On Certain Combinatorial and Topological Aspects of Based Loop Spaces

Samanta, Sandip (2025) On Certain Combinatorial and Topological Aspects of Based Loop Spaces. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Sandip Samanta (18IP021)) 18IP021.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

In this thesis, we have studied some structural questions involving loop spaces, both from a combinatorial and topological perspective. On the combinatorial side, we studied various models of associahedra and established their equivalence as abstract polytopes. We focused particularly on four models: the Stasheff complex, Loday’s cone construction, collapsed multiplihedra, and the graph cubeahedra for path graphs. Each of these models arises naturally in different contexts, and understanding their combinatorial isomorphism provides flexibility in their usage depending on the situation. Our proof provides an explicit bijective correspondence between their face posets, which helps in extending the result further to the level of geometric realization. The rest of the thesis focused on the topological side, where we explored certain structural aspects of fibrations admitting a homotopy section, with a particular emphasis on the loop space level. We revisited the notion of the James brace product associated with such fibrations and discussed its role in understanding the multiplicative structure on the loop space level. We provided a criterion that characterizes when the natural map from the product of loop spaces of base and fibre to the total space is an algebra map, in terms of the vanishing of the brace product. However, we observed that the vanishing of the brace product is not sufficient to ensure an H-splitting of the loop space fibration. In order to rectify this, we introduced and studied a more stringent operation called the generalized brace product. This operation takes values in the set of homotopy classes of maps from the smash product of suspensions and provides a stronger obstruction to the H-splitting. We proved that the vanishing of this generalized brace product is necessary and sufficient for the loop space level splitting map to be an H-map. We supported our theory by several explicit examples. In particular, we showed that in the case of sphere bundles over spheres, the vanishing of brace products not only implies H-splitting but also gives a homotopy equivalence of the total space with the product of base and fibre. This gives a new approach to classical decomposability questions in the homotopy category. We also provided an example where the brace product vanishes, but the generalized brace product does not, thereby demonstrating that the latter indeed provides a sharper invariant. Further, we discussed a correction to a result in Husemöller’s classical book, giving a correct statement involving the image of the J-homomorphism. We also investigated rational aspects of these questions. In particular, we proved that the rational vanishing of the brace product is sufficient to conclude a rational homotopy equivalence between the total space and the product of base and fibre. This provides a rational analogue of the decomposability question. Later, we related the generalized brace product to a generalized version of the Whitehead J-homomorphism and provided a new conceptual proof of a result of Milnor concerning suspensions. This connection offers a deeper understanding of how secondary operations such as brace products and J-homomorphisms interact in the context of loop space fibrations. Finally, we discussed the decomposability of certain noteworthy examples of sphere bundles over spheres using the theory that we built.

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