高维流形的广义二重结构与简单同伦型
Generalised doubles and simple homotopy types of high dimensional manifolds
AI总结：
本文刻画了高维流形中存在 h-配边但非简单同伦等价例子的基本群条件，并基于加厚体二重结构及 Whitehead 挠率公式，证明当 n≥12 为偶数且 Wh(G) 对合非平凡时必存在此类例子。
摘要：
我们刻画了当 n 足够大时，存在 n 维流形满足 h-配边（因此同伦等价）但不简单同伦等价的基本群集合。特别地，对于偶数 n ≥ 12，我们证明：只要有限展示群 G 的 Whitehead 群 Wh(G) 上的对合非平凡，就存在这样的例子。这扩展了此前的工作，此前我们构造了第一批同伦等价但不简单同伦等价的偶数维流形例子。我们的构造基于加厚体的二重结构，证明的一个关键要素是此类流形之间同伦等价的 Whitehead 挠率公式。
英文摘要：
We characterise the set of fundamental groups for which there exist $n$-manifolds that are $h$-cobordant (hence homotopy equivalent) but not simple homotopy equivalent, when $n$ is sufficiently large. In particular, for $n \ge 12$ even, we show that examples exist for any finitely presented group $G$ such that the involution on the Whitehead group $Wh(G)$ is nontrivial. This expands on previous work, where we constructed the first examples of even-dimensional manifolds that are homotopy equivalent but not simple homotopy equivalent. Our construction is based on doubles of thickenings, and a key ingredient of the proof is a formula for the Whitehead torsion of a homotopy equivalence between such manifolds.
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