Loop space decompositions of highly connected Poincaré duality complexes

Abstract

One of the main goals of homotopy theory is to determine the homotopy types of topological spaces. Furthermore, given a space, one may hope there is a way to decompose this space into simpler spaces whose homotopy theory is well understood. In particular, it can be useful to study product decompositions of the based loop space of the object we wish to consider -- doing so provides useful data about the object's homotopy groups, amongst other things. Furthermore, by considering manifolds through the lens of homotopy theory, it is natural to broaden one’s scope to Poincar\'e Duality complexes: such complexes are a topological generalisation of manifolds and have an underlying structure that is readily exploitable. Thus, by studying the loop spaces of Poincar\'e Duality complexes, we may answer questions about the homotopy theory of manifolds.

In particular, given two manifolds of the same dimension, a natural object to consider is their connected sum. This situation is often flipped: one asks the question of whether a given manifold is decomposable as a connected sum of simpler manifolds. In areas such as surgery theory and differential topology this problem is of fundamental importance, despite the fact that in dimensions higher than two the problem is often inaccessible. This thesis studies this highly geometric problem from a new topological viewpoint, using elements of classical homotopy theory together with recent results.

In expanding upon these methods, we find that the loop space decompositions of several classes of highly connected manifolds coincide with those of the loop spaces of certain connected sums, and thus we have a homotopy theoretic perspective on the above question. Indeed, we apply these results to comment on the Vigu\'e-Poirrier Conjecture, a particular long standing question from rational homotopy theory. We also prove a higher dimensional homotopy theoretic analogue to a theorem of C.T.C. Wall -- a fundamental calculation from differential topology that shows one may decompose a simply connected 6-manifold as a connected sum of two simpler manifolds -- for \((n-2)\)-connected \(2n\)-dimensional Poincar\'e Duality complexes.

Key to these discussions is a consideration of inert maps, a concept brought across from rational homotopy theory. By combining other results from this area, we provide an answer to another problem: under what circumstances does the total space of a pullback fibration over a connected sum have the rational homotopy type of a connected sum? We conclude with a reformatted version of a recent paper of the author, which gives a condition on rational cohomology to yield an affirmative answer, but only after taking based loop spaces. This takes inspiration from recent work of Jeffrey and Selick, in which they study pullback bundles of this type, but under stronger hypotheses compared to our result.

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ORCID for Stephen Theriault: orcid.org/0000-0002-7729-5527

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