Rigidity of self-maps of $V_{n,2}$ and classification of manifolds tangentially homotopy equivalent to $V_{n,2} \times S^k$

TL;DR

This paper investigates the rigidity of self-maps of Stiefel manifolds and classifies manifolds tangentially homotopy equivalent to their products with spheres, providing explicit inverses and partial classifications.

Contribution

It introduces new methods to find inverses in the structure set and classifies certain manifolds up to almost diffeomorphism, extending understanding of Stiefel manifolds and their related manifolds.

Findings

Complete classification for specific cases like V_{12,2}×S^3 and V_{16,2}×S^3.

Explicit construction of inverses in the structure set for many cases.

Identification of a large subgroup of the image of η where inverses can be found.

Abstract

We study two problems concerning the Stiefel manifolds $V_{n,2}$ and their products with spheres. First, we address a rigidity problem: we determine, for most values of~$n$, all self-maps of $V_{n,2}$ that are homotopic to an almost diffeomorphism. Second, we classify smooth closed manifolds tangentially homotopy equivalent to $V_{n,2} \times S^k$ up to almost diffeomorphism, for $k = 3, 5$ or $7 \leq k \leq n-3$, $k \neq 2^i - 2$. Our method is to find explicit inverses in the structure set via normal invariants of specific tangential homotopy equivalences. In favourable cases -- notably $V_{12,2} \times S^3$, $V_{16,2} \times S^3$, $V_{12,2} \times S^5$, $V_{10,2} \times S^5$ -- the classification is complete: every such manifold is almost diffeomorphic to $V_{n,2} \mathbin{\#} \Sigma \times S^k$ for some exotic sphere $\Sigma$. In the general case, we identify inverses for a large subgroup of $\operatorname{Im}(\eta)$ and provide a possible way forward to the remainder.