L-homology on ball complexes and products

TL;DR

This paper develops L-homology theories on ball complexes, introduces product structures, and applies them to geometric signatures and surgery obstructions, aiming to clarify product operations in topology.

Contribution

It constructs new L-homology theories on ball complexes, defines product operations, and relates these to geometric signatures and surgery obstructions.

Findings

Defined homology theories with L-spectra on ball complexes.

Established product formulas for these homology theories.

Applied results to signatures and surgery obstruction theory.

Abstract

We construct homology theories with coefficients in L-spectra on the category of ball complexes and we define products in this setting. We also obtain signatures of geometric situations in these homology groups and prove product formulae which we hope will clarify products used in the theory of the total surgery obstruction.