Homotopy Inertia Groups and Tangential Structures

TL;DR

This paper investigates the properties of homotopy inertia groups and tangential structures of certain high-dimensional smooth manifolds, providing explicit computations and establishing conditions under which these groups are trivial or related.

Contribution

It establishes conditions for equality of homotopy inertia groups, computes the concordance group of smoothings for specific manifolds, and links tangential homotopy equivalence to diffeomorphism classes.

Findings

Homotopy inertia groups are equal for manifolds with same homotopy type and vanishing odd-degree cohomology.

The homotopy inertia group of certain 8-dimensional manifolds is trivial.

The group of concordance classes of smoothings is computed for 8-dimensional manifolds.

Abstract

We show that if $M$ and $N$ have the same homotopy type of simply connected closed smooth $m$-manifolds such that the integral and mod-$2$ cohomologies of $M$ vanish in odd degrees, then their homotopy inertia groups are equal. Let $M^{2n}$ be a closed $(n-1)$-connected $2n$-dimensional smooth manifold. We show that, for $n=4$, the homotopy inertia group of $M^{2n}$ is trivial and if $n=8$ and $H^n(M^{2n};\mathbb{Z})\cong \mathbb{Z}$, the homotopy inertia group of $M^{2n}$ is also trivial. We further compute the group $\mathcal{C}(M^{2n})$ of concordance classes of smoothings of $M^{2n}$ for $n=8$. Finally, we show that if a smooth manifold $N$ is tangentially homotopy equivalent to $M^8$, then $N$ is diffeomorphic to the connected sum of $M^8$ and a homotopy $8$-sphere.