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Diffeomorphisms of odd-dimensional discs, glued into a manifold

Johannes Ebert

Algebraic & Geometric Topology 23 (2023) 2329–2345

Let μM: BDiff (D2n+1) BDiff(M), for a compact (2n+1)–dimensional smooth manifold M, be the map defined by extending diffeomorphisms on an embedded disc by the identity. By a classical result of Farrell and Hsiang, the rational homotopy groups and the rational homology of BDiff (D2n+1) are known in the concordance stable range. We prove two results on the behaviour of the map μM in the concordance stable range. Firstly, it is injective on rational homotopy groups, and secondly, it is trivial on rational homology if M contains sufficiently many embedded copies of Sn × Sn+1 int(D2n+1). We also show that μM is generally not injective on homotopy groups outside the stable range.

The homotopical statement is probably not new and follows from the theory of smooth torsion invariants. The noninjectivity outside the stable range is based on recent work by Krannich and Randal-Williams. The homological statement relies on work by Botvinnik and Perlmutter on diffeomorphisms of odd-dimensional manifolds.

diffeomorphism groups of high-dimensional manifolds, smooth torsion invariants
Mathematical Subject Classification
Primary: 57S05
Received: 19 August 2021
Revised: 14 December 2021
Accepted: 26 December 2021
Published: 25 July 2023
Johannes Ebert
Fachbereich Mathematik und Informatik
Westfälische Wilhelms-Universität Münster

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