Browsing by Browse by FOR 2008 "010112 Topology"

Let CW0 be the category of reduced CW-complexes, that is CW-complexes with 0-skeleton a point which is the base-point, and base-point preserving cellular maps.

We describe an algebraic structure on chain complexes yielding algebraic models which classify homotopy types of PD⁴-complexes. Generalizing Turaev's fundamental triples of PD³-complexes we introduce fundamental triples of PDⁿ-complexes and show that two PDⁿ complexes are orientedly homotopy equivalent if and only if their fundamental triples are isomorphic. As applications we establish a conjecture of Turaev and obtain a criterion for the existence of degree 1 maps between n-dimensional manifolds.

Turaev conjectured that the classification, realization and splitting results for Poincare duality complexes of dimension 3 (PD₃–complexes) generalize to PDₙ–complexes with (n−2)–connected universal cover for n≥3. Baues and Bleile showed that such complexes are classified, up to oriented homotopy equivalence, by the triple consisting of their fundamental group, orientation class and the image of their fundamental class in the homology of the fundamental group, verifying Turaev’s conjecture on classification. We prove Turaev’s conjectures on realization and splitting. We show that a triple (G, ω, μ), comprising a group G, a cohomology class ω∈H¹ (G;Z∕2Z) and a homology class μ∈Hₙ(G;Z^ω), can be realized by a PDₙ–complex with (n−2)–connected universal cover if and only if the Turaev map applied to μ yields an equivalence. We show that a PDₙ–complex with (n-2)–connected universal cover is a nontrivial connected sum of two such complexes if and only if its fundamental group is a nontrivial free product of groups. We then consider the indecomposable PDₙ–complexes of this type. When n is odd the results are similar to those for the case n=3. The indecomposables are either aspherical or have virtually free fundamental group. When n is even the indecomposables include manifolds which are neither aspherical nor have virtually free fundamental group, but if the group is virtually free and has no dihedral subgroup of order >2 then it has two ends.

We extend Hendriks' classification theorem and Turaev's realisation and splitting theorems for PD³-complexes to the relative case of PD³-pairs. The results for PD³-complexes are recovered by restricting the results to the case of PD³-pairs with empty boundary. Up to oriented homotopy equivalence, PD³-pairs are classified by their fundamental triple consisting of the fundamental group system, the orientation character and the image of the fundamental class under the classifying map. Using the derived module category we provide necessary and sufficient conditions for a given triple to be realised by a PD³-pair. The results on classification and realisation yield splitting or decomposition theorems for PD³-pairs, that is, conditions under which a given PD³-pair decomposes as interior or boundary connected sum of two PD³-pairs.

With the development of algebraic topology, Poincaré's discovery has been extended and reformulated. ... A 'Poincaré duality complex complex of dimension n', or PDⁿ-complex, is a finitely dominated CW-complex exhibiting n-dimensional equivariant Poincaré duality. We may thus regard Poincaré duality complexes as 'homotopy generalisations' of manifolds.

We compute the monoid of essential self-maps of SⁿxSⁿ fixing the diagonal. More generally, we consider products SxS, where S is a suspension. Essential self-maps of SxS demonstrate the interplay between the pinching action for a mapping cone and the fundamental action on homotopy classes under a space. We compute examples with non-trivial fundamental actions.

It is well-known how to compute the structure of the second homotopy group of a space, X, as a module over the fundamental group π₁X, using the homology of the universal cover and the Hurewicz isomorphism. We describe a new method to compute the third homotopy group, π₃X as a module over π₁X. Moreover, we determine π₃X as an extension of π₁X-modules derived from Whitehead's Certain Exact Sequence. Our method is based on the theory of quadratic modules. Explicit computations are carried out for pseudo-projective 3-spaces X=S¹Ue²Ue³ consisting of exactly one cell in each dimension ≤ 3.