Jeremy Brookman

We classify, up to homeomorphism, all closed manifolds having the homotopy type of a connected sum of two copies of real projective n-space. 1. Statement of results Let P = Pn(R) be real projective n-space. López de Medrano [LdM71] and C.T.C. Wall [Wal68, Wal99] classified, up to PL homeomorphism, all closed PL manifolds homotopy equivalent to P when n > 4. This was extended to the topological category by Kirby-Siebenmann [KS77, p. 331]. Four-dimensional surgery [FQ90] extends the homeomorphism classification to dimension 4. Cappell [Cap74a, Cap74c, Cap76] discovered that the situation for connected sums is much more complicated. In particular, he showed [Cap74b] that there are closed manifolds homotopy equivalent to P #P 4k+1 which are not nontrivial connected sums. Recent computations of the unitary nilpotent group for the integers by Connolly-Ranicki [CR05], Connolly-Davis [CD04], and Banagl-Ranicki [BR] show that there are similar examples in dimension 4k (see [JK] for an analysis when k = 1). In this paper we classify up to homeomorphism all closed manifolds homotopy equivalent to P#P. Any such manifold has S × S as a two-fold cover; equivalently we classify free involutions on S × S inducing a non-trivial map on H1. This paper was prompted by a question of Wolfgang Lück [Lüc, Sequence (4.10), Theorem 4.11] – what does the group automorphism Z2 ∗ Z2 → Z2 ∗ Z2 given by interchanging the Z2’s induce in L-theory? We give a complete answer to Lück’s question and apply the answer to the classify the above manifolds. By the positive solution to Kneser’s conjecture (see [Hem04]) any closed 3manifold homotopy equivalent to P #P 3 is homeomorphic to Q#R where Q and R are closed 3-manifolds homotopy equivalent to P . If the spherical space form conjecture in dimension 3 is true, then Q and R would be homeomorphic to P . Hence, conjecturally, any closed 3-manifold homotopy equivalent to P #P 3 is standard. Henceforth we assume n > 3. Note that the fundamental group of P #P 4 is small in the sense of Freedman-Quinn [FQ90], so that surgery theory applies. Let Īn (respectively J̄n) be the set of homeomorphism classes of closed manifolds homotopy equivalent to P (respectively P#P). For n even, let In = Īn and Jn = J̄n. For n odd, let In (respectively Jn) be the set of oriented homeomorphism classes of closed oriented manifolds homotopy equivalent to P (respectively P#P). The Partially supported by a grant from the National Science Foundation.