A. Given a graded E 1 -module over an E 2 -algebra in spaces, we construct an augmented semisimplicial space up to higher coherent homotopy over it, called its canonical resolution, whose graded connectivity yields homological stability for the graded pieces of the module with respect to constant and abelian coe cients. We furthermore introduce a notion of coe cient systems of nite degree in this context and show that, without further assumptions, the corresponding twisted homology groups stabilise as well.is generalises a framework of Randal-Williams and Wahl for families of discrete groups.In many examples, the canonical resolution recovers geometric resolutions with known connectivity bounds. As a consequence, we derive new twisted homological stability results for e.g. moduli spaces of high-dimensional manifolds, unordered con guration spaces of manifolds with labels in a bration, and moduli spaces of manifolds equipped with unordered embedded discs. is in turn implies representation stability for the ordered variants of the la er examples.is said to satisfy homological stability if the induced maps in homology are isomorphisms in degrees that are small relative to n. ere is a well-established strategy for proving homological stability that traces back to an argument by illen for the classifying spaces of a sequence of inclusions of groups G n . Given simplicial complexes whose connectivity increases with n and on which the groups G n act simplicially, transitively on simplices, and with stabilisers isomorphic to groups G n−k prior in the sequence, stability can o en be derived by employing a spectral sequence relating the di erent stabilisers. In [RW ], Randal-Williams and Wahl axiomatised this strategy of proof, resulting in a convenient categorical framework for proving homological stability for families of discrete groups that form a braided monoidal groupoid. eir work uni es and improves many classical stability results and has led to a number of applications since its introduction [Fri ; GW ; PW ; Ran ; SW ]. However, homological stability phenomena have been proved to occur not only in the context of discrete groups, but also in numerous non-aspherical situations, many of them of a moduli space avour, such as unordered con guration spaces of manifolds [McD ; Seg ; Seg ], the most classical example, or moduli spaces of high-dimensional manifolds [GR ; GR ] to emphasise a more recent one. e majority of the stability proofs in this context resemble the original line of argument for discrete groups, and one of the objectives of the present work is to provide a conceptualisation of this pa ern.Instead of considering the single spaces M n and the maps M n → M n+1 between them one at a time, it is bene cial to treat them as a single space M = n ≥0 M n together with a grading M : M → N 0 to the nonnegative integers, capturing the decomposition of M into the pieces M n , and a stabilisation map s : M → M that restricts to the maps M n → M n+1 , so it increases the degree by one. From the perspective of homotopy...

We construct a zig–zag from the once delooped space of pseudoisotopies of a closed 2n-disc to the once looped algebraic K-theory space of the integers and show that the maps involved are p-locally $$(2n-4)$$ ( 2 n - 4 ) -connected for $$n\,{>}\,3$$ n > 3 and large primes p. The proof uses the computation of the stable homology of the moduli space of high-dimensional handlebodies due to Botvinnik–Perlmutter and is independent of the classical approach to pseudoisotopy theory based on Igusa’s stability theorem and work of Waldhausen. Combined with a result of Randal-Williams, one consequence of this identification is a calculation of the rational homotopy groups of $$\mathrm {BDiff}_\partial (D^{2n+1})$$ BDiff ∂ ( D 2 n + 1 ) in degrees up to $$2n-5$$ 2 n - 5 .

We compute the mapping class group of the manifolds $$\sharp ^g(S^{2k+1}\times S^{2k+1})$$ ♯ g ( S 2 k + 1 × S 2 k + 1 ) for $$k>0$$ k > 0 in terms of the automorphism group of the middle homology and the group of homotopy $$(4k+3)$$ ( 4 k + 3 ) -spheres. We furthermore identify its Torelli subgroup, determine the abelianisations, and relate our results to the group of homotopy equivalences of these manifolds.

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

It is well known that Sullivan showed that the mapping class group of a simply connected highdimensional manifold is commensurable with an arithmetic group, but the meaning of "commensurable" in this statement seems to be less well known. We explain why this result fails with the now standard definition of commensurability by exhibiting a manifold whose mapping class group is not residually finite. We do not suggest any problem with Sullivan's result: rather we provide a gloss for it. Résumé. Il est notoire que Sullivan a démontré que le groupe de difféotopie d'une variété de haute dimension simplement connexe est commensurable avec un groupe arithmétique, mais la signification du terme « commensurable » dans son théorème semble bien moins connue. Nous expliquons la raison pour laquelle ce résultat n'est plus vrai en utilisant la définition désomais standard de commensurabilité en exhibant une variété dont le groupe de difféotopie n'est pas résiduellement fini. Il n'est pas question d'un problème avec le théorème de Sullivan, mais plutôt d'y ajouter une glose.