Embedding calculus for surfaces

We prove convergence of Goodwillie-Weiss' embedding calculus for spaces of embeddings into a manifold of dimension at most two, so in particular for diffeomorphisms between surfaces. We also relate the Johnson filtration of the mapping class group of a surface to a certain filtration arising from embedding calculus.Comment: 28 pages, v2 major revision: generalised main result, added section on relation to Johnson filtration, changed proofs and expositio

Stability of concordance embeddings

We prove a stability theorem for spaces of smooth concordance embeddings. From it we derive various applications to spaces of concordance diffeomorphisms and homeomorphisms

Two remarks on spaces of maps between operads of little cubes

We record two facts on spaces of derived maps between the operads $E_d$ of little $d$-cubes. Firstly, these mapping spaces are equivalent to the mapping spaces between the non-unitary versions of $E_d$. Secondly, all endomorphisms of $E_d$ are automorphisms. We also discuss variants for localisations of $E_d$ and for versions with tangential structures.Comment: 10 pages; v2: corrected minor oversigh

Finiteness properties of automorphism spaces of manifolds with finite fundamental group

Given a closed smooth manifold M of even dimension 2n≥6 with finite fundamental group, we show that the classifying space BDiff(M) of the diffeomorphism group of M is of finite type and has finitely generated homotopy groups in every degree. We also prove a variant of this result for manifolds with boundary and deduce that the space of smooth embeddings of a compact submanifold N⊂M of arbitrary codimension into M has finitely generated higher homotopy groups based at the inclusion, provided the fundamental group of the complement is finite. As an intermediate result, we show that the group of homotopy classes of simple homotopy self-equivalences of a finite CW complex with finite fundamental group is up to finite kernel commensurable to an arithmetic group

Mapping class groups of highly connected ( 4 k + 2 ) -manifolds

Funder: University of CambridgeAbstract: We compute the mapping class group of the manifolds ♯g(S2k+1×S2k+1) for k>0 in terms of the automorphism group of the middle homology and the group of homotopy (4k+3)-spheres. We furthermore identify its Torelli subgroup, determine the abelianisations, and relate our results to the group of homotopy equivalences of these manifolds