The Whitehead group and stably trivial $G$-smoothings

TL;DR

The paper addresses how equivariant smoothing of $G$-manifolds can yield infinitely many non-isotopic $G$-smoothings that remain nontrivial in the equivariant setting yet become trivial after crossing with $ obreak olinebreak imes\, obreak \mathbb{R}$. It develops a framework based on controlled $h$-cobordisms and Whitehead torsion, connecting the problem to equivariant homology and the Farrell–Jones conjecture to extract independent smoothing data from $H_0(M;Wh_1(G))$ and $H_2(M;K_{-1}(bZ[G]))$. The main result proves the existence of infinitely many stably trivial $G$-smoothings for an odd-order cyclic group $G$ acting semifreely on a compact manifold $X$, under hypotheses on the fixed set, fundamental groups, and cohomology, with distinct torsion data yielding non-isotopic smoothings. A second regime, under additional arithmetic conditions on $|G|$, derives further stably trivial smoothings from a higher $H_2$–type term, again controlled by Whitehead and negative $K$-theory data. Altogether, the work reveals a rich interplay between equivariant smoothing theory, controlled topology, and algebraic $K$-theory, showing non-injectivity phenomena for stabilization maps via concrete torsion invariants.

Abstract

A closed manifold $M$ of dimension at least $5$ has only finitely many smooth structures. Moreover, smooth structures of $M$ are in bijection with smooth structures of $M\times\mathbb{R}$. Both of these statements are false equivariantly. In this paper, we use controlled $h$-cobordisms to construct infinitely many $G$-smoothings of a $G$-manifold $X$. Moreover, these $G$-smoothings are isotopic after taking a product with $\mathbb{R}$.

Figures (2)

• Figure 1: $F$ and $G$ in the proof of Proposition \ref{['prop: swindle']}
• Figure 2: $V$ in the proof of Proposition \ref{['prop: distinguishing smooth structures']}

Theorems & Definitions (54)

• Theorem 1.1
• Remark
• Remark
• Example 1
• Example 2
• Remark
• Definition 2.1
• Conjecture 2.2
• Proposition 2.3
• Proposition 2.4