pseudo-mapping class group of X relative to $\partial X$ , that is, the quotient of $\operatorname{Homeo}(X,\partial X)$ by those homeomorphisms which are pseudo-isotopic to the identity (see Definition 6.2 and Remark 6.3). Then we have the following result.

Proposition 2.23. Let X be a smooth 4-manifold. Then the map

\[ \operatorname{cs}\colon \widetilde{\pi}_0\operatorname{Homeo}(X,\partial X)\to H^3(X,\partial X;\mathbb{Z}/2) \]

sending a representative of a pseudo-isotopy class of self-homeomorphisms to its Casson–Sullivan invariant is a crossed homomorphism. In other words, if $f,g\colon X\to X$ are representatives of pseudo-isotopy classes of self-homeomorphisms then

\[\operatorname{cs}(g\circ f)= \operatorname{cs}(f) + f^*\operatorname{cs}(g).\]

We begin by giving a method for describing the group operation on $[X\times I,\partial; K(\mathbb{Z}/2,4)]$ . The normal way to define this is by using the highly-connected map $\mathcal{B}(\operatorname{TOP}\!/\!\operatorname{O})\to \Omega K(\mathbb{Z}/2,5)$ and then using composition of loops to define the group operation (see [Reference HatcherHat02, §4.3]). However, we also have that $[X\times I,\partial; K(\mathbb{Z}/2,4),\ast]= [\Sigma X_+,\{\operatorname{pt}\};K(\mathbb{Z}/2,4),\ast]$ , where $X_+:=X\sqcup\{\operatorname{pt}\}$ and $\Sigma X_+$ denotes the reduced suspension of $X_+$ . Hence, $[X\times I,\partial; K(\mathbb{Z}/2,4)]$ has a natural operation by ‘stacking’ cylinders given by the natural group operation coming from the reduced suspension. By the suspension-loopspace adjoint relation,

\[ [\Sigma X_+,\{\operatorname{pt}\};K(\mathbb{Z}/2,4),\ast] = [X_+,\{\operatorname{pt}\};\Omega K(\mathbb{Z}/2,4),\ast] = [X_+,\{\operatorname{pt}\};\Omega \Omega K(\mathbb{Z}/2,5),\ast]\]

and the group operation given by ‘stacking’ corresponds to the group operation given by the outermost loopspace on the right. The standard group operation comes from the inner loopspace structure, but these two are equivalent; this is entirely analogous to the fact that $\pi_2$ of any space is abelian (see, for example, [Reference HatcherHat02, §4.3, p. 395–396]).

Let $\mathscr{S}_X$ denote the smooth structure on X. Consider the homotopy class of maps corresponding to $\varpi(\operatorname{cs}(f)+f^*\operatorname{cs}(g))=\varpi\operatorname{cs}(f) + (f\times \operatorname{Id})^*\varpi\operatorname{cs}(g)$ . (Here we used the definition of $\varpi$ from Definition 2.11.) We again use the terminology in Remark 2.14. This corresponds to stacking the homotopy class of $p\circ \tau_{X\times I}$ extended by the null-homotopies given by $\mathscr{S}_X$ and $f^*(\mathscr{S}_X)$ with the homotopy class of $p\circ \tau_{X\times I}$ extended by the null-homotopies given by $f^*(\mathscr{S}_X)$ and $(g\circ f)^*(\mathscr{S}_X)$ . But this is relatively homotopic to the homotopy class of $p\circ \tau_{X\times I}$ extended by the null-homotopies given by $\mathscr{S}_X$ and $(g\circ f)^*(\mathscr{S}_X)$ . The relative homotopy is given by the null-homotopy corresponding to $(f\times \operatorname{Id})^*(\mathscr{S}_X\times I)$ . This completes the proof.

Let X and X′ be four-dimensional smooth manifolds and let $f\colon X\to X'$ be an orientation-preserving homeomorphism restricting to a diffeomorphism on $\partial X$ . If we use a disc $D\subset X$ to perform a connected sum with $S^2\times S^2$ , and use the same disc in $S^2\times S^2$ and the disc $f(D)\subset X'$ to form a connected sum $X'\# (S^2\times S^2)$ , this produces a well-defined homeomorphism

\[ f_\#\colon X\# (S^2\times S^2)\to X'\# (S^2\times S^2) \]

by extending f onto the $S^2\times S^2$ summand via the identity map.

For certain manifolds, however, the connected-sum homeomorphism is well-defined. For example, if one of the connected summands is $S^2\times S^2$ and the other is simply-connected, then the connected-sum homeomorphism is well-defined. For more information, see [Reference Auckly, Kim, Melvin and RubermanAKM⁺15]. This is, unfortunately, not very useful to our purposes since the simply-connected assumption renders the Casson–Sullivan invariant trivial.

Pinching off a 4-sphere yields a standard degree-one map $S^2\times S^2\to S^4$ and by extending this map via the identity onto a 4-manifold X we obtain a degree-one collapse map $\ell\colon X\# (S^2\times S^2)\to X\# S^4$ .

Proposition 2.26. The standard degree-one map that collapses the $S^2\times S^2$ -1.5pt connected summand $\ell\colon X\#(S^2\times S^2)\to X$ (as above) induces an isomorphism

\[ \ell^* \colon H^3(X,\partial X;\mathbb{Z}/2)\to H^3(X \# (S^2\times S^2),\partial X;\mathbb{Z}/2) \]

such that $\ell^*(\operatorname{cs}(f))=\operatorname{cs}(f_\#)$ .

Since $f_\#$ restricts to the identity map on the $S^2\times S^2$ summand and the tangent bundle of $S^2\times S^2$ is stably trivial, the stable tangent microbundle $t_{(X\# (S^2\times S^2))\times I}$ is homotopic to the stable tangent microbundle $t_{X\times I}$ precomposed with the map $\ell\times \operatorname{Id}$ . This means that $\varpi\operatorname{cs}(f_\#)$ , the homotopy class of $p\circ t_{(X\# (S^2\times S^2))\times I}$ extended by the null-homotopies given by $\mathscr{S}_{X\# (S^2\times S^2)}$ and $f_{\#}^*(\mathscr{S}_{X'\# (S^2\times S^2)})$ , is relatively homotopic to the homotopy class of $p\circ t_{X\times I}\circ (\ell\times \operatorname{Id})$ extended by the null-homotopies given by precomposing the null-homotopies corresponding to $\mathscr{S}_X$ and $f^* \mathscr{S}_{X'}$ with the map $\ell$ . Since $(\ell\times \operatorname{Id})^*(\varpi\operatorname{cs}(f))=\varpi(\ell^*\operatorname{cs}(f))$ , it follows that $\ell^*(\operatorname{cs}(f))=\operatorname{cs}(f_\#)$ .

This result is stated and proved in [Reference Freedman and QuinnFQ90, §8.6]. However, the proof is somewhat dispersed in the book and many of the details are not given. For this reason, we give the full proof below.

Proof of Proposition 2.27. Let $W:=M_f$ be the mapping cylinder of f. Since $\operatorname{cs}(f)=0$ , we can by definition smooth the stable topological normal bundle of W relative to the given smoothings of the stable tangent bundles of the boundary. By Kirby and Siebenmann [Reference Kirby and SiebenmannKS77, Essay IV, Theorem 10.1], we can realise this by a smooth structure on W extending the given structures on the boundary. Note that here we have crucially used that the dimension of W is at least 5 and at most 7. This allows us to view W as a (relative) smooth h-cobordism (W, X, X′). Note that W is topologically a product (by definition) but that it is not necessarily a product smoothly, and that W already restricts to a smooth product on $\partial X$ . We aim to turn W into a smooth product cobordism via stabilisations.

We may assume in the standard way that W has only 2- and 3-handles in its relative (to X) handle decomposition (for a reference, see [Reference S., Kalmar, Kim, Powell and RayBKK⁺21, § 20.1]). Since W is topologically a product we know that the 2- and 3-handles algebraically cancel. Let $\mathcal{W}$ be a collection of immersed Whitney discs in $X\times\{1/2\}$ for the pairs of cancelling intersections of the descending manifolds for the 3-handles and the ascending manifolds for the 2-handles. If these discs were embedded disjointly then we could use Whitney moves on the discs to force the 2- and 3-handles to geometrically cancel and hence W could be made into a smooth product cobordism. Let $p\in D_1\cap D_2$ be an intersection point for two Whitney discs $D_1,D_2\in\mathcal{W}$ (potentially $D_1=D_2$ ) and let $\alpha$ be an arc in W from $X\times\{0\}$ to $X\times\{1\}$ which intersects $D_1$ and $D_2$ exactly once at p and is disjoint from all other discs in $\mathcal{W}$ . Let $q=(q_1,q_2)$ be a point in $S^2\times S^2$ . Then we form a new cobordism

\[ W':= (W\setminus \nu \alpha)\cup_{\partial\nu \alpha = (\partial\nu q)\times I} ((S^2\times S^2 \setminus \nu q)\times I).\]

We define a new set of Whitney discs $\mathcal{W}'$ to be the same as $\mathcal{W}$ except with $D_1$ and $D_2$ replaced with $D_1'$ and $D_2'$ , respectively, defined as

\begin{align*} D_1'&:= (D_1\cap (W\setminus\nu \alpha)) \cup ( (S^2\times\{q_2\})\cap (S^2\times S^2\setminus\nu q)), \\ D_2'&:= (D_2\cap (W\setminus\nu \alpha)) \cup ( (\{q_1\}\times S^2)\cap (S^2\times S^2\setminus\nu q)).\end{align*}

The Whitney discs in $\mathcal{W}'$ then pair the cancelling intersections of the 2- and 3-handles of W′. The number of intersections between $D_1'$ and $D_2'$ is one fewer than the number of intersections between $D_1$ and $D_2$ ; we have effectively removed an intersection point. This is known as the Norman trick. Repeating this procedure for all intersections between Whitney discs, eventually we produce a smooth cobordism W′′ which is topologically a product $W''\approx X\# (\#^k S^2\times S^2)\times I$ for some non-negative integer k. The set of Whitney discs for the pairs of cancelling intersections of the 2- and 3-handles $\mathcal{W}''$ consists now of disjointly embedded Whitney discs, and hence (as noted above) W′′ is diffeomorphic to a product cobordism relative to one end of the cobordism. We now make this conclusion precise.

Let $X_\#:=X\# (\#^k S^2\times S^2)$ , $X'_{\#}:= X'\# (\#^k S^2\times S^2)$ and let $\mathscr{S}$ , $\mathscr{S}'$ denote the smooth structures on $X_\#$ and $X'_{\#}$ , respectively. Now W′′ is the mapping cylinder of $f_\#$ , that is, $W'' \cong X_\# \times I$ , with the smooth structure $\mathscr{S}_{W''}$ where $\mathscr{S}_{W''}\vert_{X_\#\times\{0\}}=\mathscr{S}$ and $\mathscr{S}_{W''}\vert_{X_\#\times\{1\}}=f_\#^*(\mathscr{S}')$ . In the previous paragraph, we obtained a homeomorphism

\[ F\colon X_\#\times I \to X_\#\times I \]

such that $F^*(\mathscr{S}'\times I)=\mathscr{S}_W''$ and such that $F\vert_{X_{\#}\times\{0\}\cup \partial X_{\#}\times I}$ is the identity. By Definition 6.1 this means we have produced a pseudo-isotopy between the smooth structures $\mathscr{S}$ and $f_\#^*(\mathscr{S}')$ and, by the same argument as in the third paragraph of the proof of Proposition 6.5, this means that $f_\#$ is pseudo-isotopic to a diffeomorphism.

To summarise, Propositions 2.26 and 2.27 together tell us that the Casson–Sullivan invariant is the stable obstruction to pseudo-smoothing homeomorphisms of 4-manifolds, much as the Kirby–Siebenmann invariant is the stable obstruction to smoothing 4-manifolds.

2.4.1 Non-compact 4-manifolds

Non-compact 4-manifolds have the property that they are easier to smooth than their compact counterparts, in the sense that non-compact 4-manifolds always admit smooth structures (and hence compact 4-manifolds can always be smoothed away from a point) [Reference Freedman and QuinnFQ90, Chapter 8.2]. In light of this, one might wonder whether a stronger result than Proposition 2.27 holds if we assume our manifold is non-compact. We briefly explain what happens in this case. We begin with some definitions (cf. § 6).

Definition 2.28. Let M be a topological manifold. We say that smooth structures $\mathscr{S}$ and $\mathscr{S}'$ on M are concordant if there exists a smooth structure $\mathscr{T}$ on $M\times I$ such that $\mathscr{T}\vert_{M\times \{0\}}=\mathscr{S}$ and $\mathscr{T}\vert_{M\times \{1\}}=\mathscr{S}'$ .

We say that $\mathscr{S}$ and $\mathscr{S}'$ are sliced concordant if they are concordant, as above, such that the projection $M\times I \to I$ is a submersion with respect to the smooth structure $\mathscr{T}$ on $M\times I$ .

Proposition 2.29. Let X be a topological 4-manifold with $\mathscr{S}$ and $\mathscr{S}'$ smooth structures on X, and let $f\colon X\to X$ be a homeomorphism restricting to a diffeomorphism on $\partial X$ with respect to the smooth structures $\mathscr{S}\vert_{\partial X}$ and $\mathscr{S}'\vert_{\partial X}$ . If $\operatorname{cs}(f)=0$ , then $\mathscr{S}$ and $f^*(\mathscr{S}')$ are concordant. If X is non-compact and $\operatorname{cs}(f)=0$ , then $\mathscr{S}$ and $f^*(\mathscr{S}')$ are sliced concordant.

This proposition essentially follows (in the non-compact case) from the following theorem.

Theorem 2.30 ([Reference SiebenmannSie71, Theorem 4.4] and [Reference Freedman and QuinnFQ90, Theorem 8.7B]). There is a one-to-one correspondence between sliced concordance classes of smooth structures on a non-compact 4-manifold with homotopy classes of liftings of the stable tangent microbundle to $\operatorname{\mathcal{B}\!\operatorname{O}}$ .

Remark 2.31. Note that although in [Reference Freedman and QuinnFQ90] they state the above theorem for concordance classes rather than sliced concordance classes, the references they refer to for the proof give it for sliced concordance classes.

Proof of Proposition 2.29. The fact that $\mathscr{S}$ and $f^*(\mathscr{S}')$ are concordant is clear in the context of the proof of Proposition 2.27. In fact, this is exactly what the first paragraph of the proof establishes. If $\operatorname{cs}(f)=0$ , then by Lemma 2.15 and Proposition 2.18 the lifts of the stable tangent microbundle corresponding to $\mathscr{S}$ and $f^*(\mathscr{S})$ are homotopic, so by Theorem 2.30 these smooth structures are sliced concordant.

Remark 2.32. Note that an important part of the proof of Theorem 2.30 is that the map

\[ \operatorname{TOP}(4)/{\operatorname{O}}(4) \to {\operatorname{TOP}}/{\operatorname{O}} \]

is 5-connected [Reference Freedman and QuinnFQ90, Theorem 9.7A], and this theorem rests on the result of Quinn [Reference QuinnQui86] that $\pi_4(\operatorname{TOP}(4)/{\operatorname{O}}(4))=0$ , the proof of which was found to contain a gap. The proof, however, was recently corrected in [Reference Gabai, Gay, Hartman, Krushkal and PowellGGH⁺26].

A concordance, sliced or otherwise, between the smooth structures $\mathscr{S}$ and $f^*(\mathscr{S}')$ does not obviously give any nice statement about the properties of f itself. Hence, the author does not know of an interpretation of Proposition 2.29 in terms of the smoothability of the homeomorphism (cf. § 6).

2.5 A connected-sum formula over a circle for the Casson–Sullivan invariant

This subsection is devoted to proving a connected sum along a circle formula for the Casson–Sullivan invariant. We shall start by giving the necessary definitions.

Definition 2.33. Let $X_1$ and $X_2$ be a pair of smooth 4-manifolds and let $\gamma_i \subset X_i$ be a pair of framed, embedded circles. Then we define the connected sum over $\gamma_1,\gamma_2$ to be the smooth manifold

\[ X_1\#_{\gamma_1=\gamma_2} X_2:=(X_1\setminus\nu\gamma_1)\cup_\varphi(X_2\setminus\nu\gamma_2)\]

where the gluing is performed using the orientation-reversing map $\varphi\colon S^1\times S^2\to S^1\times S^2$ which sends $(x,y)\to (x,a(y))$ for $a\colon S^2\to S^2$ the antipodal map. For a precise description of how this gives a well-defined smooth manifold, see [Reference KosinskiKos93, § IV.4].

Let $X_1, X_2$ , and $X'_1, X'_2$ be two pairs of smooth, orientable, compact 4-manifolds and let $\gamma_i \subset X_i$ be a pair of embedded circles for $i=1$ , 2. Furthermore, let $f_i\colon X_i\to X'_i$ be a pair of homeomorphisms. Since our manifolds are orientable, these circles admit framings which we will use implicitly (the choice will have no effect from the perspective of the Casson–Sullivan invariant). Let $\nu \gamma_i$ denote the two tubular neighbourhoods. If we use these to perform the connected sum over a circle $X_1\#_{\gamma_1=\gamma_2}X_2$ , and use $f(\nu\gamma_i)$ perform the connected sum over a circle $X'_1\#_{f_1(\gamma_1)=f_2(\gamma_2)}X'_2$ , this gives a connected sum over a circle homeomorphism

\[ f_\# \colon X_1\#_{\gamma_1=\gamma_2} X_2 \to X'_1\#_{f_1(\gamma_1)=f_2(\gamma_2)} X'_2\]

defined as $f_i$ on $(X_i\setminus\nu\gamma_i)$ . A priori this homeomorphism depends on the choice of framings, but we will ignore these choices and consider any such result the connected sum over a circle homeomorphism. Like with the case of connected-sum homeomorphisms, there is an additional complication when one wants to consider self-homeomorphisms. In this case, we must first assume that the $f_i$ map $\gamma_i$ to $\gamma_i$ as free homotopy classes. Then, up to isotopy, we may assume by homotopy implies isotopy, isotopy extension [Reference Edwards and KirbyEK71], uniqueness of normal bundles [Reference Freedman and QuinnFQ90, Chapter 9.3], and the calculation of the mapping class group of $S^1\times S^2$ (see [Reference GluckGlu62]) that the $f_i$ are either the identity map on a tubular neighbourhood of the curve $\gamma_i$ or the Gluck twist map

(2.2) \begin{align} T\colon S^1 \times D^3 &\to S^1 \times D^3 \nonumber\\ (t,x) &\mapsto (t, \Phi_t(x))\end{align}

where $\Phi_t$ denotes the (positive) rotation map of $D^3$ around the (oriented) straight line from the south pole to the north pole by an angle of t (we have used the identification $S^1\cong [0,2\pi]/0\sim2\pi$ ). If the homeomorphisms $f_i$ are both of the same type as above (i.e. both are identity maps or both are twist maps on tubular neighbourhoods of $\gamma_i$ ) we may then define the connected sum of these homeomorphisms over the curves $\gamma_i$ to be

\[ f_\# \colon X_1\#_{\gamma_1=\gamma_2} X_2 \to X_1\#_{\gamma_1=\gamma_2} X_2\]

as $f_i$ on $(X_i\setminus\nu\gamma_i)$ . However, there was a choice in how to isotope the $f_i$ such that the homeomorphisms fix the curves $\gamma_i$ , and so it is unlikely that the resulting homeomorphism is well-defined, even up to isotopy. We will ignore this ambiguity, and whenever we write the connected sum over a circle for self-homeomorphisms, we shall simply mean one such choice of a homeomorphism. The above also generalises to general case, that is, for $X_1$ , $X_2$ , $X'_1$ , $X'_2$ and $\gamma_i\subset X_i$ , $\gamma'_i\subset X'_i$ being two pairs of embedded circles, but one has to assume that the homeomorphisms $f_i$ send the free homotopy classes of $\gamma_i$ to $\gamma'_i$ .

We now define a map that will be very important to us.

Definition 2.34. Let $X_{\#}:=X_1\#_{\gamma_1=\gamma_2}X_2$ . Consider the long exact sequence for the triple

\[ (X_{\#},\partial X_{\#}\sqcup \partial \overline{\nu}\gamma_1,\partial X_{\#}) \]

and the Mayer–Vietoris sequence for the decomposition

\[ (X_\#,\partial X_\# \sqcup\partial\overline{\nu}\gamma_1)=((X_1\setminus\nu\gamma_1) \cup (X_2\setminus\nu\gamma_2),\partial(X_1\setminus\nu\gamma_1)\cup\partial(X_2\setminus\nu\gamma_2)) \]

with $\mathbb{Z}/2$ -coefficients (which we will now suppress). The Mayer–Vietoris sequence yields an isomorphism

\[ \mathfrak{a}\colon H^3(X_{\#},\partial {\kern1pt}\sqcup \partial\nu\gamma_1)\to H^3(X_1\setminus\nu\gamma_1,\partial)\oplus H^3(X_2\setminus\nu\gamma_2,\partial) \]

and the sequence of the triple yields a surjectionFootnote ¹

\[ \mathfrak{b}\colon H^3(X_{\#},\partial{\kern1pt}\sqcup\partial\nu\gamma_1)\to H^3(X_{\#},\partial). \]

The inclusion $X_i\setminus\nu\gamma_i\subset X_i$ induces an isomorphism $\mathfrak{c}\colon H_1(X_1\setminus\nu\gamma_1)\oplus H_1(X_2\setminus\nu\gamma_2)\to H_1(X_1)\oplus H_1(X_2)$ . We then define a map

\[ Q\colon H^3(X_1,\partial)\oplus H^3(X_2,\partial)\to H^3(X_{\#},\partial) \]

as $Q:=\mathfrak{b}\circ \mathfrak{a}^{-1}\circ (\operatorname{PD}^{-1},\operatorname{PD}^{-1})\circ \mathfrak{c}^{-1}\circ (\operatorname{PD},\operatorname{PD})$ .

Proof. All of the maps used in the definition of Q are isomorphisms except for $\mathfrak{b}$ , which is a surjection. Hence, Q is surjective. Furthermore, we see that

\[ \ker Q = (\operatorname{PD}^{-1},\operatorname{PD}^{-1})\circ \mathfrak{c}\circ (\operatorname{PD},\operatorname{PD})\circ \mathfrak{a}(\ker\mathfrak{b}). \]

Now $\ker\mathfrak{b}$ is generated by $\operatorname{PD}^{-1}[\gamma_1]$ , and hence (by a standard naturality of cap product argument) we see that $(\operatorname{PD},\operatorname{PD})\circ \mathfrak{a}(\ker\mathfrak{b})=([\gamma_1],[\gamma_2])$ . The result follows immediately.

The rest of this subsection will be devoted to proving the following theorem.

Theorem 2.36. Let $X_1, X_2$ and $X'_1, X'_2$ be two pairs of compact, connected, smooth, orientable 4-manifolds and let $\gamma_i \subset X_i$ and $\gamma'_i\subset X'_i$ be two pairs of embedded circles, with $f_i\colon X_i\to X'_i$ a pair of homeomorphisms such that $(f_i)_*[\gamma_i]=[\gamma'_i]$ as free homotopy classes, and such that the connected-sum homeomorphism

\[ f_{\#}:=f_1\#_{\gamma_1=\gamma_2}f_2\colon X_1 \#_{\gamma_1=\gamma_2}X_2 \to X'_1\#_{\gamma'_1=\gamma'_2}X'_2\]

is defined. Let Q be the map in Definition 2.34. Then

\[\operatorname{cs}(f_\#)=Q(\operatorname{cs}(f_1),\operatorname{cs}(f_2)).\]

The proof of this theorem relies on a sequence of diagram chases, using the relevant Mayer–Vietoris exact sequences and long exact sequences of the triple. Although we have already used these, the setup here will be more complicated and so we recall them. The long exact sequence for the triple of CW-complexes $W\supset B \supset A$ in cohomology (with $\mathbb{Z}/2$ -coefficients suppressed) is

(2.3) \begin{equation}\cdots\to H^k(W,B) \to H^k(W,A)\to H^k(B,A) \to H^{k+1}(W,B) \to\cdots\end{equation}

and the fully relative Mayer–Vietoris sequence for the pair of CW-complexes $(W,Y)=(A\cup B, C\cup D)$ (with $\mathbb{Z}/2$ -coefficients suppressed) is

(2.4) \begin{equation}\cdots\to H^k(W,Y)\to \begin{matrix} H^k(A,C)\\ \oplus \\ H^k(B,D)\end{matrix} \to H^k(A\cap B,C\cap D) \to H^{k+1}(W,Y) \to\cdots\end{equation}

See [Reference HatcherHat02, p. 200/204] for these standard exact sequences.

First we give the notation for the setup.

Setup 2.37. Let $X_1, X_2$ and $X'_1, X'_2$ be two pairs of compact, connected, smooth, orientable 4-manifolds, let $\gamma_i \subset X_i$ be a pair of embedded circles, and let $f_i\colon X_i\to X'_i$ be a pair of homeomorphisms such that $(f_i)_*[\gamma_i]=[\gamma'_i]\in\pi_1(X_i)$ and such that the $f_i$ restrict to diffeomorphisms $(f_i)_0\colon \partial X_i\to \partial X'_i$ . Denote by $X_\#$ and $X'_\#$ the connected sum $X_1\#_{\gamma_1=\gamma_2} X_2$ and $X'_1\#_{\gamma'_1=\gamma'_2} X'_2$ , respectively, and by $f_\#\colon X_\#\to X'_\#$ the connected-sum homeomorphism. We then set up the following notation.

1. (i) Let $W:= M_{f_\#}$ the mapping cylinder of $f_{\#}$ .

2. (ii) Let $E:= \partial (\nu \gamma_1) \times I \subset W$ .

3. (iii) Let $F:= \partial W = (X_\# \times \{0\}) \cup (\partial X_\# \times I) \cup (X_\# \times \{1\})\subset W$ .

4. (iv) Let $A:= (X_1\setminus(\nu\gamma_1)) \times I \subset W$ and let $B:= (X_2\setminus(\nu\gamma_2)) \times I \subset W$ .

5. (v) Let $F_A:=F\cap A$ and let $F_B:=F\cap B$ .

6. (vi) Let $C:= F_A\cup E\subset A$ and let $D:= F_B\cup E\subset B$ .

7. (vii) Let $Y:= C\cup D$ .

Note that $W=A\cup B$ and $A\cap B=C\cap D=E$ . See Figure 1.

Figure 1.

A dimension-reduced picture of Setup 2.37 where the pictured tori denote the connected sum $S^1\times S^2$ . Note that the labels for C and D have been omitted.

Combining the sequences (2.3) and (2.4) for the triples (W, Y, F), $(A,C,F_A)$ , and $(B,D,F_B)$ and the pairs $(W,Y)=(A\cup B,C\cup D)$ , $(W,F)=(A\cup B, F_A\cup F_B)$ , and $(Y,F)=(C\cup D, F_A\cup F_B)$ we obtain the following commutative diagram which we will use extensively.

In what follows, for all of our cohomology groups we are using $\mathbb{Z}/2$ -coefficients but we will suppress these in the diagrams.

It will be useful to simplify the above diagram. First, note that the leftmost and rightmost groups on the second line vanish because $A\cap B=C\cap D$ (further observe that this means the remaining non-trivial horizontal map on the second line must be an isomorphism). The topmost groups and the leftmost group on the third line are all isomorphic to $H^3(E,\partial E)\cong \mathbb{Z}/2$ . We can replace all of the remaining outer groups with zeros since these maps must be zero maps (one can explicitly see this by continuing the sequences and using commutativity along with Poincaré duality). To illustrate this, we will show that the bottom vertical maps are zero. Showing that the map to $H^4(A\cap B, F_A\cap F_B)$ is zero is analogous. We start by continuing the sequences at the bottom of the diagram to obtain the following diagram.

We want to show that the maps a and b are both the zero maps. It suffices to show that d and e are injections. By Poincaré duality, e is dual to a direct sum of inclusion-induced maps

\[ H_0(E)\oplus H_0(E) \to H_0(A)\oplus H_0(B) \]

which is clearly an isomorphism since both A, B and E are connected. Hence, e is injective. Now we claim that f is the zero map, since the previous map in the Mayer–Vietoris sequence is the diagonal map and hence is injective. This means that c is injective, and so by commutativity d must be injective also, and hence a and b are both the zero maps. The preceding simplification yields the following diagram.

(2.5)

We note that the topmost horizontal map is the diagonal map $\Delta\colon x\mapsto (x,x)$ . Before giving the proof of Theorem 2.36 we consider a special case in the form of a lemma.

Proof of Theorem 2.36. We will use the notation from Setup 2.37 throughout. Consider the pair $(W,Y)=(A\cup B, C\cup D)$ . First, we will consider a special case, namely when $X_2=S^1\times S^3$ , $\gamma_2 = S^1\times\{\operatorname{pt}\}$ and $f_2=\operatorname{Id}_{X_2}$ . We will then prove the general case of the theorem via our consideration of the special case.

Let $X_2=S^1\times S^3=X'_2$ , let $\gamma_2=S^1\times\{\operatorname{pt}\}$ and let $f_2=\operatorname{Id}_{X_2}$ . Then $X_2\setminus (\nu\gamma_2)\cong S^1\times D^3$ , hence $X_\# \cong X_1$ and $f_\# = f_1$ . The map $i_B$ is then Poincaré dual to the inclusion-induced map $H_1(S^1\times S^2\times I) \to H_1(S^1\times D^3 \times I)$ and hence is injective. It follows by commutativity that r is injective. Consider any element $x\in H^4(W,F)$ . From the diagram and injectivity of r, we can see that there are two possible lifts of x in $H^4(W,Y)$ that differ by r(z) where $\mathbb{Z}/2=\mathbb{Z}/2\langle z\rangle$ . We will define a preferred lift t(x) by specifying that the element $t(x)= (t_1(x),0)\in H^4(A,C)\oplus H^4(B,D)$ , which uniquely determines t (the fact that the second coordinate can be made to be zero follows from $i_B$ being surjective in this special case). Now consider $\varpi\operatorname{cs}(f_1)\in H^4(W,F)$ . By naturality of the Kirby–Siebenmann invariant (Remark 2.9), q maps

\[ \operatorname{ks}(W,Y)=(\operatorname{ks}(A,C),\operatorname{ks}(B,D))\in H^4(W,Y) \]

to $\varpi\operatorname{cs}(f_1)$ , and, by the definition of $f_2$ , we have that $\operatorname{ks}(B,D)=0$ . This means that $t(\varpi\operatorname{cs}(f_1))=\operatorname{ks}(W,Y)$ .

We have produced a map $\mathfrak{A}\colon H^3(X_1,\partial X_1)\to H^4(A,C)$ as the composition

\[ H^3(X_1,\partial X_1)\xrightarrow{\varpi} H^4(W,F)\xrightarrow{t} H^4(A,C)\oplus H^4(B,D)\xrightarrow{\operatorname{pr}_1} H^4(A,C).\]

Similarly, by considering the reverse special case where $X_1=S^1\times S^3=X'_1$ , $f_1=\operatorname{Id}_{X_1}$ and where $f_2$ is any homeomorphism $f_2\colon X_2 \to X'_2$ , we obtain in the same way a map $\mathfrak{B}\colon H^3(X_2,\partial X_2)\to H^4(B,D)$ .

Now consider general $X_1,X_2,X'_1,X'_2,f_1,f_2$ . Write $\alpha:= \mathfrak{A}(\operatorname{cs}(f_1))$ and $\beta:= \mathfrak{B}(\operatorname{cs}(f_2)$ . Consider the element $(\alpha,\beta)\in H^4(A,C)\oplus H^4(B,D)\cong H^4(W,Y)$ and map it down using q to $q(\alpha,\beta)$ . We have that $q(\alpha,\beta)=\varpi\operatorname{cs}(f_\#)$ , as by construction $(\alpha,\beta)=(\operatorname{ks}(A,C),\operatorname{ks}(B,D))$ and by naturality of the Kirby–Siebenmann invariant, this must map to $\operatorname{ks}(W,F)=\varpi\operatorname{cs}(f_\#)$ via q, since q is inclusion-induced.

So we have constructed a map $P\colon H^3(X_1,\partial X_1)\oplus H^3(X_2,\partial X_2)\to H^3(X_{\#},\partial X_{\#})$ as the composition

\[ H^3(X_1,\partial X_1)\oplus H^3(X_2,\partial X_2) \xrightarrow{(\mathfrak{A},\mathfrak{B})} H^4(A,C)\oplus H^4(B,D) \xrightarrow{q} H^4(W,F)\xrightarrow{\varpi^{-1}} H^3(X_{\#},\partial X_{\#})\]

which sends $(\operatorname{cs}(f_1),\operatorname{cs}(f_2))\mapsto \operatorname{cs}(f_\#)$ .

It remains to show that this map is equal to the map Q from Definition 2.34. We show this now. First, analogously to in Setup 2.37, consider the long exact sequences for the triples:

1. (i) $(X_\#,\partial X_\#\sqcup\partial\overline{\nu}\gamma_1,\partial X_\#)$ ;

2. (ii) $(X_i\setminus\nu\gamma_i,\partial(X_i\setminus\nu\gamma_i),\partial X_i)$ ;

and the relative Mayer–Vietoris sequences for the pairs:

1. (i) $(X_\#,\partial X_\#)=((X_1\setminus\nu\gamma_1) \cup (X_2\setminus\nu\gamma_2),\partial X_1 \cup \partial X_2)$ ;

2. (ii) $(X_\#,\partial X_\#\sqcup\partial\overline{\nu}\gamma_1)=((X_1\setminus\nu\gamma_1) \cup (X_2\setminus\nu\gamma_2),\partial(X_1\setminus\nu\gamma_1)\cup\partial(X_2\setminus\nu\gamma_2))$ .

These sequences give the following commutative diagram analogous to (2.5) (note that we have simplified the notation by writing $\partial$ on its own to refer to the boundary of a manifold when it is clear by context which manifold is being referred to).

(2.6)

One can see that diagram (2.6) is isomorphic to (2.5) using the isomorphism given in Definition 2.11 (or the analogous isomorphisms). Since those isomorphisms come from Poincaré duality and inclusion maps, the two parallel diagrams must commute. This allows us to reinterpret the construction of our map P on the level of the 4-manifolds themselves, rather than the mapping cylinders, which then allows us to relate this map to Q.

Consider the following diagram which, aside from the lowermost map and group, splits as a direct sum of diagrams.

(2.7)

The two vertical maps are direct sums of the respective Mayer–Vietoris sequences. The top-left horizontal map comes from the direct sum of the respective long exact sequences of triples. The lowest vertical map is defined using maps from Definition 2.34. Ignoring the notated lifts t and t′ for now, we show that this diagram commutes. The commuting of the lower triangle is trivial. We now show that the top rectangle commutes. Since the rectangle is a direct sum of two diagrams, it suffices to show that one of the diagram summands commutes. We write this as follows, dropping the indices (i.e. $X_1=X$ , etc.).

(2.8)

Let $x\in H^3(X,\partial X\sqcup \partial\overline{\nu}\gamma)$ and let y denote its image in $H^3(X,\partial)$ . By the vertical isomorphism, we see that x maps to an element $(\operatorname{PD}^{-1}\alpha_1,\operatorname{PD}^{-1}\alpha_2)\in H^3(X\setminus\nu,\partial)\oplus H^3(\overline{\nu}\gamma,\partial)$ , where $\alpha_1$ is the homology class of some curve in X and $\alpha_2$ is trivial or is the homology class of $\gamma$ . Mapping this pair to the right and then up gives the class $\alpha_1+\alpha_2$ , which is the homology class of some curve in X. We now need to show that y is Poincaré dual to $\alpha_1+\alpha_2$ . Since we are using $\mathbb{Z}/2$ -coefficients, we will view these cohomology groups as the hom-duals of the respective homology groups. The element y is defined such that it evaluates on the relative cycle s in X first by considering s as a further relative cycle $s'\in H_3(X,\partial X\sqcup\partial\overline{\nu}\gamma)$ and then evaluating using x. By the isomorphism in the corresponding Mayer–Vietoris sequence in homology, any such cycle splits as a pair $(s_1,s_2)$ of relative cycles in $H_3(X\setminus\nu\gamma,\partial)$ and $H_3(\overline{\nu}\gamma,\partial)$ , respectively. Hence, evaluating x(s′) is the same as evaluating $\alpha_1(s_1)+\alpha_2(s_2)$ , and so y is Poincaré dual to $\alpha_1+\alpha_2$ . This completes the proof that the diagram commutes.

Consider again diagram (2.7) and let $(x_1,x_2)\in H^3(X_1,\partial)\oplus H^3(X_2,\partial)$ . This pair is mapped by P by sending it down the left of the diagram, first by using the preferred lift t, then using the $(\operatorname{pr}_1,\operatorname{pr}_1)$ and then finally using the lowest vertical map. The lift t is such that $t(x_1,x_2)$ is of the form $((t(x_1),0),(t(x_2),0))$ .

The lift t′ is defined as at the beginning of Definition 2.34: t′ is defined as $\mathfrak{c}^{-1}\circ (\operatorname{PD},\operatorname{PD})$ postcomposed with the inclusion as a direct sum. This is the unique lift of $(\operatorname{PD},\operatorname{PD})(x_1,x_2)$ which is of the form $(\ast,0,\ast,0)$ . This means that the pair $(x_1,x_2)$ is mapped by Q by sending it down the right of the diagram, first by using the lift t′ and then mapping $t'(x_1,x_2)$ down again to $H^3(X_{\#},\partial)$ via $(\operatorname{PD}^{-1}\circ \operatorname{pr}_1,\operatorname{PD}^{-1}\circ\operatorname{pr}_1))$ and the lowest vertical map.

To finish the proof, we need to know that the middle triangle formed by the two lifts, t and t′, commutes. The fact that the diagram (without the lifts) commutes shows that

\[ y:=((\operatorname{PD},\operatorname{PD}),(\operatorname{PD},\operatorname{PD}))(t(x_1,x_2)) \]

is a lift of $(\operatorname{PD},\operatorname{PD})(x_1,x_2)$ . Since y is also of the form $(\ast,0,\ast,0)$ (as $t(x_1,x_2)$ is) the observation in the above paragraph means that $y=t'(x_1,x_2)$ , and hence the triangle commutes.

2.6 A connected-sum formula for the Casson–Sullivan invariant

Before finishing this section we will note that, similarly to Theorem 2.36, we also have a formula for the Casson–Sullivan invariant under the actual connected-sum operation. Since all of the arguments are the same as or simpler than for connected summing over a circle, we will not give many details. Let $X_1, X_2$ and $X'_1, X'_2$ be two pairs of compact, connected topological 4-manifolds and let $f_i\colon X_i\to X'_i$ be a pair of homeomorphisms. As mentioned previously (see Remark 2.25 and the preceding paragraph), we can form a connected-sum homeomorphism $f_\#\colon X_1\# X_2 \to X'_1\# X'_2$ . If we let $q_i\colon X_1\# X_2\to X_i$ denote the collapse maps onto the ith connected summand, then we have the following formula.

Theorem 2.38. Let $X_1, X_2$ and $X'_1, X'_2$ be two pairs of compact, connected, smooth, orientable 4-manifolds with $f_i\colon X_i\to X'_i$ a pair of homeomorphisms such that the connected-sum homeomorphism

\[ f_{\#}:=f_1\# f_2\colon X_1\# X_2\to X'_1\# X'_2 \]

is defined. Let $q_i$ be the pair of collapse maps defined above. Then

\[ \operatorname{cs}(f_\#)=q_1^*\operatorname{cs}(f_1)+q_2^*\operatorname{cs}(f_2). \]

We will not give the proof here, as it is by exactly the same method as for proving Theorem 2.36 but with simpler arguments. In particular, having the degree-one collapse is very useful (such collapse maps do not always exist for connected sums over circles). This result will only be used for proving Corollary 1.6 in § 3.3.

3.1 Surgery

We now fundamentally make use of the following theorem, due to Ronnie Lee. This subsection will contain Wall’s surgery obstruction groups, the L-groups, but we will postpone recalling their definition (see Definition 4.5) until § 4 where the surgery exact sequence will be recalled. The only fact we need here is that $L_5(\mathbb{Z}[\mathbb{Z}])\cong \mathbb{Z}$ (see [Reference WallWal70, Theorem 13A.8]).

The proof of the above theorem has only been available in a handwritten note, which is not easily accessible.Footnote ² For a modernised exposition of this proof, see [Reference GalvinGal24a] (available on the author’s website).

Using this, we obtain the following corollary.

Corollary 3.4. There exists a homeomorphism

\[\sigma\colon (S^1\times S^3) \# (S^2\times S^2)\to (S^1\times S^3) \# (S^2\times S^2)\]

realising the generator of $L_5(\mathbb{Z}[\mathbb{Z}])$ . More precisely, let N be the standard cobordism between $S^1\times S^3$ and $(S^1\times S^3)\#(S^2\times S^2)$ . Then the surgery problem

\[ W = (N \cup_\sigma -N) \to (S^1\times S^3) \times I\]

has surgery obstruction the generator of $L_5(\mathbb{Z}[\mathbb{Z}])$ . Furthermore, $\sigma\vert_{\nu\theta}=\operatorname{Id}_{\nu\theta}$ , where $\theta=S^1\times\{\operatorname{pt}\}\subset S^1\times S^3$ .

3.2 Proof of Proposition 3.1

Throughout this subsection, let $\sigma\colon (S^1\times S^3) \# (S^2\times S^2)\to (S^1\times S^3) \# (S^2\times S^2)$ and W be as in Corollary 3.4. The aim is to prove Proposition 3.1 by showing that this $\sigma$ has $\operatorname{cs}(\sigma)$ the generator of $H^3((S^1\times S^3)\# (S^2\times S^2);\mathbb{Z}/2)$ . This will be done by performing operations on W given in Corollary 3.4 and keeping track of what happens to the relative Kirby–Siebenmann invariant of the cobordism along the way. We begin with some lemmas.

In what follows, we will also need the following lemma.

Lemma 3.6. Under the standard identification

\[ H^4(W,\partial W;\mathbb{Z}/2) \cong H^4(M_\sigma,\partial M_{\sigma};\mathbb{Z}/2)\xrightarrow{\varpi^{-1}} H^3((S^1\times S^3)\# (S^2\times S^2);\mathbb{Z}/2)\]

(see Definition 2.11) we have that $\operatorname{ks}(W,\partial W)=\varpi\operatorname{cs}(\sigma)$ where $M_\sigma$ denotes the mapping cylinder.

Proof. Note that N is smoothable relative to the standard smooth structure on the boundary. It is clear that we have a homeomorphism relative to the boundary

\[ W\approx N \cup_{(S^1\times S^3)\# (S^2\times S^2)} (((S^1\times S^3)\#(S^2\times S^2))\times I) \cup_{\sigma}-N;\]

that is, inserting a product $(S^1\times S^3\#S^2\times S^2)\times I$ to the right end of N does not change the relative homeomorphism type of W. We now see that the identification

\[ H^4(W,\partial W;\mathbb{Z}/2) \xrightarrow{\cong} H^4(M_\sigma,\partial M_\sigma;\mathbb{Z}/2) \]

gives $\operatorname{ks}(W,\partial W)=\operatorname{ks}(M_\sigma)=\varpi\operatorname{cs}(\sigma)$ where $M_\sigma$ is the mapping cylinder of $\sigma$ (note that here we write ‘ $=$ ’ as there is only one isomorphism between groups isomorphic to $\mathbb{Z}/2$ .).

Lemma 3.7 [Reference Freedman and QuinnFQ90, Proof of 11.6A]. Let $S^1\times E_8 \to S^1\times S^4$ be the surgery problem which has surgery obstruction the inverse of the generator of $L_5(\mathbb{Z}[\mathbb{Z}])$ , let $\gamma\subset W$ be an embedded curve representing the generator of $H_1(W)$ , and let $\gamma'$ be the embedded curve $S^1\times\{\operatorname{pt}\}\subset S^1\times E_8$ . Then the connected-sum surgery problem

\[ W\#_{\gamma=\gamma'}(S^1\times E_8)\to (S^1\times S^3)\times I\]

has vanishing surgery obstruction.

We can now prove the proposition.

Proof of Proposition 3.1. Start with the cobordism W and modify the surgery obstruction to be trivial using Lemma 3.7. Since the result has vanishing surgery obstruction, we can surger it relative to the boundary to an s-cobordism W′. Since $\pi_1(W')\cong \pi_1(S^1\times S^3)\cong\mathbb{Z}$ is good we can use the s-cobordism theorem [Reference Freedman and QuinnFQ90, Theorem 7.1A], which gives that W′ is homeomorphic to the mapping cylinder $M_f$ for some homeomorphism $f\colon S^1\times S^3\to S^1\times S^3$ . By Lemma 3.5, $\operatorname{cs}(f)=0$ and hence $\operatorname{ks}(W',\partial W')=0$ .

We now need to keep track of how we modified the Kirby–Siebenmann invariant throughout this process. The main tool for doing so will be the long exact sequence of the triple (see (2.3)) with $\mathbb{Z}/2$ -coefficients suppressed throughout. When we used Lemma 3.7 to kill the surgery obstruction, the connected sum over a circle was induced by a relative cobordism C between $W\sqcup (E_8\times S^1)$ and $W\#_{\gamma=\gamma'}(S^1\times E_8)$ . Let $W_\#:= W\#_{\gamma=\gamma'}(S^1\times E_8)$ . This relative cobordism C is formed by attaching a single 1-handle (at one point on $\gamma$ and at one point on $\gamma'$ ) and then a single 2-handle which attaches by going along $\gamma$ , then the 1-handle, then $\gamma'$ , and then back along the 1-handle. From the long exact sequence of the triple $(C,W\sqcup S^1\times E_8, \partial W)$ we have the sequence

\[ H^4(C,W\sqcup S^1\times E_8) \to H^4(C,\partial W) \to H^4(W\sqcup S^1\times E_8,\partial W) \to H^5(C,W\sqcup S^1\times E_8).\]

The outer groups $H^4(C,W\sqcup S^1\times E_8)$ and $H^5(C,W\sqcup S^1\times E_8)$ must both vanish since the cobordism C was made by only attaching 1- and 2-handles. Hence, we get an isomorphism

(3.1) \begin{equation} H^4(C,\partial W) \xrightarrow{\cong} H^4(W\sqcup S^1\times E_8,\partial W)\cong H^4(W,\partial W)\oplus H^4(S^1\times E_8)\cong \mathbb{Z}/2\oplus \mathbb{Z}/2.\end{equation}

By considering the long exact sequence of the triple $(C,W_{\#},\partial W_{\#})$ we have the following commutative diagram.

The leftmost vertical isomorphism again comes from the relative handle decomposition of C, and the rightmost vertical isomorphism is clear by the definition of $W_\#$ . The middle vertical isomorphism comes from (3.1) and the fact that $H^*(C,\partial W_{\#})\cong H^*(C,\partial W)$ . The leftmost horizontal map can be seen to be the diagonal map since the generator of $H^4(C,W_{\#})$ is Poincaré–Lefschetz dual to the annulus with boundaries homologous to the generators of $H_1(W)$ and $H_1(S^1\times E_8)$ . The rightmost horizontal map is then given by exactness. Using this and naturality of the Kirby–Siebenmann invariant, we can deduce that

\[ \operatorname{ks}(W_{\#},\partial W_{\#})=\operatorname{ks}(W,\partial W)+\operatorname{ks}(S^1\times E_8), \]

where we can naturally identify the groups as there is only one isomorphism between groups isomorphic to $\mathbb{Z}/2$ .

Using a similar argument, one can see that the surgeries used to surger $W_{\#}$ to the s-cobordism W′ do not alter the Kirby–Siebenmann invariant (the relative handle decomposition for the relative cobordism given by the trace of any given surgery consists of only a single handle, so the computation is greatly simplified). Hence, since by naturality $\operatorname{ks}(S^1\times E_8)=1$ ,

\[ \operatorname{ks}(W',\partial W')=\operatorname{ks}(W_{\#},\partial W_{\#})=\operatorname{ks}(W,\partial W) +\operatorname{ks}(S^1\times E_8)= \operatorname{ks}(W,\partial W)+1. \]

As we already concluded that $\operatorname{ks}(W',\partial W')=0\in \mathbb{Z}/2$ , this implies that $\operatorname{ks}(W,\partial W)=1$ , and hence, by Lemma 3.6, $\operatorname{cs}(\sigma)$ is the generator of $H^3((S^1\times S^3)\# (S^2\times S^2);\mathbb{Z}/2)$ , as claimed.

3.3 An interesting example

We can use the objects and tools that we have developed so far to illustrate an interesting example involving the Casson–Sullivan invariant that demonstrates its dependence on smooth structures, as is expected by § 2.3. This will also prove Corollary 1.6.

Let $\mathscr{S}$ denote the standard smooth structure on X and (in an abuse of notation) also denote the induced formal smooth structure (see Remark 2.7). Then $g\colon X_{\mathscr{S}} \to X_{\mathscr{S}}$ is (isotopic to) a diffeomorphism, and hence $\operatorname{cs}(g)=0$ with respect to the smooth structure $\mathscr{S}$ . However, let $f^*(\mathscr{S})$ be the smooth structure on X induced by f. Using Proposition 2.17, we can compute the Casson–Sullivan invariant of g with respect to the smooth structure $f^*(\mathscr{S})$ . As in Proposition 2.17, let $a\in [X,\operatorname{TOP}\!/\!\operatorname{O}]$ be the unique element such that $a\cdot \mathscr{S} = f^*(\mathscr{S})$ . We then have that

\begin{align*}\operatorname{cs}(g)&= \delta(g,f^*(\mathscr{S})) \\ &= a + g^*(a)+\delta(g,\mathscr{S}) \\ &= a+ g^*(a) \\ &= \delta(f,\mathscr{S})+g^*(\delta(f,\mathscr{S})) \\ &= \operatorname{cs}(f)+g^*(\operatorname{cs}(f)),\end{align*}

which is equal to the element $(1,1)\in \mathbb{Z}/2\oplus \mathbb{Z}/2\cong H^3(X;\mathbb{Z}/2)$ by Theorem 2.38 and Proposition 3.1. So g is no longer smoothable with respect to the smooth structure $f^*(\mathscr{S})$ .

Corollary 1.6 follows immediately from this example.

Proof of Corollary 1.6. Take $\mathscr{S}':=f^*(\mathscr{S})$ . Let $X_{\mathscr{S}}$ denote X with the standard smooth structure, and let $X_{\mathscr{S}'}$ denote X with the smooth structure $\mathscr{S}'$ . By definition, $f\colon X_{\mathscr{S}}\to X_{\mathscr{S}'}$ is a diffeomorphism, so the two smooth structures are diffeomorphic (see Definition 6.1). By the calculation in Theorem 3.9, g is not stably pseudo-smoothable with respect to the smooth structure $\mathscr{S}'$ , since its Casson–Sullivan invariant is non-trivial.

4.1 The surgery exact sequence

For a reference, see [Reference RanickiRan02, Chapter 13] in high dimensions, and for the four-dimensional surgery exact sequence, see [Reference Freedman and QuinnFQ90, Chapter 11] or [Reference S., Kalmar, Kim, Powell and RayBKK⁺21, Chapter 22]. We will use the surgery sequence in both the smooth and topological categories, so to simplify the notation we will use $\operatorname{CAT}$ to stand in for both $\operatorname{DIFF}$ and $\operatorname{TOP}$ .

The surgery exact sequence consists of three types of objects, which we now recall: the structure set, the normal invariants, and the L-groups.

Definition 4.1. Let $(M,\partial M)$ be a connected $\operatorname{CAT}$ n-manifold with (potentially empty) boundary. Then the $\operatorname{CAT}$ structure set of M, denoted by $\mathcal{S}_{\operatorname{CAT}}(M,\partial M)$ is the set of all equivalence classes of pairs $(N,\varphi)$ where N is a $\operatorname{CAT}$ n-manifold and $\varphi$ a homotopy equivalence $\varphi\colon N\xrightarrow{\simeq} M$ such that the restriction $\varphi\vert_{\partial N}\colon \partial N \to \partial M$ is a $\operatorname{CAT}$ isomorphism. The equivalence relation is that $(N,\varphi)\sim (N',\varphi')$ if there exists a (relative) $\operatorname{CAT}$ h-cobordism $(W;N,N')$ with a homotopy equivalence

\[\Phi \colon W \to M\times [0,1]\]

such that $\Phi\vert_{N}=\varphi\colon N\to M\times\{0\}$ and $\Phi\vert_{N'}=\varphi'\colon N' \to M\times\{1\}$ and such that

\[ \Phi\vert_{\partial N\times [0,1]}=(\varphi\vert_{\partial N}\times \operatorname{Id}_{[0,1]})\colon \partial N\times[0,1] \to \partial M\times [0,1]. \]

Similarly, there is a simple $\operatorname{CAT}$ structure set, denoted $\mathcal{S}_{\operatorname{CAT}}^s$ , which is defined analogously to the regular structure set but with all homotopy equivalences replaced with simple homotopy equivalences (and hence all h-cobordisms replaced with s-cobordisms). Sometimes will we write $\mathcal{S}_{\operatorname{CAT}}^h$ to specify that we mean the regular structure set.

We now define the normal invariants.

Proposition 4.2. Let G(k) be the monoid of homotopy equivalences $S^{k-1}\to S^{k-1}$ , let G denote the direct limit of the inclusions $G(k)\hookrightarrow G(k+1)$ . Then there are fibration sequences

\[ G/\!\operatorname{O} \to \operatorname{\mathcal{B}\!\operatorname{O}} \to \operatorname{\mathcal{B} G} \to \mathcal{B}(G/\!\operatorname{O})\]

and

\[ G/\!\operatorname{TOP} \to \operatorname{\mathcal{B}\!\operatorname{TOP}} \to \operatorname{\mathcal{B} G} \to \mathcal{B}(G/\!\operatorname{TOP}).\]

We now define Wall’s surgery obstruction groups.

In the even case (if $n\equiv 0,2\pmod{4}$ ), $L_n(\mathbb{Z}[\pi])$ is defined as the set of stable equivalence classes of $(-1)^{n/2}$ -quadratic forms over stably free $\mathbb{Z}[\pi]$ -modules.

In the odd case (if $n\equiv 1,3\pmod{4}$ ), $L_n(\mathbb{Z}[\pi])$ is defined as the set of stable equivalence classes of $(-1)^{(n-1)/2}$ -quadratic formations over stably free $\mathbb{Z}[\pi]$ -modules.

For a reference on (quadratic) forms and formations see [Reference RanickiRan81, § 1.6]. There are also simple L-groups, denoted by $L_n^s(\mathbb{Z}[\pi])$ . We will not give the definition here, but for a reference see [Reference LückLüc25, § 9.10].

We can now state the surgery exact sequence, due to Browder, Novikov, Sullivan and Wall (and Freedman and Quinn in dimension 4).

Theorem 4.7 ([Reference WallWal70, Theorem 10.8] and [Reference Freedman and QuinnFQ90, Theorem 11.3A]). Let M be a compact, connected, oriented n-dimensional $\operatorname{CAT}$ manifold. Then for $\operatorname{CAT}=\operatorname{DIFF}$ and $n\geqslant 5$ we have the following exact sequence of pointed sets, which continues to the left in the obvious manner.

And for $\operatorname{CAT}=\operatorname{TOP}$ and $n\geqslant 5$ (and $n=4$ provided that $\pi$ is a good group; see Remark 1.8) we have the following analogous exact sequence of abelian groups.

Furthermore, there are simple versions of both of these exact sequences where the L-groups are replaced by the simple L-groups and the structure sets are replaced by the simple structure sets.

4.2 Forming mapping cylinders from the surgery exact sequence

We now specialise to the case of interest. Let X be a compact, connected, smooth 4-manifold with boundary $\partial X$ . The obstruction to lifting an element of $\mathcal{N}_{\operatorname{TOP}}$ to $\mathcal{N}_{\operatorname{DIFF}}$ is given by the map $\xi_*$ induced by the fibration

\[ G/\operatorname{O} \to G/\operatorname{TOP} \xrightarrow{\xi} \mathcal{B}(\operatorname{TOP}\!/\!\operatorname{O}).\]

We will consider the following augmented part of the $\operatorname{TOP}$ and $\operatorname{DIFF}$ surgery exact sequences for X.

The idea is to construct a mapping cylinder for a homeomorphism with non-trivial Casson–Sullivan invariant from this sequence. The way we will do this is by finding an element $N\in \mathcal{N}_{\operatorname{TOP}}(X\times I,\partial)$ which has vanishing surgery obstruction $\theta(N)$ , but has $\xi_*(N)\neq 0$ . First, we need to understand the map $\xi_*$ more, which we do via the following two lemmas.

Lemma 4.8. We have an isomorphism

\[ [(X\times I,\partial), G/\operatorname{TOP}] \cong H^2(X\times I,\partial;\mathbb{Z}/2)\oplus H^4(X\times I,\partial;\mathbb{Z}). \]

Proof. This follows from the work of Sullivan [Reference Sullivan, Ranicki, Casson, Sullivan, Armstrong, Rourke and CookeSul96], which can be found in [Reference RanickiRan02]. Instead, though, we will refer to [Reference Madsen and MilgramMM79] for its exposition on this topic. In particular, it follows from [Reference Madsen and MilgramMM79, Remark 4.36] that

\[ (G/\operatorname{TOP}[2])_5 \simeq K(\mathbb{Z}_{(2)},4) \times K(\mathbb{Z}/2,2)\]

where $G/\operatorname{TOP}[2]$ denotes the 2-localisation of $G/\operatorname{TOP}$ and $(G/\operatorname{TOP})_5$ denotes its fifth Postnikov stage. From this, one can use [Reference Madsen and MilgramMM79, 4.35] to see that

\[ (G/\operatorname{TOP})_5 \simeq K(\mathbb{Z},4) \times K(\mathbb{Z}/2,2)\]

and, together with the fact that $X\times I$ is a five-dimensional CW-complex, a standard obstruction-theoretic argument completes the proof of the lemma.

Lemma 4.9. We have a commutative diagram

where m is the map sending $(x,y) \mapsto (\operatorname{red}_2(y))$ .

Morita [Reference MoritaMor72, Proposition 3] gives us that m is the map sending $(x,y)\mapsto \operatorname{Sq}^2(x)+\operatorname{red}_2(y)$ . Now it suffices to show that $\operatorname{Sq}^2(x)=0$ . We have a commutative diagram (by the naturality of Steenrod squares)

and the lower horizontal map must vanish since $\operatorname{Sq}^i\colon H^j(-;\mathbb{Z}/2)\to H^{j+i}(-;\mathbb{Z}/2)$ is the zero map for $i\lt j$ .

Now we construct the mapping cylinder.

Proof. Assume that we have an element $N\in\mathcal{N}_{\operatorname{TOP}}(X\times I,\partial)$ . By Remark 4.4 this means we can consider N to be a manifold $(N,\partial N)$ together with $\partial N$ homeomorphic to $\partial (X\times I)$ , and hence $\partial N$ has an induced smooth structure given by X. Furthermore, we assume that $\theta(N)=0$ . This means that we can lift N to an element (also denoted by N) in the structure set $\mathcal{S}^s_{\operatorname{TOP}}(X\times I, \partial)$ . The homeomorphism $\partial N \approx \partial (X\times I)\cong X\times \{0\}\cup \partial X\times I \cup X\times\{1\}$ induces a decomposition $\partial N = \partial_+ N \cup \partial_0 N \cup \partial_- N$ . If we define $D:= \partial (\partial_+ N)$ , then we may assume that $\partial_0 N \cong D\times I$ . By the relative s-cobordism theorem [Reference Freedman and QuinnFQ90, Theorem 7.1A] (which applies since X has good fundamental group by assumption) there exists a homeomorphism relative to $\partial_+N \cup (D\times I)$ ,

\[ (N,\partial_+ N) \xrightarrow{\approx} (\partial_+ N \times I,\partial_+ N\times\{0\}),\]

such that this homeomorphism restricts to the identity on $\partial_+N \cup (D\times I)$ . Let $\widetilde{f}$ denote the restriction of this homeomorphism to $\partial_- N\to \partial_+N\times\{1\}$ . Since $N\in\mathcal{S}^s_{\operatorname{TOP}}(X\times I, \partial)$ , we also have a (simple) homotopy equivalence, restricting to a homeomorphism on the boundary

\[ (N;\partial_+ N,\partial_0 N,\partial_- N) \xrightarrow{\simeq} (X\times I; X\times \{0\},\partial X\times I,X\times\{1\}).\]

Putting this together, we have a commutative diagram

where f is defined such that the diagram commutes.

It follows that our constructed element $N\in\mathcal{S}^s_{\operatorname{TOP}}(X\times I, \partial)$ is homeomorphic to the mapping cylinder $M_{f}:=(X\times I) \cup_{f} X$ , restricting to a diffeomorphism on the boundary by construction.

4.3 Proofs of Theorems 1.7 and 1.9

We now define the technical ‘realisability condition’ which was mentioned in the introduction.

Definition 4.11. Let X be a closed, connected, smooth, orientable 4-manifold with $\pi_1(X)\cong\pi$ where $\pi$ is a good group. We say that X satisfies the Casson–Sullivan realisability condition if the surgery obstruction map (after reidentifying the normal invariants using Lemma 4.8)

\[\theta\colon H^2(X\times I,\partial;\mathbb{Z}/2)\oplus H^4(X\times I,\partial;\mathbb{Z}) \to L_5(\mathbb{Z}[\pi])\]

is such that for every $y\in H^4(X\times I,\partial;\mathbb{Z})$ there exists an $x\in H^2(X\times I,\partial;\mathbb{Z}/2)$ such that $\theta(x,y)=0$ .

We begin with a simple but essential observation.

Now it only remains to prove the main theorem, but most of the work has already been done.

Proof of Theorem 1.7. Let X be as in the statement of the theorem and let $\eta\in H^3(X;\mathbb{Z}/2)$ .

Recall the diagram from Lemma 4.9. We claim that for any $z\in H^4(X\times I,\partial;\mathbb{Z}/2)$ there exists a $y\in H^4(X\times I,\partial; \mathbb{Z})$ such that $m(0,y)=\operatorname{red}_2(y)=z$ . This is readily seen by considering the Bockstein exact sequence

\[ H^4(X\times I,\partial;\mathbb{Z})\xrightarrow{\operatorname{red}_2} H^4(X\times I,\partial;\mathbb{Z}/2)\to H^5(X\times I,\partial;\mathbb{Z})\xrightarrow{\cdot 2} H^5(X\times I,\partial;\mathbb{Z}).\]

The first map is surjective since the last map is injective. This completes the proof of the claim.Footnote ³

So, given any $\eta\in H^3(X,\partial X;\mathbb{Z}/2)$ , we can find an element $y\in H^4(X\times I,\partial;\mathbb{Z})$ such that

\[ \varpi^{-1}m(0,y)=\eta. \]

By the realisability condition, there exists an $x\in H^2(X\times I,\partial ;\mathbb{Z}/2)$ such that $\theta(x,y)=0$ . Define $N_\eta$ such that $N_\eta$ maps to (x,y) under the isomorphism given in Lemma 4.8. By Lemma 4.9, it follows that $\xi_*(N_\eta)=\varpi\eta$ . Hence, by Proposition 4.10 and Lemma 4.12, we have that there exists a homeomorphism $f\colon X\to X$ with $\operatorname{cs}(f)=\eta$ .

We finish this section by proving Theorem 1.9, which follows easily from the construction of the non-pseudo-smoothable homeomorphisms produced by Theorem 1.7.

Proof of Theorem 1.9. Let X be as in the statement of the theorem. Then, as in the above proof of Theorem 1.7, for every non-zero $\eta\in H^3(X;\mathbb{Z}/2)$ there exists a mapping cylinder $M_f=(X\times I)\cup_f X$ such that $\operatorname{cs}(f)=\eta\neq0$ . Hence, f is not stably pseudo-isotopic to a diffeomorphism by Propositions 2.22 and 2.26. However, now also note that $M_f\in\mathcal{S}^s_{\operatorname{TOP}}(X\times I, \partial)$ and hence we have a homotopy equivalence

\[ ((X\times I)\cup_f X;X\times \{0\}, X\times\{1\}) \simeq (X\times I;X\times\{0\},X\times\{1\})\]

which by construction restricts to the identity $\operatorname{Id}_X$ on $X\times\{0\}$ and f on $X\times \{1\}$ . By postcomposing this homotopy equivalence with the projection to X, this produces a homotopy between f and $\operatorname{Id}_X$ .

4.4 Applications

In light of the previous subsection, it is natural to ask for which 4-manifolds Theorems 1.7 and 1.9 apply. This subsection is devoted to giving such examples. In all that follows, let X be a closed, connected, smooth, orientable 4-manifold with $\pi_1(X)\cong\pi$ .

Let $\Gamma$ be a finite cyclic group. Then it follows from Bak [Reference BakBak76] that $L^s_5(\mathbb{Z}[\Gamma])=0$ . Hence, for $\pi\cong \Gamma$ we have that the surgery obstruction vanishes trivially and hence Theorems 1.7 and 1.9 apply. We can say more, however.

Proposition 4.13 follows from a result of Hambleton, Milgram, Taylor and Williams, of which we now restate the relevant part (adapted to our situation).

Theorem 4.15 [Reference Hambleton, Milgram, Taylor and WilliamsHMT⁺88, Theorem A]. Let M be a five-dimensional manifold with boundary $\partial M$ such that $\pi_1(M)\cong \pi$ is finite and $\operatorname{im}(SK_1(\mathbb{Z}[\rho])\to SK_1(\mathbb{Z}[\pi]))=0$ , where $\rho$ is the Sylow 2-subgroup of $\pi$ . Then the surgery obstruction map

\[\theta\colon [(M,\partial M),G/\operatorname{TOP}] \to L_5^s(\mathbb{Z}[\pi])\]

is given by

\[ x \mapsto \kappa^s_3(c_*(\operatorname{Arf}_3(x))).\]

In the above theorem, $\kappa^s_3$ denotes the map (constructed in [Reference Hambleton, Milgram, Taylor and WilliamsHMT⁺88, § 1])

\[\kappa^s_3\colon H_3(\mathcal{B}\pi;\mathbb{Z}/2)\to L^s_5(\mathbb{Z}[\pi]),\]

where $c\colon M\to \mathcal{B}\pi$ is defined as the classifying map for the universal cover of M, and $\operatorname{Arf}_3$ denotes the (suitable) Arf invariant.

Before we prove Proposition 4.13, we state a few groups $\Gamma$ for which $SK_1(\Gamma)$ vanishes (this is the work of many mathematicians, and we direct the reader to [Reference OliverOli88, pp. 3–4] and the citations within for references):

1. (i) $\Gamma = \mathbb{Z}/2n$ ;

2. (ii) $\Gamma = \mathbb{Z}/2^n \times \mathbb{Z}/2$ ;

3. (iii) $\Gamma = (\mathbb{Z}/2)^n$ ;

4. (vi) $\Gamma = D_{2n}$ , the dihedral group of order 4n.

So Proposition 4.13 applies to any of the above groups. It also follows that Theorems 1.7 and 1.9 also apply to any $\Gamma$ that has any of the above groups as its Sylow 2-subgroup. For example, let $\rho$ be isomorphic to one of the above groups and let O be an odd-order group. Then $\Gamma:=\rho\times O$ has $\rho$ as its Sylow 2-subgroup and hence Proposition 4.13 also applies to $\Gamma$ .

Proof of Proposition 4.13. Let X satisfy the hypotheses of Proposition 4.13 and let $b\in H^4(X\times I;\mathbb{Z})$ . We need to show that there exists an $a\in H^2(X\times I;\mathbb{Z}/2)$ such that $\theta(a,b)=0$ . We claim that we can simply take $a=0$ regardless of b.

Note that $X\times I$ is a five-dimensional manifold with boundary and by assumption we have that $\operatorname{im}(SK_1(\mathbb{Z}[\rho])\to SK_1(\mathbb{Z}[\Gamma]))=0$ , hence Theorem 4.15 applies to $X\times I$ . Theorem 4.15 then tells us that the surgery obstruction map $\theta$ factors as

\[ [(X\times I,\partial),G/\operatorname{TOP}] \to H^2(X\times I,\partial; \mathbb{Z}/2)\cong H_3(X;\mathbb{Z}/2)\to H_3(\mathcal{B}\pi;\mathbb{Z}/2) \xrightarrow{\kappa_3^s} L_5^s(\mathbb{Z}[\pi]),\]

where the first map is the projection, using Lemma 4.8. Hence, $\theta(a,b)$ does not depend on b, and so for all $b\in H^4(X\times I,\partial;\mathbb{Z}/2)$ we have that $\theta(0,b)=\theta(0,0)=0$ .

4.5 Partial unstable realisation of the Casson–Sullivan invariant

The purpose of this subsection is to find examples where we can partially realise the Casson–Sullivan invariant using the method described in § 4.2. The way we will do this is by considering the assembly maps for the surgery obstruction map $\theta$ and comparing them using spectral sequences to surgery obstruction maps for which we have full realisation. In all that follows, let X be a closed, connected, smooth, orientable 4-manifold with $\pi_1(X)\cong\pi$ .

First we give the relevant notation. Let $\varepsilon\in\{-\infty\}\cup\{\dots,-1, 0, 1, 2\}$ be the decoration, and recall that $\varepsilon=2$ refers to $\varepsilon=s$ and $\varepsilon=1$ refers to $\varepsilon=h$ . For more information about decorations, see [Reference LückLüc25]. Let $\mathbb{L}^\varepsilon_\bullet(R)$ denote the quadratic L-theory spectrum of a ring R with decoration $\varepsilon$ (this is a spectrum that has homotopy groups $\pi_k(\mathbb{L}^\varepsilon_\bullet(R))=L^\varepsilon_k(R)$ ) and let $\mathbb{L}_\bullet:=\mathbb{L}_\bullet(\mathbb{Z})$ (note that, as suggested by the notation, this is independent of our choice of decoration). We will use $\mathbb{L}_\bullet\langle1\rangle$ to denote the 1-connective cover of $\mathbb{L}_\bullet$ (a spectrum which has $\pi_k(\mathbb{L}_\bullet\langle1\rangle)=0$ for all $k\lt1$ and has $\pi_k(\mathbb{L}_\bullet\langle1\rangle)=\pi_k(\mathbb{L}_\bullet)$ for $k\geqslant 1$ ). Then these spectra determine a generalised (co)homology theory and we have the following factorisation of the surgery obstruction map $\theta^\varepsilon$ (note that the notation used will be explained below).

The fact that $\theta^{\varepsilon}$ factors first as maps a then b is due to Ranicki and Quinn [Reference QuinnQui70, Reference RanickiRan92]. Further factoring through the homology of $\mathcal{B}\pi$ is contained in [Reference Hambleton, Milgram, Taylor and WilliamsHMT⁺88, Appendix], though they attribute this to [Reference RanickiRan92]. The denoted even further factorisation of the assembly map is described in [Reference Davis and LückDL98]. We now describe the maps.

The isomorphism a arises via the definition of cohomology with coefficients in a spectrum: since we have that $(\mathbb{L}_\bullet\langle 1\rangle)_0\simeq G/\operatorname{TOP}$ , we have that

\[ H^0(X\times I,\partial; \mathbb{L}_\bullet)\cong [X\times I, \partial; G/\operatorname{TOP}]\cong \mathcal{N}_{\operatorname{TOP}}(X\times I,\partial). \]

The isomorphism b is given by Ranicki–Sullivan duality. To define the map c, we factor it as the composition of maps given in the following diagram where $\widetilde{X\times I}$ denotes the universal cover of $X\times I$ .

To define the map d, recall that we have a map $\mathbb{L}^\varepsilon_\bullet\langle1\rangle\to \mathbb{L}^\varepsilon_\bullet$ by the definition of a 1-connective cover. Then d is the induced map on homology via this map. The definition of the isomorphism e follows from the definition of the $\pi$ -equivariant homology of the $\operatorname{Orb}(\pi)$ -space $\ast_{\mathcal{TRIV}}$ with coefficients in a spectrum. For this details on this, see [Reference Davis and LückDL98, Example 5.5]. The isomorphism g is again given by the definition of the $\pi$ -equivariant homology. The map $\sigma^\varepsilon$ is what we will call the $\varepsilon$ -assembly map. Since we need to use the s-cobordism theorem to construct our mapping cylinders, we will need to consider the case $\varepsilon=s$ , for which this assembly map is certainly not an isomorphism in general. Composing all of these maps together, we get a factorisation of $\theta^s$ .

Now we want to use this formalism to try to understand $\theta^s$ for some groups for which we cannot use Proposition 4.13. Let $\Gamma$ be a finite cyclic group such that we have a non-trivial homomorphism $h\colon\Gamma\to \pi$ . In what follows we will condense the ‘assembly map’ by setting $\overline{\sigma}:=g\circ \sigma^s\circ e$ .

Consider the following diagram.

By naturality of the assembly maps, if a normal invariant is hit by the map $h_*$ after being mapped down to $H_5(\mathcal{B}\pi;\mathbb{L}^s_\bullet)$ , then its surgery obstruction must be zero. It follows that if the map $h_*$ is surjective then $\theta^s$ is the zero map.

Everything we have presented so far could in principle be used to study any case where we have a non-trivial homomorphism $\Gamma\to \pi$ , but we now specialise to the case where $\pi=\mathbb{Z}\times\Gamma$ , where we will be able to make concrete findings.

To try to understand this map $h_*$ better we will use the Atiyah–Hirzebruch spectral sequence (AHSS). Recall that the AHSS is a homology spectral sequence that computes the generalised homology of a space in terms of the regular homology of that space with coefficients in the generalised homologies of a point. In our particular case it computes, for a space K,

\[ E^2_{p,q} = H_p(K;\pi_q(\mathbb{L}_\bullet\langle1\rangle)) \implies E^\infty_{p,q}=H_{p+q}(K;\mathbb{L}_\bullet\langle1\rangle).\]

Recall that $\pi_q(\mathbb{L}_\bullet\langle1\rangle)=L_q(\mathbb{Z})$ for $q\gt0$ and is zero otherwise. We first wish to compute $H_5(\mathcal{B}G;\mathbb{L}_\bullet\langle1\rangle)$ for the cases $G=\pi$ and $G=\Gamma$ . We show the relevant terms below of the $E^3$ -page, along with the relevant third-page differentials, in Figure 2.

Figure 2.

The $E^3$ -page of the spectral sequence for computing $H_*(\mathcal{B}G;\mathbb{L}_\bullet\langle1\rangle)$ with the relevant terms and differentials shown for computing $H_5(\mathcal{B}G;\mathbb{L}_\bullet\langle1\rangle)$ .

The conclusions we can draw from this spectral sequence are contained in the following lemma.

Lemma 4.16. Let $\Gamma$ be a finite cyclic group of order 2n, let $\pi\cong \mathbb{Z}\times\Gamma$ and let $d_G$ denote the differential $d^{4,3}_{3}$ denoted in Figure 2. Then we have the commutative diagram

where the rows are exact, and $H_3(\mathcal{B}\Gamma;\mathbb{Z}/2)\cong \mathbb{Z}/2$ , $\operatorname{coker} d_\Gamma \cong \mathbb{Z}/2n$ or $\mathbb{Z}/n$ , $H_3(\mathcal{B}\pi;\mathbb{Z}/2)\cong \mathbb{Z}/2\oplus \mathbb{Z}/2$ , and $\operatorname{coker} d_\pi \cong \mathbb{Z}\oplus \mathbb{Z}/2n$ or $\mathbb{Z}\oplus\mathbb{Z}/n$ .

Finally, the diagonal $p+q=5$ computes the associated graded for $H_5(\mathcal{B}G;\mathbb{L}_\bullet\langle1\rangle)$ and hence we have the short exact sequences which form the rows of the stated commutative diagram. The vertical maps are then induced by the obvious inclusion map $\Gamma\to \pi$ and the diagram commutes by naturality of spectral sequences. This completes the proof.

We have that $H^4(X\times I,\partial;\mathbb{Z}/2)\cong H^3(X;\mathbb{Z}/2)\cong \mathbb{Z}/2\oplus \mathbb{Z}/2$ (where the first summand is generated by the Poincaré dual to a curve representing the generator of $\mathbb{Z}\subset\mathbb{Z}\times\Gamma$ and the second is generated by the Poincaré dual to a curve representing the generator of $\mathbb{Z}\subset\mathbb{Z}\times\Gamma$ ) and the aim is now to realise the element (0,1) as the Casson–Sullivan invariant of a homeomorphism $f\colon X\to X$ . Following the realisation procedure laid out in §§ 4.2 and 4.3, by Proposition 4.10 and Lemma 4.12 it suffices to show that there exists an element $y\in \mathcal{N}_{\operatorname{TOP}}(X\times I,\partial)\cong [(X\times I,\partial),G/\operatorname{TOP}]$ such that the map m from Lemma 4.9 sends y to $(0,1)\in H^4(X\times I,\partial;\mathbb{Z}/2)$ and such that $\theta(y)=0$ .

Lemma 4.17. Let $\Gamma$ be a finite cyclic group of order 2n, $\pi=\mathbb{Z}\times\Gamma$ and let $\iota\colon \Gamma\to\pi$ be the obvious inclusion map. The leftmost vertical map in the commutative diagram from Lemma 4.16 is then $\iota_*$ and is given by

\begin{align*}\iota_*\colon \begin{cases} \mathbb{Z}/2n \\ \mathbb{Z}/n\end{cases} \to& \begin{cases} \mathbb{Z}\oplus \mathbb{Z}/2n \\ \mathbb{Z}\oplus\mathbb{Z}/n \end{cases} \\ a \mapsto& (0,a).\end{align*}

Since $\iota(a)=(0,a)$ , the commutativity of the above diagram gives the result regardless of the isomorphism types of the cokernels given in Lemma 4.16.