Morse theory with homotopy coherent diagrams

TL;DR

This paper develops a homotopy coherent diagram framework for Morse theory on noncompact manifolds, enabling the computation of Morse homology via colimits of parametrized Morse data.

Contribution

It introduces a novel homotopy coherent diagram approach to Morse theory on noncompact manifolds, extending classical methods with higher-dimensional parameter spaces.

Findings

Constructed a family of Morse data parametrized by cubes of arbitrary dimensions.

Defined a chain complex as a colimit of the homotopy coherent diagram.

Proved the chain complex computes Morse homology.

Abstract

We study Morse theory on noncompact manifolds equipped with exhaustions by compact pieces, defining the Morse homology of a pair which consists of the manifold and related geometric/homotopy data. We construct a collection of Morse data parametrized by cubes of arbitrary dimensions. From this collection, we obtain a family of linear maps subject to some coherency conditions, which can be packaged into a homotopy coherent diagram. We introduce a chain complex which is a colimit for the diagram and show that it computes the Morse homology.