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Exotic non-orientable four-manifolds with prescribed fundamental group

Part of: Low-dimensional topology Differential topology

Published online by Cambridge University Press:  26 March 2026

RAFAEL TORRES
RAFAEL TORRES*
Affiliation:
Scuola Internazionale Superiori di Studi Avanzati (SISSA), Via Bonomea 265, 34136, Trieste, Italy. e-mail: rtorres@sissa.it
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Abstract

We show that any finitely presented group with an index two subgroup is realised as the fundamental group of a closed smooth non-orientable four-manifold that admits an exotic smooth structure, which is obtained by performing a Gluck twist. The orientation 2-covers of these four-manifolds are diffeomorphic. These two smooth structures remain inequivalent after adding arbitrarily many copies of the product of a pair of 2-spheres and stabilise after adding a single copy of the complex projective plane.

MSC classification

Primary: 57R55: Differentiable structures
Secondary: 57M05: Fundamental group, presentations, free differential calculus

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 181 , Issue 1 , July 2026 , pp. 773 - 780
Copyright
© The Author(s), 2026. Published by Cambridge University Press on behalf of The Cambridge Philosophical Society

1. Introduction

Kreck showed [ Reference Kreck14 , theorem 1] that for any finitely presented group pi $\pi$ with a non-trivial element w 1 element of upper H Superscript 1 Baseline left parenthesis pi semicolon double struck upper Z divided by 2 right parenthesis $w_1\in H^1(\pi; {\mathbb{Z}}/2)$ , there is a closed smooth four-manifold upper M Subscript pi $M_{\pi}$ with fundamental group pi $\pi$ and orientation-character w 1 $w_1$ such that the connected sums upper M Subscript pi Baseline number sign upper K Baseline 3 $M_{\pi}\,{\mathbin{\#}}\, K3$ and upper M Subscript pi Baseline number sign 11 left parenthesis upper S squared times upper S squared right parenthesis $M_{\pi}\,{\mathbin{\#}}\,11(S^2\times S^2)$ are homeomorphic, but not diffeomorphic (see [ Reference Kasprowski, Powell and Ray12 , section 5·5]). The K3-surface is denoted by K3 [ Reference Gompf and Stipsicz9 , section 1·3]. The purpose of this paper is to strengthen and extend Kreck’s result as follows.

Theorem A. Let G be a finitely presented group that has a subgroup of index two and denote its deficiency by def(G). There are closed smooth non-orientable four-manifolds upper M Subscript upper G $M_G$ and upper N Subscript upper G $N_G$ with fundamental group G and that satisfy the following properties:

  1. (i) the Euler characteristic of upper M Subscript upper G $M_G$ is chi left parenthesis upper M Subscript upper G Baseline right parenthesis equals 4 minus 2 normal d normal e normal f left parenthesis upper G right parenthesis $\chi(M_G) = 4 - 2\,\mathrm{def} (G)$ ;

  2. (ii) there is a homeomorphism

    upper M Subscript upper G Baseline right arrow upper N Subscript upper G Baseline semicolon \begin{equation*}M_G\rightarrow N_G;\end{equation*}

  3. (iii) there is no diffeomorphism

    upper M Subscript upper G Baseline number sign left parenthesis n minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis right arrow upper N Subscript upper G Baseline number sign left parenthesis n minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis \begin{equation*}M_G\,{\mathbin{\#}}\,(n - 1)(S^2\times S^2)\rightarrow N_G\,{\mathbin{\#}}\,(n - 1)(S^2\times S^2)\end{equation*}
    for any n element of double struck upper N $n\in {\mathbb{N}}$ ;

  4. (iv) there is a diffeomorphism

    ModifyingAbove upper M With caret Subscript upper G Baseline right arrow ModifyingAbove upper N With caret Subscript upper G \begin{equation*}\widehat{M}_G\rightarrow \widehat{N}_G\end{equation*}
    between their orientation 2-covers ModifyingAbove upper M With caret Subscript upper G Baseline right arrow upper M Subscript upper G $\widehat{M}_G\rightarrow M_G$ and ModifyingAbove upper N With caret Subscript upper G Baseline right arrow upper N Subscript upper G $\widehat{N}_G\rightarrow N_G$ , which admit a Spin structure;

  5. (v) the four-manifold upper N Subscript upper G $N_G$ is obtained by performing a Gluck twist along a smoothly embedded 2-sphere upper S right arrow with hook upper M Subscript upper G $S\hookrightarrow M_G$ ;

  6. (vi) there is a diffeomorphism

    upper M Subscript upper G Baseline number sign double struck upper C double struck upper P squared right arrow upper N Subscript upper G Baseline number sign double struck upper C double struck upper P squared period \begin{equation*}M_G\,{\mathbin{\#}}\, \mathbb{CP}^2\rightarrow N_G\,{\mathbin{\#}}\, \mathbb{CP}^2.\end{equation*}

The first and last three items of Theorem A do not appear in Kreck’s work. An advantage of the construction used in the proof of Theorem A over the examples of Kreck is that the Euler characteristic of the four-manifolds considered here is considerably smaller. For the sake of completeness, we show in Section 3·7 that the four-manifolds upper M Subscript pi Baseline number sign upper K Baseline 3 $M_{\pi}\,{\mathbin{\#}}\, K3$ and upper M Subscript pi Baseline number sign 11 left parenthesis upper S squared times upper S squared right parenthesis $M_{\pi}\,{\mathbin{\#}}\, 11(S^2\times S^2)$ also satisfy the properties listed in (iv) and (vi). Cappell–Shaneson [ Reference Cappell and Shaneson4 ] showed that if X is a closed smooth four-manifold such that its fundamental group contains an orientation-reversing element of order two, then there is a closed smooth non-orientable four-manifold Q and a simple homotopy equivalence f colon upper Q right arrow upper X $f: Q\rightarrow X$ whose smooth normal invariant is the nontrivial element in the kernel of left bracket upper X semicolon upper G divided by upper O right bracket right arrow left bracket upper X comma upper G divided by upper T upper O upper P right bracket $[X; G/O]\rightarrow [X, G/TOP]$ . The four-manifold Q, however, might well be diffeomorphic to X as shown by Akbulut in the case upper X equals upper S squared times double struck upper R normal upper P squared $X = S^2\times {{\mathbb R\mkern-0.5mu\mathrm P}}^2$ [ Reference Akbulut, Gordon and Kirby1 ]. We use the eta $\eta$ -invariant of the twisted Dirac operator to show that the four-manifolds of Theorem A are not diffeomorphic through work of Stolz; see Section 2. The construction of the exotic smooth structure upper N Subscript upper G $N_G$ of Theorem A uses an exotic smooth structure A due to Akbulut [ Reference Akbulut, Gordon and Kirby1 , Reference Akbulut2 ] that arises from performing a Gluck twist to the connected sum left parenthesis upper S cubed ModifyingAbove times With tilde upper S Superscript 1 Baseline right parenthesis number sign left parenthesis upper S squared times upper S squared right parenthesis $(S^3\widetilde{\times} S^1)\,{\mathbin{\#}}\,(S^2\times S^2)$ of the non-orientable 3-sphere bundle over the circle upper S cubed ModifyingAbove times With tilde upper S Superscript 1 $S^3\widetilde{\times} S^1$ with a copy of upper S squared times upper S squared $S^2\times S^2$ ; see Section 3·1. A result of Gompf [ Reference Gompf8 ] implies that the orientation 2-cover of A is diffeomorphic to left parenthesis upper S Superscript 1 Baseline times upper S cubed right parenthesis number sign 2 left parenthesis upper S squared times upper S squared right parenthesis $(S^1\times S^3) \,{\mathbin{\#}}\, 2(S^2\times S^2)$ as pointed out by Torres in [ Reference Torres20 , proposition 9]. Hence, Theorem A provides a multitude of new examples of exotic orientation-reversing free involutions on four-manifolds [ Reference Cappell and Shaneson4 , Reference Fintushel and Stern5 , Reference Gompf8 , Reference Torres20 ].

Kreck’s examples and the four-manifolds of Theorem A satisfy the identity w 1 squared equals w 2 $w_1^2 = w_2$ , where w Subscript i $w_i$ is the ith Stiefel-Whitney class. Recently, Kasprowski-Powell [ Reference Kasprowski and Powell11 ] used Kreck’s construction to produce four-manifolds that satisfy the second and third items of Theorem A and for which w 1 squared not equals w 2 $w_1^2\neq w_2$ .

2. A spectral invariant to distinguish smooth structures

This section contains a brief discussion on normal upper P normal i normal n Superscript plus $\mathrm{Pin}^+$ -structures and on the eta $\eta$ -invariant of a closed smooth four-manifold that is equipped with such a structure. The reader is directed towards [ Reference Kirby and Taylor13 , Reference Stolz19 ] for background results. In the sequel, a normal upper P normal i normal n Superscript plus $\mathrm{Pin}^+$ -structure phi Subscript upper M $\phi_M$ on a smooth four-manifold M will be denoted by left parenthesis upper M comma phi Subscript upper M Baseline right parenthesis $(M, \phi_M)$ . We also refer to the pair left parenthesis upper M comma phi Subscript upper M Baseline right parenthesis $(M, \phi_M)$ as a upper P i n Superscript plus ${Pin}^+$ -manifold. For any such four-manifold M with first cohomology group upper H Superscript 1 Baseline left parenthesis upper M semicolon double struck upper Z divided by 2 right parenthesis equals double struck upper Z divided by 2 $H^1(M; {\mathbb{Z}}/2) = {\mathbb{Z}}/2$ , there are two normal upper P normal i normal n Superscript plus $\mathrm{Pin}^+$ -structures left brace left parenthesis upper M comma plus or minus phi Subscript upper M Baseline right parenthesis right brace $\{(M, \pm \phi_M)\}$ , and their classes left brace left bracket left parenthesis upper M comma plus or minus phi Subscript upper M Baseline right parenthesis right bracket right brace $\{[(M, \pm \phi_M)]\}$ are mutual inverses in the fourth upper P i n Superscript plus ${Pin}^+$ -bordism group normal upper Omega 4 Superscript upper P i n Super Superscript plus Baseline equals double struck upper Z divided by 16 $\Omega^{{Pin}^+}_4 = {\mathbb{Z}}/16$ [ Reference Stolz19 , corollary 6·4], [ Reference Kirby and Taylor13 , theorem 5·2, p. 190]. This group is generated by the real projective 4-space [ Reference Kirby and Taylor13 , theorem 5·2], and eta left parenthesis double struck upper R normal upper P Superscript 4 Baseline comma plus or minus phi Subscript double struck upper R normal upper P Sub Superscript 4 Subscript Baseline right parenthesis equals plus or minus 1 divided by 8 mod 2 double struck upper Z $\eta({{\mathbb R\mkern-0.5mu\mathrm P}}^4, \pm \phi_{{{\mathbb R\mkern-0.5mu\mathrm P}}^4}) = \pm {1}/{8} \mod 2{\mathbb{Z}}$ [ Reference Stolz19 , corollary 5·4]. The following theorem is paramount for our purposes.

Theorem 2·1 Stolz [Reference Stolz 19 ]. Let left parenthesis upper M comma phi Subscript upper M Baseline right parenthesis $(M, \phi_M)$ be a closed smooth upper P i n Superscript plus ${Pin}^+$ - four-manifold.

  1. (i) The invariant eta left parenthesis upper M comma phi Subscript upper M Baseline right parenthesis mod 2 double struck upper Z $\eta(M, \phi_M) \mod 2{\mathbb{Z}}$ is a complete upper P i n Superscript plus ${Pin}^+$ -bordism invariant.

  2. (ii) If M is simply connected, then eta left parenthesis upper M comma phi Subscript upper M Baseline right parenthesis equals left parenthesis 1 divided by 16 right parenthesis sigma left parenthesis upper M right parenthesis mod 2 double struck upper Z $\eta(M, \phi_M) = ({1}/{16})\sigma(M)\,\mod\,2{\mathbb{Z}}$ , where sigma left parenthesis upper M right parenthesis $\sigma(M)$ is the signature.

Stolz used these properties along with the modified surgery theory methods developed by Kreck [ Reference Kreck14 , Reference Kreck16 ] to show that Cappell–Shaneson’s exotic double struck upper R normal upper P Superscript 4 ${{\mathbb R\mkern-0.5mu\mathrm P}}^4$ is not stably diffeomorphic to the standard one [ Reference Stolz19 , theorems B and 7·4]. By building heavily on these results, we record the following generalisation of Stolz’s result.

Theorem 2·2. Let upper M 1 $M_1$ and upper M 2 $M_2$ be closed non-orientable smooth four-manifolds that admit a upper P i n Superscript plus ${Pin}^+$ -structure and that have the same Euler characteristic. If for every upper P i n Superscript plus ${Pin}^+$ -structure left brace left parenthesis upper M Subscript i Baseline comma phi Subscript upper M Sub Subscript i Subscript Baseline right parenthesis colon i equals 1 comma 2 right brace $\{(M_i, \phi_{M_i}): i = 1, 2\}$ , the values of the corresponding eta $\eta$ -invariants satisfy

(2·1) eta left parenthesis upper M 1 comma phi Subscript upper M 1 Baseline right parenthesis not equals eta left parenthesis upper M 2 comma phi Subscript upper M 2 Baseline right parenthesis mod 2 double struck upper Z comma \begin{equation}\eta(M_1, \phi_{M_1}) \neq \eta(M_2, \phi_{M_2}) \mod 2{\mathbb{Z}},\end{equation}

then there is no diffeomorphism

(2·2) upper M 1 number sign left parenthesis n minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis right arrow upper M 2 number sign left parenthesis n minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis \begin{equation}M_1\,{\mathbin{\#}}\,(n - 1)(S^2\times S^2) \rightarrow M_2\,{\mathbin{\#}}\,(n - 1)(S^2\times S^2)\end{equation}

for any n element of double struck upper N $n\in {\mathbb{N}}$ .

Proof. Recall that a normal 1-type xi colon upper B right arrow upper B upper O $\xi\,:\,B\rightarrow BO$ of a four-manifold M determines a bordism theory normal upper Omega 4 Superscript xi $\Omega^\xi_4$ of four-manifolds that admit a lift of their stable normal bundle across the fibration xi $\xi$ [ Reference Kreck14 , Reference Kreck16 ]. A choice of normal 1-smoothing determines a class in the group normal upper Omega 4 Superscript xi $\Omega^\xi_4$ . Different normal 1-smoothings need not define the same class in normal upper Omega 4 Superscript xi $\Omega^\xi_4$ . The main ingredient in the proof of Theorem 2·2 is the following result.

Theorem 2·3 Kreck [Reference Kreck 16 , theorem C]. Let upper M 1 $M_1$ and upper M 2 $M_2$ be two closed smooth four-manifolds with the same Euler characteristic and that share the same normal 1-type xi colon upper B right arrow upper B upper O $\xi: B\rightarrow BO$ . The four-manifolds upper M 1 $M_1$ and upper M 2 $M_2$ admit normal 1-smoothings that are bordant in normal upper Omega 4 Superscript xi $\Omega^\xi_4$ , if and only if there is an n element of double struck upper N $n\in {\mathbb{N}}$ such that the connected sums upper M 1 number sign left parenthesis n 1 minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis $M_1\,{\mathbin{\#}}\, (n_1 - 1)(S^2\times S^2)$ and upper M 2 number sign left parenthesis n 2 minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis $M_2\,{\mathbin{\#}}\,(n_2 - 1)(S^2\times S^2)$ are diffeomorphic.

We now argue that Theorem 2·2 is a corollary of Theorem 2·3. As it was observed by Kasprowski-Land-Powell-Teichner [ Reference Kasprowski, Land, Powell and Teichner10 , theorem 1·1], there is a one-to-one correspondence between stable diffeomorphism classes of 4-manifolds with the same normal 1-type and normal upper Omega 4 Superscript xi Baseline divided by upper A u t left parenthesis xi right parenthesis $\Omega^\xi_4/{Aut}(\xi)$ . In the presence of a upper P i n Superscript plus ${Pin}^+$ -structure, there is a map normal upper Omega 4 Superscript xi Baseline divided by upper A u t left parenthesis xi right parenthesis right arrow left parenthesis double struck upper Z divided by 16 right parenthesis divided by left parenthesis double struck upper Z divided by 2 right parenthesis $\Omega^\xi_4/{Aut}(\xi) \rightarrow ({\mathbb{Z}}/16)/({\mathbb{Z}}/2)$ [ Reference Kreck14 , section 5], [ Reference Kreck16 ]. Moreover, the four-manifolds upper M 1 $M_1$ and upper M 2 $M_2$ of Theorem 2·2 do not have the same normal 1-type since they both admit a upper P i n Superscript plus ${Pin}^+$ -structure [ Reference Kreck16 , section 2, proposition 2], and the eta $\eta$ -invariant detects the nine equivalence classes

StartSet 0 comma plus or minus 1 comma plus or minus 2 comma plus or minus 3 plus or minus 4 comma plus or minus 5 comma plus or minus 6 comma plus or minus 7 comma 8 EndSet element of normal upper Omega 4 Superscript upper P i n Super Superscript plus Superscript Baseline divided by left parenthesis x tilde negative x right parenthesis period \begin{equation*}\{0, \pm 1, \pm 2, \pm 3\pm 4, \pm 5, \pm 6, \pm 7, 8\} \in \Omega^{{Pin}^+}_4/(x\sim -x).\end{equation*}

3. Proof of Theorem A

3·1. A result of Akbulut

For the sake of clarity in the proof of Theorem A, we now summarise the result of Akbulut and the observations on it due to Torres that were mentioned in the introduction. A Gluck twist along a smoothly embedded 2-sphere upper S subset of upper X $S\subset X$ with tubular neighbourhood nu left parenthesis upper S right parenthesis equals upper D squared times upper S squared $\nu(S) = D^2\times S^2$ is the cut-and-paste construction upper X Subscript upper S Baseline equals left parenthesis upper X minus nu left parenthesis upper S right parenthesis right parenthesis union Subscript phi Baseline left parenthesis upper D squared times upper S squared right parenthesis $X_S = (X\setminus \nu(S))\cup_{\varphi} (D^2\times S^2)$ , where the gluing diffeomorphism is phi left parenthesis theta comma x right parenthesis equals left parenthesis theta comma r Subscript theta Baseline left parenthesis x right parenthesis right parenthesis $\varphi(\theta, x) = (\theta, r_\theta(x))$ for every left parenthesis theta comma x right parenthesis element of upper S Superscript 1 times upper S squared $(\theta, x)\in S^1\times S^2$ and x right arrow from bar r Subscript theta Baseline left parenthesis x right parenthesis $x\mapsto r_{\theta}(x)$ is the nontrivial element in pi 1 left parenthesis upper S upper O left parenthesis 3 right parenthesis right parenthesis equals double struck upper Z divided by 2 $\pi_1({SO}(3)) = {\mathbb{Z}}/2$ [ Reference Akbulut3 , section 6·1].

Theorem 3·1. Akbulut [Reference Akbulut 2 ]. There is a smoothly embedded 2-sphere

(3·1) normal upper Sigma right arrow with hook left parenthesis upper S cubed ModifyingAbove times With tilde upper S Superscript 1 Baseline right parenthesis number sign left parenthesis upper S squared times upper S squared right parenthesis \begin{equation}\Sigma\hookrightarrow (S^3\widetilde{\times} S^1)\,{\mathbin{\#}}\,(S^2\times S^2)\end{equation}

such that performing a Gluck twist along it yields a four-manifold A that is homeomorphic, but not diffeomorphic to left parenthesis upper S cubed ModifyingAbove times With tilde upper S Superscript 1 Baseline right parenthesis number sign left parenthesis upper S squared times upper S squared right parenthesis $(S^3\widetilde{\times} S^1)\,{\mathbin{\#}}\,(S^2\times S^2)$ and that satisfies the following properties:

  1. (i) Torres [ Reference Torres20 ]. The value of the eta $\eta$ -invariant for both upper P i n Superscript plus ${Pin}^+$ -structures left parenthesis upper A comma plus or minus phi Subscript upper A Baseline right parenthesis $(A, \pm \phi_A)$ is eta left parenthesis upper A comma plus or minus phi Subscript upper A Baseline right parenthesis equals 1 mod 2 double struck upper Z $\eta(A, \pm \phi_A) = 1 \mod 2{\mathbb{Z}}$ ;

  2. (ii) Torres [ Reference Torres20 ]. The orientation 2-cover of A is diffeomorphic to the connected sum left parenthesis upper S Superscript 1 Baseline times upper S cubed right parenthesis number sign 2 left parenthesis upper S squared times upper S squared right parenthesis $(S^1\times S^3)\,{\mathbin{\#}}\,2(S^2\times S^2)$ .

3·2. Construction of a four-manifold with prescribed fundamental group

We occupy ourselves with a proof of the following theorem in this section. The deficiency of a group G is denoted by def(G).

Theorem 3·2. Let G be a finitely presented group that has a subgroup of index two. There is a closed smooth non-orientable four-manifold M(G) with Euler characteristic chi left parenthesis upper M left parenthesis upper G right parenthesis right parenthesis equals 2 minus 2 normal d normal e normal f left parenthesis upper G right parenthesis $\chi(M(G)) = 2 - 2\mathrm{def}(G)$ , second Stiefel-Whitney class w 2 left parenthesis upper M left parenthesis upper G right parenthesis right parenthesis equals 0 $w_2(M(G)) = 0$ , whose fundamental group is pi 1 left parenthesis upper M left parenthesis upper G right parenthesis right parenthesis approximately equals upper G $\pi_1(M(G)) \cong G$ and such that left bracket left parenthesis upper M left parenthesis upper G right parenthesis comma phi Subscript upper M left parenthesis upper G right parenthesis Baseline right parenthesis right bracket equals 0 element of normal upper Omega 4 Superscript upper P i n Super Superscript plus $[(M(G), \phi_{M(G)})] = 0\in \Omega^{{Pin}^+}_4$ for every upper P i n Superscript plus ${Pin}^+$ -structure left parenthesis upper M left parenthesis upper G right parenthesis comma phi Subscript upper M left parenthesis upper G right parenthesis Baseline right parenthesis $(M(G), \phi_{M(G)})$ .

A statement similar to Theorem 3·2 appears in [ Reference Kreck14 , p. 253].

Proof. We are indebted to Daniel Kasprowski for suggesting the following argument. Let K(G) be a 2-complex with pi 1 left parenthesis upper K left parenthesis upper G right parenthesis right parenthesis equals upper G $\pi_1(K(G)) = G$ associated to a presentation of G that realises its deficiency so that its Euler characteristic is chi left parenthesis upper K left parenthesis upper G right parenthesis right parenthesis equals 1 minus normal d normal e normal f left parenthesis upper G right parenthesis $\chi(K(G)) = 1 - \mathrm{def}(G)$ . Choose K(G) to have non-trivial first Stiefel-Whitney class and second Stiefel-Whitney class w 1 squared left parenthesis upper K left parenthesis upper G right parenthesis right parenthesis plus w 2 left parenthesis upper K left parenthesis upper G right parenthesis right parenthesis equals 0 $w_1^2(K(G)) + w_2(K(G)) = 0$ . A result of Wall says that there is a five-dimensional thickening Y(G) of K(G) whose Stiefel-Whitney classes are given by w Subscript i Baseline left parenthesis nu left parenthesis upper Y left parenthesis upper G right parenthesis right parenthesis equals w Subscript i Baseline left parenthesis upper K left parenthesis upper G right parenthesis right parenthesis $w_i(\nu(Y(G)) = w_i(K(G))$ for i equals 1 comma 2 $i = 1, 2$ and for which there is an isomorphism pi 1 left parenthesis partial differential upper Y left parenthesis upper G right parenthesis right parenthesis right arrow pi 1 left parenthesis upper K left parenthesis upper G right parenthesis right parenthesis $\pi_1(\partial Y(G)) \rightarrow \pi_1(K(G))$ [ Reference Wall21 , proposition 5·1]. The boundary partial differential upper Y Subscript upper G Baseline equals upper M left parenthesis upper G right parenthesis $\partial Y_G = M(G)$ is a closed smooth non-orientable four-manifold with fundamental group G, Euler characteristic chi left parenthesis upper M left parenthesis upper G right parenthesis right parenthesis equals 2 minus 2 d e f left parenthesis upper G right parenthesis $\chi(M(G)) = 2 - 2{def}(G)$ (see [ Reference Kreck and Schafer15 , proposition II·2]), and vanishing second Stiefel-Whitney class w 2 left parenthesis upper M left parenthesis upper G right parenthesis right parenthesis equals 0 $w_2(M(G)) = 0$ . There is a total of k equals r k Subscript double struck upper Z divided by 2 Baseline left parenthesis upper H Superscript 1 Baseline left parenthesis upper M left parenthesis upper G right parenthesis semicolon double struck upper Z divided by 2 right parenthesis right parenthesis $k = {rk}_{{\mathbb{Z}}/2}(H^1(M(G); {\mathbb{Z}}/2))$ (the rank of the cohomology group) upper P i n Superscript plus ${Pin}^+$ -structures left brace left parenthesis upper M left parenthesis upper G right parenthesis comma phi Subscript upper M left parenthesis upper G right parenthesis Superscript i Baseline right parenthesis colon i equals 1 comma ellipsis comma 2 Superscript k Baseline right brace $\{(M(G), \phi^i_{M(G)})\,:\, i = 1, \ldots, 2^k\}$ [ Reference Stolz19 , corollary 6·4]. Each of these upper P i n Superscript plus ${Pin}^+$ -structures arises from a upper P i n Superscript plus ${Pin}^+$ -structure on Y(G). Thus, left bracket left parenthesis upper M left parenthesis upper G right parenthesis comma phi Subscript upper M left parenthesis upper G right parenthesis Superscript i Baseline right parenthesis right bracket equals 0 element of normal upper Omega 4 Superscript upper P i n Super Superscript plus $[(M(G), \phi^i_{M(G)})] = 0\in \Omega^{{Pin}^+}_4$ for every 1 less than or equals i less than or equals 2 Superscript k $1\leq i \leq 2^k$ .

3·3. Two constructions of the four-manifold upper M Subscript upper G $M_G$ of Theorem A

The four-manifold of Theorem A is defined to be

(3·2) upper M Subscript upper G Baseline colon equals upper M left parenthesis upper G right parenthesis number sign left parenthesis upper S squared times upper S squared right parenthesis period \begin{equation}M_G: = M(G)\,{\mathbin{\#}}\, (S^2\times S^2).\end{equation}

Every upper P i n Superscript plus ${Pin}^+$ -structure left parenthesis upper M Subscript upper G Baseline comma phi Subscript upper M Sub Subscript upper G Subscript Baseline right parenthesis $(M_G, \phi_{M_G})$ arises from upper P i n Superscript plus ${Pin}^+$ -structures

left parenthesis upper M left parenthesis upper G right parenthesis comma phi Subscript upper M left parenthesis upper G right parenthesis Baseline right parenthesis normal a normal n normal d left parenthesis upper S squared times upper S squared comma phi Subscript upper S squared times upper S squared Baseline right parenthesis comma \begin{equation*}(M(G), \phi_{M(G)}) \quad \mathrm{and}\quad \big(S^2\times S^2, \phi_{S^2\times S^2}\big),\end{equation*}

and we have that

(3·3) eta left parenthesis upper M Subscript upper G Baseline comma phi Subscript upper M Sub Subscript upper G Subscript Baseline right parenthesis equals 0 mod 2 double struck upper Z \begin{equation}\eta(M_G, \phi_{M_G}) = 0 \mod 2{\mathbb{Z}}\end{equation}

and

left bracket left parenthesis upper M Subscript upper G Baseline comma phi Subscript upper M Sub Subscript upper G Subscript Baseline right parenthesis right bracket equals left bracket left parenthesis upper M left parenthesis upper G right parenthesis comma phi right parenthesis right bracket equals 0 element of normal upper Omega 4 Superscript upper P i n Super Superscript plus \begin{equation*}[(M_G, \phi_{M_G})] = [(M(G), \phi)] = 0\in \Omega^{{Pin}^+}_4\end{equation*}

by construction. A slight modification of the assemblage (3·2), which yields a useful expression of the four-manifold upper M Subscript upper G $M_G$ is as follows. Let alpha 0 subset of left parenthesis upper S cubed ModifyingAbove times With tilde upper S Superscript 1 Baseline right parenthesis number sign left parenthesis upper S squared times upper S squared right parenthesis $\alpha_0\subset (S^3\widetilde{\times} S^1)\,{\mathbin{\#}}\,(S^2\times S^2)$ be a simple loop whose homotopy class generates the group pi 1 left parenthesis left parenthesis upper S cubed ModifyingAbove times With tilde upper S Superscript 1 Baseline right parenthesis number sign left parenthesis upper S squared times upper S squared right parenthesis right parenthesis equals double struck upper Z $\pi_1((S^3\widetilde{\times} S^1)\,{\mathbin{\#}}\,(S^2\times S^2)) = {\mathbb{Z}}$ . Let gamma subset of upper M left parenthesis upper G right parenthesis $\gamma \subset M(G)$ be an orientation-reversing simple loop whose homotopy class is a generator of pi 1 left parenthesis upper M left parenthesis upper G right parenthesis right parenthesis equals upper G $\pi_1(M(G)) = G$ . The tubular neighbourhoods of the two loops alpha $\alpha$ and gamma $\gamma$ are diffeomorphic to upper D cubed ModifyingAbove times With tilde upper S Superscript 1 $D^3\widetilde{\times} S^1$ and

left parenthesis upper S cubed ModifyingAbove times With tilde upper S Superscript 1 Baseline right parenthesis number sign left parenthesis upper S squared times upper S squared right parenthesis minus nu left parenthesis alpha 0 right parenthesis equals left parenthesis upper D cubed ModifyingAbove times With tilde upper S Superscript 1 Baseline right parenthesis number sign left parenthesis upper S squared times upper S squared right parenthesis period \begin{equation*}(S^3\widetilde{\times} S^1)\,{\mathbin{\#}}\,(S^2\times S^2)\setminus \nu(\alpha_0) = (D^3\widetilde{\times} S^1)\,{\mathbin{\#}}\,(S^2\times S^2).\end{equation*}

We can glue these two four-manifolds along these loops to produce

(3·4) upper M Subscript upper G Baseline equals left parenthesis upper M left parenthesis upper G right parenthesis minus nu left parenthesis gamma right parenthesis right parenthesis union left parenthesis left parenthesis upper D cubed ModifyingAbove times With tilde upper S Superscript 1 Baseline right parenthesis number sign left parenthesis upper S squared times upper S squared right parenthesis right parenthesis equals upper M left parenthesis upper G right parenthesis number sign left parenthesis upper S squared times upper S squared right parenthesis period \begin{equation}M_G = (M(G)\setminus \nu(\gamma))\cup ((D^3\widetilde{\times}S^1)\,{\mathbin{\#}}\,(S^2\times S^2))= M(G)\,{\mathbin{\#}}\,(S^2\times S^2).\end{equation}

3·4. Construction of the four-manifold upper N Subscript upper G $N_G$ of Theorem A, its orientation 2-cover and the existence of a homeomorphism

Let A be the four-manifold of Theorem 3·1 constructed by Akbulut that is homeomorphic, but not diffeomorphic to left parenthesis upper S cubed ModifyingAbove times With tilde upper S Superscript 1 Baseline right parenthesis number sign left parenthesis upper S squared times upper S squared right parenthesis $(S^3\widetilde{\times} S^1)\,{\mathbin{\#}}\, (S^2\times S^2)$ [ Reference Akbulut, Gordon and Kirby1 , theorem 1]. Let alpha subset of upper A $\alpha\subset A$ be a simple loop whose homotopy class generates the fundamental group pi 1 left parenthesis upper A right parenthesis equals double struck upper Z $\pi_1(A) = {\mathbb{Z}}$ . Let gamma subset of upper M left parenthesis upper G right parenthesis $\gamma \subset M(G)$ be an orientation-reversing simple loop whose homotopy class is a generator of pi 1 left parenthesis upper M left parenthesis upper G right parenthesis right parenthesis equals upper G $\pi_1(M(G)) = G$ , where M(G) is the closed smooth non-orientable 4-manifold of Theorem 3·2. Build

(3·5) upper N Subscript upper G Baseline colon equals left parenthesis upper M left parenthesis upper G right parenthesis minus nu left parenthesis gamma right parenthesis right parenthesis union left parenthesis upper A minus nu left parenthesis alpha right parenthesis right parenthesis \begin{equation}N_G: = (M(G)\setminus \nu(\gamma))\cup (A\setminus \nu(\alpha))\end{equation}

such that the induced upper P i n Superscript plus ${Pin}^+$ -structures induced on the boundary components match. There are homeomorphisms

(3·6) upper N Subscript upper G Baseline right arrow upper M left parenthesis upper G right parenthesis number sign left parenthesis upper S squared times upper S squared right parenthesis right arrow upper M Subscript upper G Baseline \begin{equation}N_G\rightarrow M(G)\,{\mathbin{\#}}\,(S^2\times S^2)\rightarrow M_G\end{equation}

by construction given that the four-manifold A is homeomorphic to the connected sum left parenthesis upper S cubed ModifyingAbove times With tilde upper S Superscript 1 Baseline right parenthesis number sign left parenthesis upper S squared times upper S squared right parenthesis $(S^3\widetilde{\times} S^1)\,{\mathbin{\#}}\,(S^2\times S^2)$ . Moreover, the orientation 2-cover ModifyingAbove upper N Subscript upper G Baseline With caret right arrow upper N Subscript upper G $\widehat{N_G}\rightarrow N_G$ is diffeomorphic to orientation 2-cover of (3·4) given that the orientation 2-cover of A is diffeomorphic to left parenthesis upper S Superscript 1 Baseline times upper S cubed right parenthesis number sign 2 left parenthesis upper S squared times upper S squared right parenthesis $(S^1\times S^3)\,{\mathbin{\#}}\,2(S^2\times S^2)$ by the third clause of Theorem 3·1.

The existence of a smoothly embedded 2-sphere upper S right arrow with hook upper M Subscript upper G $S\hookrightarrow M_G$ with trivial tubular neighbourhood nu left parenthesis upper S right parenthesis equals upper D squared times upper S squared $\nu(S) = D^2\times S^2$ such that performing a Gluck twist to it produces the 4-manifold upper N Subscript upper G $N_G$ follows from Theorem 3·1. Indeed, the 2-sphere

(3·7) normal upper Sigma right arrow with hook left parenthesis upper S cubed ModifyingAbove times With tilde upper S Superscript 1 Baseline right parenthesis number sign left parenthesis upper S squared times upper S squared right parenthesis \begin{equation}\Sigma\hookrightarrow (S^3\widetilde{\times} S^1)\,{\mathbin{\#}}\,(S^2\times S^2)\end{equation}

that was constructed by Akbulut is disjoint from the simple loop alpha 0 subset of left parenthesis upper S cubed ModifyingAbove times With tilde upper S Superscript 1 Baseline right parenthesis number sign left parenthesis upper S squared times upper S squared right parenthesis $\alpha_0\subset (S^3\widetilde{\times} S^1)\,{\mathbin{\#}}\,(S^2\times S^2)$ used in the construction of upper M Subscript upper G $M_G$ in (3·4) by a general position argument: the sum of the dimensions of these two submanifolds is strictly less than the dimension of the ambient manifold. It follows from the construction of upper N Subscript upper G $N_G$ in (3·5), that this four-manifold is obtained by performing a Gluck twist to a smoothly embedded 2-sphere in upper M Subscript upper G $M_G$ .

3·5. The upper P i n Superscript plus ${Pin}^+$ -bordism class of upper N Subscript upper G $N_G$ and distinguishing the diffeomorphism types of upper M Subscript upper G Baseline number sign left parenthesis n minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis $M_G\,{\mathbin{\#}}\,(n - 1)(S^2\times S^2)$ and upper N Subscript upper G Baseline number sign left parenthesis n minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis $N_G\,{\mathbin{\#}}\,(n - 1)(S^2\times S^2)$ for n element of double struck upper N $n\in {\mathbb{N}}$

From the construction (3·5), it follows that there is a upper P i n Superscript plus ${Pin}^+$ -bordism between upper N Subscript upper G $N_G$ and the disjoint union of upper M left parenthesis upper G right parenthesis square cup upper A $M(G)\sqcup A$ with their respective upper P i n Superscript plus ${Pin}^+$ -structures. This observation and Theorem 3·2 yield the following proposition.

Proposition 3·3. For every choice of normal upper P normal i normal n Superscript plus $\mathrm{Pin}^+$ -structure left parenthesis upper M Subscript upper G Baseline comma phi Subscript upper M Sub Subscript upper G Subscript Baseline right parenthesis $(M_G, \phi_{M_G})$ on upper M Subscript upper G $M_G$ , we have that

(3·8) eta left parenthesis upper M Subscript upper G Baseline comma phi Subscript upper M Sub Subscript upper G Subscript Baseline right parenthesis equals 0 mod 2 double struck upper Z \begin{equation}\eta(M_G, \phi_{M_G}) = 0 \mod 2{\mathbb{Z}}\end{equation}

and

(3·9) left bracket left parenthesis upper M Subscript upper G Baseline comma phi Subscript upper M Sub Subscript upper G Subscript Baseline right parenthesis right bracket equals 0 element of normal upper Omega 4 Superscript normal upper P normal i normal n Super Superscript plus Superscript Baseline period \begin{equation}[(M_G, \phi_{M_G})] = 0 \in \Omega^{\mathrm{Pin}^+}_4.\end{equation}

For every choice of normal upper P normal i normal n Superscript plus $\mathrm{Pin}^+$ -structure left parenthesis upper N Subscript upper G Baseline comma phi Subscript upper N Sub Subscript upper G Subscript Baseline right parenthesis $(N_G, \phi_{N_G})$ , we have that

(3·10) eta left parenthesis upper N Subscript upper G Baseline comma phi Subscript upper N Sub Subscript upper G Subscript Baseline right parenthesis equals 1 mod 2 double struck upper Z \begin{equation}\eta(N_G, \phi_{N_G}) = 1 \mod 2{\mathbb{Z}}\end{equation}

and

(3·11) left bracket left parenthesis upper N Subscript upper G Baseline comma phi Subscript upper N Sub Subscript upper G Subscript Baseline right parenthesis right bracket equals left bracket left parenthesis upper A comma plus or minus phi Subscript upper A Baseline right parenthesis right bracket equals 8 element of normal upper Omega 4 Superscript normal upper P normal i normal n Super Superscript plus Superscript Baseline period \begin{equation}[(N_G, \phi_{N_G})] = [(A, \pm \phi_{A})] = 8 \in \Omega^{\mathrm{Pin}^+}_4.\end{equation}

Moreover, for every n element of double struck upper N $n\in {\mathbb{N}}$ and for every choice of normal upper P normal i normal n Superscript plus $\mathrm{Pin}^+$ -structures on the connected sums

left parenthesis upper M Subscript upper G Baseline number sign left parenthesis n minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis comma phi Subscript upper M Sub Subscript upper G Subscript Superscript prime Baseline right parenthesis a n d left parenthesis upper N Subscript upper G Baseline number sign left parenthesis n minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis comma phi Subscript upper N Sub Subscript upper G Subscript Superscript prime Baseline right parenthesis comma \begin{equation*}(M_G\,{\mathbin{\#}}\,(n - 1)(S^2\times S^2), \phi_{M_G}') \quad {and}\quad (N_G\,{\mathbin{\#}}\,(n - 1)(S^2\times S^2), \phi_{N_G}'),\end{equation*}

we have

(3·12) left bracket left parenthesis upper M Subscript upper G Baseline number sign left parenthesis n minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis comma phi Subscript upper M Sub Subscript upper G Subscript Superscript prime Baseline right parenthesis right bracket equals left bracket left parenthesis upper M Subscript upper G Baseline comma phi Subscript upper M Sub Subscript upper G Subscript Baseline right parenthesis right bracket equals 0 element of normal upper Omega 4 Superscript normal upper P normal i normal n Super Superscript plus \begin{equation}[(M_G\,{\mathbin{\#}}\,(n - 1)(S^2\times S^2), \phi_{M_G}')] = [(M_G, \phi_{M_G})] = 0\in \Omega^{\mathrm{Pin}^+}_4\end{equation}

and

(3·13) left bracket left parenthesis upper N Subscript upper G Baseline number sign left parenthesis n minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis comma phi Subscript upper N Sub Subscript upper G Subscript Superscript prime Baseline right parenthesis right bracket equals left bracket left parenthesis upper N Subscript upper G Baseline comma phi Subscript upper N Sub Subscript upper G Subscript Baseline right parenthesis right bracket equals 8 element of normal upper Omega 4 Superscript normal upper P normal i normal n Super Superscript plus Superscript Baseline period \begin{equation}[(N_G\,{\mathbin{\#}}\,(n - 1)(S^2\times S^2), \phi_{N_G}')] = [(N_G, \phi_{N_G})] = 8\in \Omega^{\mathrm{Pin}^+}_4.\end{equation}

The value (3·8) was computed in Section 3·2, while the value (3·10) follows from the discussion in Section 3·4 since the eta $\eta$ -invariant is additive under disjoint unions. These values determined the upper P i n Superscript plus ${Pin}^+$ -bordism classes [ Reference Kirby and Taylor13 , introduction]. Theorem 2·1 states that the value of the eta $\eta$ -invariant of left parenthesis left parenthesis n minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis comma phi Subscript n Baseline right parenthesis $((n - 1)(S^2\times S^2), \phi_n)$ is

(3·14) eta left parenthesis left parenthesis n minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis comma phi Subscript n Baseline right parenthesis equals sigma left parenthesis left parenthesis n minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis right parenthesis equals 0 mod 2 double struck upper Z \begin{equation}\eta((n - 1)(S^2\times S^2), \phi_n) = \sigma((n - 1)(S^2\times S^2)) = 0 \mod 2{\mathbb{Z}}\end{equation}

for every n element of double struck upper N $n\in {\mathbb{N}}$ , where sigma left parenthesis left parenthesis n minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis equals 0 $\sigma((n - 1)(S^2\times S^2) = 0$ is the signature of the connected sum of n minus 1 $n - 1$ copies of upper S squared times upper S squared $S^2\times S^2$ .

3·6. A proof of Theorem A

The four-manifold upper M Subscript upper G $M_G$ is constructed in (3·2), while the four-manifold upper N Subscript upper G $N_G$ is constructed in (3·5). It is straight-forward to see that chi left parenthesis upper M Subscript upper G Baseline right parenthesis equals 4 minus 2 d e f left parenthesis upper G right parenthesis $\chi(M_G) = 4 - 2{def}(G)$ . The constructions (3·4) and (3·5) imply that upper M Subscript upper G $M_G$ is homeomorphic to upper N Subscript upper G $N_G$ as it was discussed in Section 3·4, where we also argued the existence of a diffeomorphism between the orientation 2-covers of upper M Subscript upper G $M_G$ and upper N Subscript upper G $N_G$ and the Gluck twist construction of upper N Subscript upper G $N_G$ . Proposition 3·3 and Theorem 2·2 imply that there is no diffeomorphism upper M Subscript upper G Baseline number sign left parenthesis n minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis right arrow upper N Subscript upper G Baseline number sign left parenthesis n minus 1 right parenthesis left parenthesis upper S squared times upper S squared right parenthesis $M_G\,{\mathbin{\#}}\,(n - 1)(S^2\times S^2)\rightarrow N_G\,{\mathbin{\#}}\,(n - 1)(S^2\times S^2)$ for any n element of double struck upper N $n\in {\mathbb{N}}$ . Given that upper M Subscript upper G $M_G$ and upper N Subscript upper G $N_G$ are related by a Gluck twist, there is a diffeomorphism between upper M Subscript upper G Baseline number sign double struck upper C double struck upper P squared $M_G\,{\mathbin{\#}}\,\mathbb{CP}^2$ and upper N Subscript upper G Baseline number sign double struck upper C double struck upper P squared $N_G\,{\mathbin{\#}}\,\mathbb{CP}^2$ [ Reference Gompf and Stipsicz9 , exercise 5·2·7 (b)]. This concludes the proof of Theorem A.

3·7. Properties of Kreck’s examples

As the referee kindly pointed out, weaker versions of (iv) and (vi) of Theorem A immediately follow from coupling Kreck’s examples [ Reference Kreck14 , theorem 1] with a result of Gompf [ Reference Gompf6 ], which says that any two closed non-orientable homeomorphic four-manifolds become diffeomorphic after taking connected sums with a sufficiently large number of copies of the complex projective plane. In this section, we show that Kreck’s examples upper M 1 colon equals upper M Subscript pi Baseline number sign upper K Baseline 3 $M_1:= M_{\pi}\,{\mathbin{\#}}\, K3$ and upper M 2 colon equals upper M Subscript pi Baseline number sign 11 left parenthesis upper S squared times upper S squared right parenthesis $M_2:= M_{\pi}\,{\mathbin{\#}}\, 11(S^2\times S^2)$ also satisfy the properties listed in Theorem A (iv) and (vi), independently of the choice of upper M Subscript pi $M_\pi$ . The existence of a diffeomorphism ModifyingAbove upper M With caret Subscript 1 Baseline right arrow ModifyingAbove upper M With caret Subscript 2 $\widehat{M}_1\rightarrow \widehat{M}_2$ between the orientation 2-covers follows from the existence of a diffeomorphism upper K Baseline 3 number sign ModifyingAbove upper K Baseline 3 With bar right arrow 22 left parenthesis upper S squared times upper S squared right parenthesis $K3\,{\mathbin{\#}}\, \overline{K3}\rightarrow 22(S^2\times S^2)$ [Reference Gompf7, corollary 2]. This establishes the property in (iv) for these examples. An argument using Kirby calculus shows the existence of a diffeomorphism upper K Baseline 3 number sign double struck upper C double struck upper P squared right arrow 4 double struck upper C double struck upper P squared number sign 19 ModifyingAbove double struck upper C double struck upper P squared With bar $K3\,{\mathbin{\#}}\, \mathbb{CP}^2\rightarrow 4\mathbb{CP}^2\,{\mathbin{\#}}\, 19\overline{\mathbb{CP}^2}$ [ Reference Gompf and Stipsicz9 , exercise 8·3·4(d)] (see also [ Reference Gompf7 , Reference Mandlebaum17 , Reference Moishezon18 ]). Thus, the four-manifolds upper M Subscript pi Baseline number sign upper K Baseline 3 number sign double struck upper C double struck upper P squared $M_{\pi}\,{\mathbin{\#}}\, K3\,{\mathbin{\#}}\, \mathbb{CP}^2$ and upper M Subscript pi Baseline number sign 11 left parenthesis upper S squared times upper S squared right parenthesis number sign double struck upper C double struck upper P squared $M_{\pi}\,{\mathbin{\#}}\, 11(S^2\times S^2)\,{\mathbin{\#}}\, \mathbb{CP}^2$ are diffeomorphic to upper M Subscript pi Baseline number sign 23 double struck upper C double struck upper P squared $M_{\pi}\,{\mathbin{\#}}\, 23 \mathbb{CP}^2$ .

Acknowledgements

We thank Daniel Kasprowski and the referee for pointing out several inaccuracies in an earlier version of the paper, and their help to correct them. In particular, Daniel Kasprowski suggested the argument used to prove Theorem 3·2. Zahvaljujemo se ekipi PRH v Tolminu, Kobaridu in Ladri za njihovo velikodušno gostoljubje med pisanjem tegačlanka.

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