arXiv subjects
Publications and source records attributed to Beatrice Bleile.
In earlier work we presented necessary conditions for a fundamental triple to be that of a 3-dimensional Poincaré duality pair with aspherical boundary components. We provide a construction which shows that the necessary conditions are sufficient.
Baues and Bleille showed that, up to oriented homotopy equivalence, a Poincare duality complex of dimension $n \ge 3$ with $(n-2)$-connected universal cover, is classified by its fundamental group, orientation class and the image of its fundamental class in the homology of the fundamental group. We generalise Turaev's results for the case $n=3$, by providing necessary and sufficient conditions for a triple $(G, ω, μ)$, comprising a group, $G$, $ω\in H^{1}(G;\mathbb{Z}/2\mathbb{Z})$ and $μ\in H_{3}(G, \mathbb{Z}^ω)$ to be realised by a Poincaré duality complex of dimension $n$ with $(n-2)$-connected universal cover, and by showing that such a complex is a connected sum of two such complexes if and only if its fundamental group is a free product of groups.
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, $π_1X$, using the homology of the universal cover and the Hurewicz isomorphism. We describe a new method to compute the third homotopy group, $π_3 X$, as a module over $π_1 X$. Moreover, we determine $π_3 X$ as an extension of $π_1 X$-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 $X = S^1 \cup e^2 \cup e^3$ consisting of exactly one cell in each dimension $\leq 3$.
We compute the monoid of essential self-maps of of the product of two n-spheres fixing the diagonal. More generally, we consider products S x S, where S is a suspension. Essential self-maps of S x S 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.
We compare the structure of a mapping cone in the category Top^D of spaces under a space D with differentials in algebraic models like crossed complexes and quadratic complexes. Several subcategories of Top^D are identified with algebraic categories. As an application we show that there are exactly 16 essential self--maps of S^2 x S^2 fixing the diagonal.
We describe an algebraic structure on chain complexes yielding algebraic models which classify homotopy types of Poincare duality complexes of dimension 4. Generalizing Turaev's fundamental triples of Poincare duality complexes of dimension 3, we introduce fundamental triples for Poincare duality complexes of dimension n > 2 and show that two Poincare duality 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.
We extend Hendriks' classification theorem and Turaev's realisation and splitting theorems for Poincare duality complexes in dimension three to the relative case of Poincare duality pairs. The results for Poincare duality complexes are recovered by restricting the results to the case of Poincare duality pairs with empty boundary. Up to oriented homotopy equivalence, three-dimensional Poincare duality 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 three-dimensional Poincare duality pair. The results on classification and realisation yield splitting or decomposition theorems for three-dimensional Poincare duality pairs, that is, conditions under which a given three-dimensional Poincare duality pair decomposes as interior or boundary connected sum of two three-dimensional Poincare duality pairs.