On the homotopy type of Poincaré spaces

Abstract

We study the homotopy type of finite-oriented Poincaré spaces (and, in particular, of closed topological manifolds) in even dimension. Our results relate polarized homotopy types over a stage of the Postnikov tower with the concept of CW-tower of categories due to Baues. This fact allows us to obtain a new formula for the top-dimensional obstruction for extending maps to homotopy equivalences. Then we complete the paper with an algebraic characterization of high-dimensional handlebodies.Mathematics Subject Classification (1991). 57N65, 57R67

Poincaré duality complexes in dimension four

“…In fact, in [6], Corollary 3.2 is obtained under the condition that either the fundamental group is finite or the second rational homology of the 2-type is nonzero. Corresponding conditions were used in [5] for oriented PD 2n -complexes with .n 1/-connected universal covers, and Teichner extended the approach of [6] to the nonoriented case in his thesis [16]. Our result shows that the conditions on finiteness and rational homology used in these papers are not necessary.…”

“…such that fÁg C g n 1˛n 1 D fÁg; (3)(4)(5)(6)(7)(8)(9)(10)(11) .p / i .f ı p 0 / i D˛id C d˛i C1 for i n 1; (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) where g n 1 is the attaching map of .n 1/-cells in P . Define˛by˛n C1 OEe 0 D x (see (3)(4)(5)) and…”

“…In the oriented case, special cases of Corollary 3.2 were proved by Hambleton and Kreck [6] and Cavicchioli and Spaggiari [5]. In fact, in [6], Corollary 3.2 is obtained under the condition that either the fundamental group is finite or the second rational homology of the 2-type is nonzero.…”

“…Two closed n-dimensional manifolds or two PD ncomplexes, respectively, are orientedly homotopy equivalent if and only if their fundamental triples are isomorphic. Theorem 3.1 also yields a criterion for the existence of a map of degree one between PD n -complexes, recovering Swarup's result for maps between 3-manifolds and Hendriks' result for maps between PD 3 -complexes.In the oriented case, special cases of Corollary 3.2 were proved by Hambleton and Kreck [6] and Cavicchioli and Spaggiari [5]. In fact, in [6], Corollary 3.2 is obtained under the condition that either the fundamental group is finite or the second rational homology of the 2-type is nonzero.…”

Homotopy self-equivalences of 4-manifolds with PD2 fundamental group

“…We can represent any α ∈ H 3 (M; Z/2) as α : F → SO (3). Define a self-equivalence ϕ as ϕ : F × S 2 → F × S 2 such that ϕ(x, y) = x, α(x)y .…”