On diffeomorphisms of even-dimensional discs

Kupers, Alexander; Randal-Williams, Oscar

doi:10.1090/jams/1040

Cited by 3 publications

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“…k SO k is the H -space Diff fr @ .D k / of framing-preserving diffeomorphisms. It is the loop space of the classifying space BDiff fr @ .D k /, which features in the recent work of Kupers and Randal-Williams [13] on the rational homotopy groups of Diff @ .D k /. We see that d is rationally trivial because the Alexander trick implies that d becomes nullhomotopic after composition with the natural map k SO k !…”

Section: Discussionmentioning

confidence: 73%

The derivative map for diffeomorphism of disks: an example

Crowley,

Schick,

Steimle

2023

Geom. Topol.

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We prove that the derivative map d W Diff @ .D k / ! k SO k , defined by taking the derivative of a diffeomorphism, can induce a nontrivial map on homotopy groups. Specifically, for k D 11 we prove that the following homomorphism is nonzero:As a consequence we give a counterexample to a conjecture of Burghelea and Lashof by giving an example of a nontrivial vector bundle E over a sphere which is trivial as a topological R k -bundle (the rank of E is k D 11 and the base sphere is S 17 ).The proof relies on a recent result of Burklund and Senger which determines the homotopy 17-spheres bounding 8-connected manifolds, the plumbing approach to the Gromoll filtration due to Antonelli, Burghelea and Kahn, and an explicit construction of low-codimension embeddings of certain homotopy spheres. 57R50, 57S05; 57R60

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Section: Discussionmentioning

confidence: 73%

The derivative map for diffeomorphism of disks: an example

Crowley,

Schick,

Steimle

2023

Geom. Topol.

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The Disc-structure space

Krannich,

Kupers

2024

Forum of Mathematics, Pi

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We study the $\mathscr {D}\mathrm {isc}$ -structure space $S^{\mathscr {D}\mathrm {isc}}_\partial (M)$ of a compact smooth manifold M. Informally speaking, this space measures the difference between M, together with its diffeomorphisms, and the diagram of ordered framed configuration spaces of M with point-forgetting and point-splitting maps between them, together with its derived automorphisms. As the main results, we show that in high dimensions, the $\mathscr {D}\mathrm {isc}$ -structure space a) only depends on the tangential $2$ -type of M, b) is an infinite loop space, and c) is nontrivial as long as M is spin. The proofs involve intermediate results that may be of independent interest, including an enhancement of embedding calculus to the level of bordism categories, results on the behaviour of derived mapping spaces between operads under rationalisation, and an answer to a question of Dwyer and Hess in that we show that the map $\mathrm {BTop}(d)\rightarrow \mathrm {BAut}(E_d)$ is an equivalence if and only if d is at most $2$ .

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On automorphisms of high-dimensional solid tori

Bustamante,

Randal-Williams

2024

Geom. Topol.

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On diffeomorphisms of even-dimensional discs

Abstract: We determine π ∗ ( B Diff ∂ ⁡ ( D 2 n ) ) ⊗ Q \pi _(B\operatorname {Diff}_\partial (D^{2n})) \otimes \mathbb {Q} for 2 n ≥ 6 2n \geq 6 completely in degrees ∗ ≤ 4 n − 10 \leq 4n-10 , far beyond the… Show more

Cited by 3 publications

References 63 publications

The derivative map for diffeomorphism of disks: an example

The derivative map for diffeomorphism of disks: an example

The Disc-structure space

On automorphisms of high-dimensional solid tori

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On diffeomorphisms of even-dimensional discs

Abstract: We determine π ∗ ( B Diff ∂ ⁡ ( D 2 n ) ) ⊗ Q \pi _*(B\operatorname {Diff}_\partial (D^{2n})) \otimes \mathbb {Q} for 2 n ≥ 6 2n \geq 6 completely in degrees ∗ ≤ 4 n − 10 * \leq 4n-10 , far beyond the… Show more

Cited by 3 publications

References 63 publications

The derivative map for diffeomorphism of disks: an example

The derivative map for diffeomorphism of disks: an example

The Disc-structure space

On automorphisms of high-dimensional solid tori

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About

Abstract: We determine π ∗ ( B Diff ∂ ⁡ ( D 2 n ) ) ⊗ Q \pi _(B\operatorname {Diff}_\partial (D^{2n})) \otimes \mathbb {Q} for 2 n ≥ 6 2n \geq 6 completely in degrees ∗ ≤ 4 n − 10 \leq 4n-10 , far beyond the… Show more