Diffeomorphisms of discs and the second Weiss derivative of BTop(-)

Abstract

A. We compute the rational homotopy groups in degrees up to approximately 3 2 𝑑 of the group of di eomorphisms of a closed 𝑑-dimensional disc xing the boundary. Based on this we determine the optimal rational concordance stable range for high-dimensional discs, describe the rational homotopy type of 𝐵Top(𝑑) in a range, and calculate the second rational derivative of the functor 𝐵Top(−) in the sense of Weiss' orthogonal calculus. C 1. Introduction 1 2. Automorphisms, self-embeddings, and mapping class groups 9 3. Homotopy automorphisms and mapping spaces 18 4. The bre to homotopy automorphisms 27 5. Block embeddings of a point 34 6. Claspers 41 7. Homology of self-embedding spaces 49 8. Rational homotopy groups of di eomorphism groups of discs 59 9. The rational homotopy type of 𝐵STop(𝑑) and orthogonal calculus 65 Appendix A. Some cohomology of 𝐺 𝑉 73 Appendix B. Relation to the work of Watanabe 75 Glossary of notation 81 References 81

Corrigendum: On Kontsevich's characteristic classes for higher‐dimensional sphere bundles II: Higher classes

“…Remark 4. Lemma A also follows from[4, Proposition 6.4] by Krannich and Randal-Williams, proved by different method. Their class 𝛼 ∈ 𝜋 𝑘−1 ( BDif f (𝑉, 𝜕)) ⊗ ℚ is symmetric in the three handles by definition, whereas it is not obvious for our 𝛽.…”

Some rational homology computations for diffeomorphisms of odd-dimensional manifolds

“…holds in a range of degrees. This range which is nowadays known to be roughly 2n, by recent work of Krannich [43] and Krannich-Randal-Williams [42]. An instance of Morlet's lemma of disjunction states that the homotopy fibre of…”

“…In loc.cit., Borel showed that M (SL g (R), (k, l)) ≥ g − 2, but left C(SL g (R), (k, l)) implicit. A relatively naive counting argument given in the proof of [42,Theorem 7.3] shows that C(SL g (R), (k, l)) ≥ 1 8 g 2 − max(k, l)− 1, which implies that (6.18) is an isomorphism when p ≤ f k,l (g), and f k,l is a function with lim g→∞ f k,l (g) = ∞.…”

The Disc-structure space

“…As explained in the outline in Section 1.2.5, the main ingredient besides Theorem 7.24 is work of Boavida de Brito-Weiss' [BdBW18] and work of Fresse-Turchin-Willwacher [FTW17]. We also make use of results of Krannich, Kupers, Randal-Williams, and Watanabe [KrRW21,KuRW21,Wat09], though this can be avoided in most cases (see Remark 8.14).…”

“…For 𝑑 ≠ 4, Corollary 8.8 also follows by combining[Mil09, Lemma 10, p. 188] with [KS77, Essay V. §5.0(1)]. The advantage of the proof above is that it applies to 𝑑 = 4.The second result on BTop(𝑑) we will use follows from works of Krannich, Kupers, Randal-Williams, and Watanabe[KrRW21,KuRW21,Wat09]. It concerns two commutative squaresBO(2𝑛) 𝐾 (Q, 2𝑛) × BO BO(2𝑛 + 1) 𝐾 (Q, 4𝑛) × BO BTop(2𝑛) 𝐾 (Q, 2𝑛) × BTop BTop(2𝑛 + 1) 𝐾 (Q, 4𝑛)arrows are induced by the inclusion O(𝑑) ⊂ Top(𝑑) and the horizontal arrows by the stabilisation map, the Euler class 𝑒 ∈ H 2𝑛 (BTop(2𝑛); Q), and the odd-dimensional analogue of its square 𝐸 ∈ H 2𝑛+1 (BTop(2𝑛 + 1); Q) (see [KrRW21, Sections 1.2.2 and 8.1.1] for further information on this class).…”