Proper Homotopy Nonrigidity of Open Contractible Manifolds

2026/07/20 by Donghan Kim
#math.GT #math.AT

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Abstract

We study whether proper homotopy equivalence classifies open contractible manifolds that arise as interiors of compact contractible manifolds. In dimensions three and four it does: properly homotopy equivalent such interiors are homeomorphic. In every dimension N≥6 it does not. More precisely, one proper homotopy type contains infinitely many pairwise nonhomeomorphic smooth open contractible N-manifolds when N is even or when N≥9 is odd, and at least two in dimension seven. Dimension five is the only unresolved case.

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