Stable splitting of mapping spaces via nonabelian Poincar\'e duality

TL;DR

This paper introduces a novel method using nonabelian Poincaré duality to achieve stable splittings of mapping spaces and sections on manifolds, extending previous results to non-parallelizable cases and incorporating group actions.

Contribution

It presents a new approach to stable splitting of mapping spaces via nonabelian Poincaré duality, generalizing to non-parallelizable manifolds and actions.

Findings

Derived stable splitting of compactly supported mapping spaces.

Extended splitting to sections of bundles with group actions.

Method applicable to non-parallelizable manifolds.

Abstract

We use nonabelian Poincar\'e duality to recover the stable splitting of compactly supported mapping spaces, $\rm{Map_c}$$(M,\Sigma^nX)$, where $M$ is a parallelizable $n$-manifold. Our method for deriving this splitting is new, and naturally extends to give a more general stable splitting of the space of compactly supported sections of a certain bundle on $M$ with fibers $\Sigma^nX$, twisted by the tangent bundle of $M$. This generalization incorporates possible $O(n)$-actions on $X$ as well as accommodating non-parallelizable manifolds.