Manifolds homotopy equivalent to $RP^4 \# RP^4$

Abstract

Recent computations of UN il-groups by Connolly, Ranicki and Davis are used to study splittability of homotopy equivalences between 4-dimensional manifolds with infinite dihedral fundamental groups.The problem of splitting a manifold into a connected sum is one of the most natural, yet one of the most difficult problems in manifold topology. It was extensively studied by S. Cappell [4][5][6][7][8][9], who also provided an elegant solution. To be more specific: let f : M n → X n = X n 1 #X n 2 , n 5, be a homotopy equivalence of closed topological manifolds, where X n is a connected sum of manifolds X n i , i = 1, 2. The homotopy equivalence f is splittable if it is homotopic to a map (necessarily a homotopy equivalence which we continue to call f ), transverse regular to the separating sphere S n −1 ⊂ X n , such that the restriction of f to Y = f −1 (S n −1 ), and the two components of f −1 (X n − S n −1 ) are homotopy equivalences.Cappell's solution to the splitting problem for f : M n → X n is in terms of certain exotic UN il groups introduced and studied by him. Despite the fact that these groups are very difficult to deal with, he was able to perform various computations which lead to many important topological results ([7, 8]). The simplest example of a group G for which UN il groups can be nontrivial is that of an infinite dihedral group D ∞ % Z/2 * Z/2. In this case, Cappell was able to show that UN il 0 (Z/2 * Z/2) = 0 and that UN il 2 (Z/2 * Z/2) contains an infinite dimensional vector space over Z/2 (cf. [11]. Here we use the notation UN il * (Z/2 * Z/2) = UN il * (Z; Z, Z)). The odd-dimensional UN il groups UN il 1 (Z/2 * Z/2) and UN il 3 (Z/2 * Z/2) resisted calculations for quite some time. (It was known, however ([13]), that in this case the UN il-groups are either trivial or infinitely generated.) Recently F. Connolly and A. Ranicki, using rather complicated algebraic machinery, have shown that UN il 1 (Z/2 * Z/2) = 0 and UN il 3 (Z/2 * Z/2) contains an infinite dimensional vector space over Z/2 [12], and † Supported in part by NSF grants INT-9505890 and INT-98115

Free involutions on S 1 × S n

“…This case was considered in [11] and a complete answer appeared in [2, Theorem 2]. Here we only indicate some of the most interesting qualitative results.…”

“…The case R P n+1 # R P n+1 is in many ways the most interesting one, but it was treated in detail in [2] and [11], so except for a few comments in Sect. 3.2 we refer to these papers for precise statements.…”

Smooth structures on nonorientable four-manifolds and free involutions

“…There exists a manifold that is homeomorphic but not diffeomorphic to Remark 32. There are infinitely many non-orientable closed topological 4-manifolds homotopy equivalent to a connected sum that are not homeomorphic to a trivial connected sum as in Item (5) of Theorem 3 [7,22,6]. It is proven in [24, Theorem 3.4] (cf.…”

Homotopy invariance of 4-manifold decompositions: Connected sums

“…Now, since Γ ∈ N DL, by [21,32], the TOP s-cobordism of Theorem 3.2(1) is a product. (The second case was applied in [25,4].) We effectively delete the last phrase in his proof.…”

“…Given a homotopy decomposition into a connected sum, a homeomorphism decomposition need not exist. There exist infinitely many examples of non-orientable closed topological 4-manifolds homotopy equivalent to a connected sum (X = RP 4 #RP 4 ) that are not homeomorphic a non-trivial connected sum [25,4]. Hence # is not always a bijection in the case π 1 (X) = D ∞ ∈ N DL.Remark 1.11.…”