AI 中文总结
研究开可缩流形的恰当同伦非刚性，通过特定条件构造出无限多个内部恰当同伦等价但两两非同胚的紧致可缩光滑\(N\)维流形，其边界有特定性质，揭示了与\(\mathbb{R}^n\)恰当同伦型刚性不同的情况。
AI 中文摘要
斯塔尔林斯对欧几里得空间的刻画表明，对于\(n\geq5\)，\(\mathbb{R}^n\)的恰当同伦型在拓扑上是刚性的。我们表明这种现象是例外的。对于每个\(N\geq6\)的偶数整数，存在一个恰当同伦型，包含无限多个两两非同胚的光滑开可缩\(N\)维流形。更一般地，设\(N = 2d\geq6\)，且\(\pi\)是有限超完美群。若\(\pi\)的有理复表示环的约化\((-1)^d\)特征子空间非零，则存在无限多个紧致可缩光滑\(N\)维流形，其内部恰当同伦等价但两两非同胚。它们的边界是同伦等价的整同调\((N - 1)\)维球面，基本群为\(\pi\)，但两两不是拓扑\(h\)-配边的。
英文摘要
We study whether proper homotopy equivalence classifies open contractible manifolds arising as interiors of compact contractible manifolds. In dimensions three and four, it does: any two such interiors that are properly homotopy equivalent are homeomorphic. In every even dimension $N\geq 6$ and every odd dimension $N\geq 9$, however, it does not: a single proper homotopy type contains infinitely many pairwise nonhomeomorphic smooth open contractible $N$-manifolds. The cases of dimensions five and seven remain unresolved.
Comments38 pages, 1 figures
