String Topology, Euler Class and TNCZ free loop fibrations

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 $ΩM\buildrel{i}\over\hookrightarrow LM\buildrel{ev}\over\twoheadrightarrow M$. For any class $a\in H^{\ast }(LM)$ of positive degree, we prove that the cup product $\chi (M)a\cup e{v}^{\ast }(\omega )$ is null. In particular, if $i^{\ast }:H^{\ast }(LM;{\mathbb{F}}_p)\twoheadrightarrow H^{\ast }(\Omega M;{\mathbb{F}}_p)$ is onto then $\chi (M)$ is divisible by $p$ (or $M$ is a point).

36 pages

References in corpus (3)

• Cap Products in String Topology
• TQFT string operations in open-closed string topology
• String spectral sequence

Cited by in corpus (1)

• New construction of the brane coproduct and vanishing of cup products on sphere spaces