String Topology, Euler Class and TNCZ free loop fibrations

TL;DR

This paper investigates the relationship between the Euler characteristic of a manifold and the cohomology of its free loop space, proving that certain cup products vanish and deriving divisibility conditions for the Euler characteristic.

Contribution

It establishes a new vanishing result for cup products involving the Euler class in the free loop space cohomology, linking topological invariants to algebraic properties of loop fibrations.

Findings

Cup product with Euler class vanishes for positive degree classes in LM

If the induced map on cohomology is surjective, then Euler characteristic is divisible by the prime

Results imply constraints on the topology of manifolds via loop space cohomology

Abstract

Let $M$ be a connected, closed oriented manifold. Let $\omega\in H^m(M)$ be its orientation class. Let $\chi(M)$ be its Euler characteristic. Consider the free loop fibration $\Omega M\buildrel{i}\over\hookrightarrow LM\buildrel{ev}\over\twoheadrightarrow M$. For any class $a\in H^*(LM)$ of positive degree, we prove that the cup product $\chi(M)a\cup ev^*(\omega)$ is null. In particular, if $i^*:H^*(LM;\mathbb{F}_p)\twoheadrightarrow H^*(\Omega M;\mathbb{F}_p)$ is onto then $\chi(M)$ is divisible by $p$ (or $M$ is a point).