2.1. Notation

Let $\Gamma$ be the group $(\mathbb {Z}_{/{p}})^2$, where $p \neq 2$ is a prime. The goal is to consider free actions of $\Gamma$ on $S^{2n-1} \times S^{2n-1}$.

Let $R = (r_1,\ldots,r_n,r'_1,\ldots,r'_n)$ and $Q = (q_1,\ldots,q_n,q'_1,\ldots,q'_n)$ be elements of $(\mathbb {Z}_{/{p}})^{2n}$ so that the span of $R$ and $Q$ yield a $(\mathbb {Z}_{/{p}})^2$ inside $(\mathbb {Z}_{/{p}})^{2n}$. The parameters $r_1, \ldots r_n$ and $r'_1,\ldots,r'_n$ and $q_1,\ldots,q_n$ and $q'_1,\ldots,q'_n$ are called ‘rotation numbers’ in analogy with the case of a lens space. To get an action on a product of spheres, note that the unit $(2n-1)$-sphere in $\mathbb {R}^{2n} \cong \mathbb {C}^n$ gives rise to a product of spheres $S^{2n-1} \times S^{2n-1}$ in the product $\mathbb {C}^n \times \mathbb {C}^n$, and then $R$ acts on $\mathbb {C}^n \times \mathbb {C}^n$ preserving these spheres via

\begin{align*} R \cdot (z,z') & = (r,r') \cdot (z,z') \\ & = (r,r') \cdot (z_1,\ldots,z_n,z'_1,\ldots,z'_n) \\ & = \left( e^{2\pi i r_1/p} z_1, \ldots, e^{2\pi i r_1/p} z_n, e^{2\pi i r'_1/p} z'_1, \ldots e^{2\pi i r'_n/p} z'_n \right), \end{align*}

and similarly for $Q$. As a result, the group $(\mathbb {Z}_{/{p}})^{2} \cong \langle R, Q \rangle$ acts on $S^{2n-1} \times S^{2n-1}$. This action may not be free, but when the resulting action is free, we call these the standard linear examples. Classifying these examples is the goal of this paper; we will not consider other possibilities for $(\mathbb {Z}_{/{p}})^2$ actions on $S^{2n-1} \times S^{2n-1}$.

The tiniest bit of representation theory reveals that this construction involving $4n$ rotation numbers captures every free linear action of $(\mathbb {Z}_{/{p}})^2$ on $S^{2n-1} \times S^{2n-1}$.

Proof. Recall that ‘equivalent’ for $(\mathbb {Z}_{/{p}})^2$-representations $V$ and $W$ means there is an equivariant isomorphism of vector spaces $V \to W$.

By virtue of its being abelian, $\Gamma$'s irreducible complex representations are one-dimensional. Let $\chi$ be a character of an irreducible complex representation. Since the map $x \mapsto x^2$ is a bijection and since $p \neq 2$, the Frobenius–Schur indicator of $\chi$ is

\[ \frac{1}{|\Gamma|} \sum_{\gamma \in \Gamma} \chi(\gamma^2) = \frac{1}{|\Gamma|} \sum_{\gamma \in \Gamma} \chi(\gamma) = \begin{cases} 1 & \text{if } \chi \mbox{is trivial,} \\ 0 & \mbox{otherwise.} \end{cases} \]

Consequently, a real irreducible representation of $\Gamma$ is either the trivial representation or a complex representation [Reference Serre12, p. 108]. In other words, a non-trivial real irreducible representation has character $\chi + \bar {\chi }$ for some complex irreducible representation with character $\chi$. Then for $\Gamma = (\mathbb {Z}_{/{p}})^2$ generated by $(1,0)$ and $(0,1)$, there are integers $r$ and $q$, not both zero, so that

\begin{align*} \chi(1,0) & = e^{2\pi r i/p}, \\ \chi(0,1) & = e^{2\pi q i/p}, \end{align*}

and consequently $\chi + \bar {\chi }$ is the character of the real two-dimensional representation for which $(1,0)$ acts on $\mathbb {R}^2$ by rotating through the angle $2\pi r/p$ and $(0,1)$ by the angle $2\pi q/p$. Write $W_{r,q}$ for this representation.

To finish, suppose $\Gamma := \mathbb {Z}_{/{p}} \times \mathbb {Z}_{/{p}}$ and $\Gamma$ acts on $\mathbb {R}^{2n} \oplus \mathbb {R}^{2n}$ preserving the decomposition, so the representation decomposes as $V_1 \oplus V_2$ with $\dim V_1 = \dim V_2 = 2n$. Then by Maschke, the representation on $V_i$ decomposes into a sum of irreducible real representations. Each real representation is either trivial or $W_{r,q}$ for some $r,q$. After gathering together any trivial representations in pairs as $W_{0,0}$, we conclude that every linear representation of $\Gamma$ has the form

\[ W_{r_1,q_1} \oplus \cdots \oplus W_{r_n,q_n} \oplus W_{r'_1,q'_1} \oplus \cdots \oplus W_{r'_n,q'_n}, \]

i.e. the form given at the beginning of § 2.

For convenience, define

\[ W_{R,Q} := W_{r_1,q_1} \oplus \cdots \oplus W_{r_n,q_n} \oplus W_{r'_1,q'_1} \oplus \cdots \oplus W_{r'_n,q'_n}, \]

for the pair of $2n$-tuples $R = (r_1,\ldots,r_n,r'_1,\ldots,r'_n)$ and $Q = (q_1,\ldots,q_n,q'_1,\ldots,q'_n)$. Also note that proposition 2.1 generalizes to describe linear representations of $(\mathbb {Z}_{/{p}})^m$ on $(S^{2n-1})^m$.

As a warning, note again that the action given by these linear representations need not be a free action. There is a criterion, however, which characterizes when the action is free.

Proof. Let $W_{R,Q} \cong \mathbb {R}^{2n} \oplus \mathbb {R}^{2n}$ be a representation of $\Gamma$ as above. The action on $S^{2n-1} \times S^{2n-1}$ associated to $W_{R,Q}$ is free exactly when, for all vectors $v = (v_1,v_2) \in \mathbb {R}^{2n} \oplus \mathbb {R}^{2n}$ with both $v_1 \neq 0$ and $v_2 \neq 0$, the vector $v$ has trivial stabilizer. Stated the other way, this means that for all $\gamma \in \Gamma$, the restriction of $W_{R,Q}$ from $\Gamma$ to $\langle \gamma \rangle$ decomposes as a sum of representations—but there is a trivial representation in the decomposition of at most one term of $\mathbb {R}^{2n} \oplus \mathbb {R}^{2n}$.

Let $\gamma = (\gamma _1,\gamma _2) \in (\mathbb {Z}_{/{p}})^2$. Then the two-dimensional real representation $W_{r,q}$ restricted to $\langle \gamma \rangle$ is the representation on $\mathbb {R}^2$ by which $\gamma$ acts by rotation through an angle of $2 \pi (\gamma _1 r + \gamma _2 q) / p$. So the action is free exactly when, for all non-trivial $(\gamma _1,\gamma _2)$, there are some $i$'s so that

\[ \gamma_1 r_i + \gamma_2 q_i \equiv 0 \mod p, \]

or some $j$'s so that

\[ \gamma_1 r'_j + \gamma_2 q'_j \equiv 0 \mod p, \]

but no pair $(i,j)$ which satisfies both conditions. This is exactly the condition given in the lemma.

In analogy with the lens space case, when the action given by these linear representations is free, we write the quotient as $L(p,p; R, Q)$.

3.1. The first $k$-invariant and the homotopy type

The first non-trivial $k$-invariant for a $(2n-1)$-dimensional lens space with rotation numbers $r_1,\dots, r_n$ is

\[ k(L)=\prod_{i=1}^n (r_i a)\in H^{2n}(\mathbb{Z}_{/{p}};\mathbb{Z}) \]

where $a\in H^2(\mathbb {Z}_{/{p}};\mathbb {Z})$ is the generator [Reference Eilenberg and MacLane5]. We think of the $(r_i a)$ as ‘rotation classes’ and then the $k$-invariant is just the cup product of the rotation classes. The first non-trivial $k$-invariant in the case of linear $\mathbb {Z}_{/{p}} \times \mathbb {Z}_{/{p}}$ action on a product of equidimensional odd spheres can be defined similarly.

To begin, we provide the integral cohomology of the fundamental group, i.e. the cohomology of $K(\mathbb {Z}_{/{p}}\times \mathbb {Z}_{/{p}},1)$, as the generators are needed to describe the first non-trivial $k$-invariant. The ring structure is well known and can be found in [Reference Fowler and Thatcher6], [Reference Chapman3], and [Reference Long9].

Proposition 3.1 The integral cohomology ring of $K(\mathbb {Z}_{/{p}}\times \mathbb {Z}_{/{p}},1)$ is

\[ H^\star(K(\mathbb{Z}_{/{p}}\times\mathbb{Z}_{/{p}},1);\mathbb{Z})\cong \mathbb{Z}[a,b,c]/(pa,pb,pc,c^2) \]

where $|a|=|b|=2$ and $|c|=3$.

More specifically, the isomorphism in proposition 3.1 is chosen so that $a$ and $b$ correspond to the standard basis $(1,0)$ and $(0,1)$ of $\mathbb {Z}_{/{p}}\times \mathbb {Z}_{/{p}}$. There is a copy of $\mathbb {Z}_{/{p}} \times \mathbb {Z}_{/{p}} \subset \mathbb {Z}[a,b] /(pa,pb)$ contained in the cohomology ring, and an automorphism $\varphi : \mathbb {Z}_{/{p}}\times \mathbb {Z}_{/{p}} \to \mathbb {Z}_{/{p}}\times \mathbb {Z}_{/{p}}$ gives rise to an automorphism $H^\star (\varphi )$ which, when restricted to the copy of $\mathbb {Z}_{/{p}}\times \mathbb {Z}_{/{p}}$, can be identified with $\varphi$.

Just as the first $k$-invariant for a lens space is given by the product of the rotation numbers modulo $p$, the first non-trivial $k$-invariant in the case of $L(p,p; R, Q)$ is the product of rotation classes in $H^2(\mathbb {Z}_{/{p}}\times \mathbb {Z}_{/{p}}; \mathbb {Z})$.

Lemma 3.2 Lemma 5.1 in [Reference Fowler and Thatcher6]

Let $L = L(p,p;R,Q)$ and suppose $p > n$. Then $k(L) \in H^{2n}(\mathbb {Z}_{/{p}}\times \mathbb {Z}_{/{p}}); \mathbb {Z}^2)$ is

\[ \left( \prod_{i=1}^n (r_i a + q_i b), \prod_{i=1}^n (r'_i a + q'_i b) \right), \]

where $a$ and $b$ are generators of $H^2(\mathbb {Z}_{/{p}}\times \mathbb {Z}_{/{p}}; \mathbb {Z})$ as in proposition 3.1.

Combining this with Theorem 3.3 in [Reference Fowler and Thatcher6], we have the following result.

Proposition 3.3 Theorem 3.3 in [Reference Fowler and Thatcher6]

Assume the prime $p>3$ satisfies $p > n + 1$. For free linear actions of $\mathbb {Z}_{/{p}} \times \mathbb {Z}_{/{p}}$ on $S^{2n-1} \times S^{2n-1}$ with a specified identification of the quotient's fundamental group with $\mathbb {Z}_{/{p}} \times \mathbb {Z}_{/{p}}$, the quantity

\[ k(L)=\left( \prod_{i=1}^n (r_i a + q_i b), \prod_{i=1}^n (r'_i a + q'_i b) \right) \in H^{2n}(\mathbb{Z}_{/{p}} \times \mathbb{Z}_{/{p}}; \mathbb{Z}^2), \]

modulo automorphisms of $\mathbb {Z}^2$, determines the homotopy type of the quotient.

Changing the specified identification of the fundamental group with $\mathbb {Z}_{/{p}} \times \mathbb {Z}_{/{p}}$ amounts to applying the same automorphism to $a$ and $b$, as in the discussion following proposition 3.1.

We compare proposition 3.3 to the corresponding statement for $\mathbb {Z}_{/{p}}$ actions on $S^n$. In the lens space case, the homotopy classification (with a specified generator of $\pi _1$) boils down to the product of the rotation classes, i.e. rotation numbers, up to automorphisms of $\mathbb {Z}$, i.e. up to sign. Often, the classification of lens spaces up to homotopy equivalence is described using a formula such as

(3.1)\begin{equation} t^n r_1\dots r_n\equiv{\pm} r'_1\dots r'_n \pmod p, \end{equation}

where $t$ is an element of $\mathbb {Z}_{/{p}}$ relatively prime to $p$, and $r$ and $r'$ are the rotation numbers corresponding to two lens spaces. This ‘numeric’ formula might be contrasted with the cohomological perspective presented in proposition 3.3. The rotation numbers are now described as rotation classes; the power $t^n$ accounts for a change of identification of $\pi _1$ with $\mathbb {Z}_{/{p}} \times \mathbb {Z}_{/{p}}$, which would be played by replacing $a$ and $b$ in the formula in proposition 3.3. The sign in equation (3.1) corresponds to a choice of automorphism of $\mathbb {Z}$, the top dimensional cohomology group, which in proposition 3.3 becomes an automorphism of $\mathbb {Z}^2$.

3.2. Simple homotopy classification

In the lens space case, the simple homotopy type is determined by the rotation numbers. Specifically, $L(p;r_1,\dots, r_n)\simeq _s L(p;r_1'\dots r_n')$ if and only if for some $k\in \mathbb {Z}_{/{p}}$ and permutation $\sigma$, $r_i\equiv k r'_{\sigma (i)}$ for all $i$ [Reference Milnor10].

In the case of $L(p,p;R,Q)$, because the quotients are even dimensional, the homotopy classification and the simple homotopy classification coincide.

The proof carries through almost exactly the same as the proof for Proposition 2 in [Reference Thatcher14]. We include a version here for completeness.

Proof. The spaces $X$ and $Y$ are both simple Poincaré complexes since they are both manifolds, hence they are finite, connected, CW complexes with fundamental classes $[X]$ and $[Y]$, respectively, and chain homotopy equivalences $\phi _X:C^{2n+2m-2-*}(X)\to C_*(X)$ and $\phi _Y:C^{2n+2m-2-*}(Y)\to\! C_*(Y)$ satisfying $\tau (\phi _X)\!=0$ and $\tau (\phi _Y)=0$. Here $\tau$, the torsion of the chain equivalence, vanishes by the fact these are simple Poincaré complexes.

Similar to the work in [Reference Davis and Löffler4] with $\pi _1X\cong G$ finite and acting trivially on the cohomology of the universal cover $\tilde {X}$, we have the chain-homotopy commutative diagram

and so we have

\begin{align*} 0=\tau (\phi_Y)=\tau (f_*\circ\phi_X\circ f^*)& =\tau (f_*)+\tau (\phi_X)+\tau (f^*)\\ & =\tau (f_*)+\tau (f^*)=\tau (f)+({-}1)^{2n+2m-2}\overline{\tau (f)}, \end{align*}

where $\overline {\tau (f)}$ is the result of applying the involution on $\operatorname {Wh}(\pi _1Y)$ to $\tau (f)$. Since $2n+2m-2$ is even and the involution on $\operatorname {Wh}(G)$ is trivial when $G$ is a finite abelian group [Reference Bak1],

\[ \tau (f)={-}\overline{\tau (f)}={-}\tau (f). \]

As $\operatorname {Wh}(G)$ is torsion-free by assumption, $\tau (f) =0$.

For $\Gamma =\mathbb {Z}_{/{p}}\times \mathbb {Z}_{/{p}}$ actions on $S^{2n-1}\times S^{2n-1}$, provided $p > 3$ the action on cohomology is trivial, and so we immediately get the following result.

4.1. Classification by characteristic classes

Again $\Gamma$ acts on $S^{2n-1} \times S^{2n-1}$ via a decomposition-preserving orthogonal representation on $V \oplus V$. Let $\alpha : \Gamma \to SO(V \oplus V)$ be this representation. In the special case $\Gamma = \mathbb {Z}_{/{p}} \times \mathbb {Z}_{/{p}}$, combining knowledge of the representations of this abelian group with $\dim V = 2n$ refines the map $\alpha$ to

\[ \Gamma \to SO(2)^{n} \times SO(2)^{n} \subset SO(4n). \]

This description can be related to the above description: inside each $SO(2)$ factor is a copy of $\mathbb {Z}_{/{p}}$, so the map $\alpha$ amounts to a linearization of the above description in terms of $R$ and $Q$, which are both vectors of $2n$ elements of $\mathbb {Z}_{/{p}}$.

By constructing $BG$ appropriately, regard $(S^{2n-1} \times S^{2n-1})/\Gamma$ as a subcomplex of $BG$, and therefore build the map

\[ B\alpha |_{(S^{2n-1} \times S^{2n-1})/\Gamma} : (S^{2n-1} \times S^{2n-1})/\Gamma \to BSO(4n) \]

The main theorem in [Reference Szczarba13] implies that

\[ T(S^{2n-1} \times S^{2n-1}/\Gamma) \oplus \mathbb{R}^2 \]

is a $(4n)$-dimensional vector bundle over $(S^{2n-1} \times S^{2n-1})/\Gamma$ with classifying map $B\alpha |_{(S^{2n-1} \times S^{2n-1})/\Gamma }$.

The above description is explicit enough to compute the Pontrjagin classes. Suppose $R = (r_1,\ldots,r_n,r'_1,\ldots,r'_n)$ and $Q = (q_1,\ldots,q_n,q'_1,\ldots,q'_n)$ be elements of $(\mathbb {Z}_{/{p}})^{2n}$ so that the map $\alpha$ can be defined via $\alpha (1,0) = R$ and $\alpha (0,1) = Q$ and the inclusion $\mathbb {Z}_{/{p}} \subset SO(2)$.

Consider the effect of $\alpha : B\Gamma \to BSO(4n)$ on cohomology, namely

\[ \alpha^\star : H^\star(BSO(4n);\mathbb{Z}) \to H^\star(\Gamma;\mathbb{Z}). \]

Corresponding to each $SO(2)$ factor in the left-hand term of $SO(2)^n \times SO(2)^n \subset SO(4n)$, there is $v_i \in H^2(BSO(4n);\mathbb {Z})$; similarly each $SO(2)$ in the right-hand factor results in $v'_i \in H^2(BSO(4n);\mathbb {Z})$. Then $\alpha ^\star (v_i) = r_i a + q_i b$ and $\alpha ^\star (v'_i) = r'_i a + q'_i b$.

Proposition 4.1 The total Pontrjagin class in $H^\star (B(SO(2)^n \times SO(2)^n);\mathbb {Z})$ is given by

\[ \prod_{i=1}^n \left( 1 + {v_i}^2 \right)\left( 1 + {v'_i}^2 \right) \]

implying that

\[ p( \left(S^{2n-1} \times S^{2n-1}\right)/\Gamma) = \prod_{i=1}^n \left( 1 + \left(r_i \bar{a} + q_i \bar{b}\right)^2 \right) \left( 1 + \left(r'_i \bar{a} + q'_i \bar{b}\right)^2 \right). \]

There is the map from the quotient $(S^{2n-1} \times S^{2n-1})/\Gamma$ to the classifying space $B\Gamma$ which, in cohomology, provides a map from $H^\star (B\Gamma )$ to the cohomology of the quotient. We have denoted $a$ and $b$ as the generators of $H^2(B\Gamma ;\mathbb {Z})$, and in what follows, let $\bar {a}$ and $\bar {b}$ denote the image of the generators under the map

\[ H^\star(B\Gamma) \to H^\star(\left(S^{2n-1} \times S^{2n-1}\right)/\Gamma). \]

Compare the computation in proposition 4.1 with the computation for the total Pontrjagin class of the lens space $L$, which is given by

\[ p( L) = \prod_{i=1}^n \left( 1 + (r_i \bar{a})^2 \right) \]

where the $r_i$ are the rotation numbers.

4.2. Surgery

To understand the possible actions of $\Gamma =\mathbb {Z}_{/{p}} \times \mathbb {Z}_{/{p}}$ on $S^{2n-1} \times S^{2n-1}$, we consider the surgery exact sequence

\[ \cdots \to L^s_{4n-1}(\Gamma) \to S^{\operatorname{Cat},s}(X) \to [X, G/\hspace{-1.5pt}\operatorname{Cat}] \to L_{4n-2}^s(\Gamma). \]

where $X=(S^{2n-1}\times S^{2n-1})/\Gamma$ and $G/\hspace {-1.5pt}\operatorname {Cat}$ is $G/\hspace {-1.5pt}\operatorname {Top}$ or $G/\hspace {-1.5pt}\operatorname {PL}$. We note that this is the simple surgery exact sequence and that the $L_m^s$ are simple L-groups.

In general, for a group $\Gamma$ of odd order, the simple L-groups are $L_m^s(\mathbb {Z} \Gamma )=\Sigma \oplus 8\mathbb {Z}, 0, \Sigma \oplus \mathbb {Z}/2,0$ for $m \equiv 0,1,2,3 \pmod 4$, where $\Sigma$ has no torsion [Reference Bak1]. Since $4n-2\equiv 2 \pmod 4$, our surgery exact sequence becomes:

\[ 0 \to S^{\operatorname{Cat},s}(X) \to [X, G/\hspace{-1.5pt}\operatorname{Cat}] \to \Sigma\oplus \mathbb{Z}/2, \]

and our classification comes down to determining the set of normal invariants, $[X,G/\hspace {-1.5pt}\operatorname {Cat}]$, and the map $[X, G/\hspace {-1.5pt}\operatorname {Cat}] \to L_{4n-2}^s(\Gamma )=\Sigma \oplus \mathbb {Z}/2$, for $X=L(p,p;Q,R)$. We start first with the set of normal invariants.

4.3. Normal invariants

We first need a couple of lemmas. We note that they are similar to lemmas in [Reference Thatcher14].

Lemma 4.2 Suppose that $\Gamma =\mathbb {Z}_{/{p}}\times \mathbb {Z}_{/{p}}$ and that $Y$ is a $\operatorname {Cat}$-manifold for $\operatorname {Cat} = \operatorname {PL}$ or $\operatorname {Top}$ and with a free $\Gamma$-action such that $\Gamma$ acts trivially on $H^\star (Y ; \mathbb {Z}[1/p])$. Then

\[ [Y/\Gamma, G/\hspace{-1.5pt}\operatorname{Cat}] \left[ 1/p \right] \cong [Y, G/\hspace{-1.5pt}\operatorname{Cat}] \left[1/p \right]. \]

Corollary 4.3 Suppose $\Gamma =\mathbb {Z}_{/{p}}\times \mathbb {Z}_{/{p}}$ acts freely on $S^{2n-1} \times S^{2n-1}$. Then for $\operatorname {Cat}$ being $\operatorname {Top}$ or $\operatorname {PL}$,

\[ \left[(S^{2n-1} \times S^{2n-1})/\Gamma,G/\hspace{-1.5pt}\operatorname{Cat}\right] \left[ 1/p \right] \cong \mathbb{Z}_{{/}2}. \]

Proof. By lemma 4.2,

\begin{align*} \left[(S^{2n-1} \times S^{2n-1})/\Gamma,G/\hspace{-1.5pt}\operatorname{Cat} \right] \left[ 1/p \right] & \cong \left[S^{2n-1} \times S^{2n-1},G/\hspace{-1.5pt}\operatorname{Cat}\right] \left[ 1/p \right] \\ \cong \pi_{2n-1}\left( G/\hspace{-1.5pt}\operatorname{Cat}\left[ 1/p \right] \right) & \oplus \pi_{2n-1}\left( G/\hspace{-1.5pt}\operatorname{Cat}\left[ 1/p \right] \right) \oplus \pi_{4n-2}\left( G/\hspace{-1.5pt}\operatorname{Cat}\left[ 1/p \right] \right). \end{align*}

To identify maps out of the product $S^{2n-1} \times S^{2n-1}$ with the given sum of homotopy groups, we need to verify that the Whitehead products vanish; this follows from the fact that $G/\hspace {-1.5pt}\operatorname {Cat}$ is a path-connected H-space and that localizations of such spaces are path-connected H-spaces. The odd-dimensional homotopy groups of $G/\hspace {-1.5pt}\operatorname {Cat}$ are trivial, so the localized group $\pi _{2n-1}( G/\hspace {-1.5pt}\operatorname {Cat}[ 1/p ] )$ also vanishes. Since $4n-2 \equiv 2 \pmod 4$ and since $\pi _{4n-2}(G/\hspace {-1.5pt}\operatorname {Cat}) = \mathbb {Z}_{/2}$, we have $\pi _{4n-2}( G/\hspace {-1.5pt}\operatorname {Cat}[ 1/p ] ) = \mathbb {Z}_{/2}$ and the conclusion follows.

In [Reference Thatcher14], lemma 6 gives that for odd primes $p > 2n$, there is a $(2n+2)$-equivalence

\[ (BO)_{(p)} \to \prod_{0 < j < n/2} K(\mathbb{Z}_{(p)}, 4j). \]

Using the description of $BO$ from Theorem 7.4 of the survey article [Reference Lance8], lemma 6 from [Reference Thatcher14] can be improved:

Lemma 4.4 For an odd prime $p$, there is a $(2p+1)$-equivalence

\begin{align*} (BO)_{(p)} & \to K( \mathbb{Z}_{(p)}, 4 ) \times K( \mathbb{Z}_{(p)}, 8 ) \times \cdots \times K( \mathbb{Z}_{(p)}, 2(p-1) ) \\ & \simeq \prod_{i=1}^{(p-1)/2} K( \mathbb{Z}_{(p)}, 4i ). \end{align*}

Proof. For an odd prime $p$, there is an equivalence of $H$-spaces

\[ (BO)_{(p)} \simeq \prod_{i=0}^{(p-3)/2} \Omega^{4i} W, \]

where $W$ has trivial homotopy except for $\pi _{2i(p-1)} (W) = \mathbb {Z}_{(p)}$ for positive integers $i$ [Reference Lance8]. For $p$ an odd prime, $4(p-1) > 2p+1$, and therefore

\[ W \to K( \mathbb{Z}_{(p)}, 2(p-1) ) \]

is a $(2p+1)$-equivalence. Moreover, for any $0 \leq i \leq (p-3)/2$, then $4(p-1) - 4i \geq 2p + 2> 2p+1$, and so

\[ \Omega^{4i} W \to K( \mathbb{Z}_{(p)}, 2(p-1) - 4i ) \]

is a $(2p+1)$-equivalence. Therefore,

\begin{align*} (BO)_{(p)} & \to \prod_{i=0}^{(p-3)/2} \Omega^{4i} K( \mathbb{Z}_{(p)}, 2(p-1) ) \\ & \simeq \prod_{i=0}^{(p-3)/2} K( \mathbb{Z}_{(p)}, 2(p-1) - 4i ) \\ & \simeq \prod_{i=1}^{(p-1)/2} K( \mathbb{Z}_{(p)}, 4i ) \end{align*}

is a $(2p+1)$-equivalence.

For example, when $p = 3$, we have that

\[ (BO)_{(3)} \simeq K(\mathbb{Z}_{(3)}, 4) \]

is a 7-equivalence.

Lemma 4.5 Suppose that $\Gamma =\mathbb {Z}_{/{p}}\times \mathbb {Z}_{/{p}}$ acts freely on $S^{2n-1} \times S^{2n-1}$ with quotient $X$. Then

\[ \left[X,G/\hspace{-1.5pt}\operatorname{Cat}\right] \cong \mathbb{Z}_{{/}2} \oplus \bigoplus_i H^{4i}(X;\mathbb{Z}_{/{p}}) \]

for $\operatorname {Cat} =$ $\operatorname {PL}$ or $\operatorname {Top}$.

Proof. Localizing $G/\hspace {-1.5pt}\operatorname {Cat}$ at $p$ and away from $p$ gives rise to a localization square

and hence then following exact sequence

\[ 0 \to \left[X,G/\hspace{-1.5pt}\operatorname{Cat}\right] \to \left[X,G/\hspace{-1.5pt}\operatorname{Cat}\right] \left[ 1/p \right] \oplus \left[X,G/\hspace{-1.5pt}\operatorname{Cat}\right]_{(p)} \to \left[X,G/\hspace{-1.5pt}\operatorname{Cat}\right]_{(0)} \to 0. \]

Since $[X,G/\hspace {-1.5pt}\operatorname {Cat}]_{(0)} \cong \bigoplus _i H^{4i}(X;\mathbb {Q}) \cong 0$,

the short exact sequence from the localization square yields

\[ \left[X,G/\hspace{-1.5pt}\operatorname{Cat}\right] \cong \left[X,G/\hspace{-1.5pt}\operatorname{Cat}\right]\left[ 1/p \right] \oplus \left[X,G/\hspace{-1.5pt}\operatorname{Cat}\right]_{(p)}. \]

By corollary 4.3,

\[ \left[X,G/\hspace{-1.5pt}\operatorname{Cat}\right]\left[ 1/p \right] \cong \mathbb{Z}_{{/}2}, \]

a Kervaire–Arf invariant. But $BO_{(p)} \cong G/\hspace {-1.5pt}\operatorname {Cat}_{(p)}$, so both localizations are a product of Eilenberg–MacLane spaces and the lemma then follows from the calculation of $[X,G/\hspace {-1.5pt}\operatorname {Cat}]_{(p)}$.

4.4. The structure set

Recall that the surgery exact sequence for $X=L(p,p;Q,R)$ becomes

\[ 0 \to \mathcal{S}^{\operatorname{Cat},s}(X) \to [X, G/\hspace{-1.5pt}\operatorname{Cat}] \to \Sigma \oplus\mathbb{Z}_{{/}2}. \]

From lemma 4.5, we have that the right most map in the surgery exact sequence above is $[X, G/\hspace {-1.5pt}\operatorname {Cat}]\cong \mathbb {Z}_{/2} \oplus H^{4i}(X;\mathbb {Z}_{/{p}}) \to \Sigma \oplus \mathbb {Z}_{/2}$. Both $G/\hspace {-1.5pt}\operatorname {Top}$ and $G/\hspace {-1.5pt}\operatorname {PL}$ have H-space structures that make this map into a homomorphism. Since $\Sigma$ is torsion-free, and $H^{4i}(X;\mathbb {Z}_{/{p}})$ is trivial or $p$-torsion for all $i$, all of the $p$-torsion in $H^{4i}(X;\mathbb {Z}_{/{p}})$ must come from the structure set.

Consider the map $X\to S^{4n-2}$ given by sending everything but a disk to a point. This induces a commutative diagram

By the Kerviare–Arf invariant, $\pi _{4n-2}(G/\hspace {-1.5pt}\operatorname {Cat})$ maps isomorphically into the $L$-group of the trivial group, and we see that the $\mathbb {Z}_{/2}$ in the set of normal invariants maps isomorphically onto the $\mathbb {Z}_{/2}$ in $L_{4n-2}(\Gamma )$.

From this discussion, we conclude that $S^{\operatorname {Cat},s}(X)=\bigoplus _i H^{4i}(X;\mathbb {Z}_{/{p}})$, and the classification is determined by the Pontrjagin classes. This yields theorem 4.6

Since $f$ is a homotopy equivalence, the rings $H^\star (X;\mathbb {Z}_{/p})$ and $H^\star (Y;\mathbb {Z}_{/p})$ are isomorphic, and can be computed using knowledge of the $k$-invariant. Specifically, one can compute the cohomology of $L(p,p;R,Q)$ using the Serre spectral sequence on the Borel fibration $S^{2n-1} \times S^{2n-1} \to L(p,p;R,Q) \to B\Gamma$ and the relationship between the $k$-invariant and the transgression.