Related papers: Diffeomorphism groups of solid tori and the ration…

We obtain explicit formulas for the rational homotopy groups of generalised symmetric spaces, i.e., the homogeneous spaces for which the isotropy subgroup appears as the fixed point group of some finite order automorphism of the group. In…

For a fixed integer $d\geq 1$, we show that two quasitoric manifolds over a product of $d$-simplices are homotopy equivalent after appropriate localization, provided that their integral cohomology rings are isomorphic.

We consider an area-preserving diffeomorphism of a compact surface, which is assumed to be an irrational rotation near each boundary component. A finite set of periodic orbits of the diffeomorphism gives rise to a braid in the mapping…

We calculate the rational representation-ring-graded stable stems for rank 1 groups, SU(2), SO(3), Pin (2), O(2), Spin(2) and SO(2), in the same spirit as the calculations for finite groups in arXiv:2205.02382 with J.D.Quigley. This…

For $X$ a connected finite simplicial complex we consider $\Delta^d(X,n)$ the space of configurations of $n$ ordered points of $X$ such that no $d+1$ of them are equal, and $B^d(X,n)$ the analogous space of configurations of unordered…

We prove that in dimensions not equal to 4, 5, or 7, the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses…

We establish a homotopy-theoretic description of the homology of stable moduli spaces of $(2n+1)$-dimensional manifold triads $(N, \partial^h N, \partial^v N)$ with fixed $\partial^v N$, whenever $n \geq 3$ and $(N, \partial^h N)$ is…

We show that the Bousfield-Kan spectral sequence which computes the rational homotopy groups of the space of long knots in ${\mathbb R}^d$, where $d\ge 4$, collapses at the $E^2$ page. The main ingredients in the proof are Sinha's…

Given a closed smooth manifold $M$ of even dimension $2n\ge6$ with finite fundamental group, we show that the classifying space ${\rm BDiff}(M)$ of the diffeomorphism group of $M$ is of finite type and has finitely generated homotopy groups…

We study closed, connected, spin 4-manifolds up to stabilisation by connected sums with copies of $S^2 \times S^2$. For a fixed fundamental group, there are primary, secondary and tertiary obstructions, which together with the signature…

Let $T= S^1\times D^2$ be the solid torus, $\mathcal{F}$ the Morse-Bott foliation on $T$ into $2$-tori parallel to the boundary and one singular circle $S^1\times 0$, which is the central circle of the torus $T$, and…

We express the rational homotopy type of the mapping spaces $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$ of the little discs operads in terms of graph complexes. Using known facts about the graph homology this allows us to compute…

We completely describe the algebraic part of the rational cohomology of the Torelli groups of the manifolds $\#^g S^n \times S^n$ relative to a disc in a stable range, for $2n \geq 6$. Our calculation is also valid for $2n=2$ assuming that…

We discuss how stability is related to the D-topology of mapping spaces, equipped with the functional diffeology. Indeed, we show that stable classes of mapping spaces are D-open. After a reformulation of the classical stability theorem of…

In our previous works, we constructed diffeomorphisms of compact 4-manifolds $X$ by surgeries on theta-graphs embedded in $X$. In this paper, we consider the case $X=M\times I$, where $M$ is a spherical 3-manifold. For some of such $X$, we…

Characteristic classes of oriented vector bundles can be identified with cohomology classes of the disjoint union of classifying spaces BSO_n of special orthogonal groups SO_n with n=0,1,... A characteristic class is stable if it extends to…

The main theorem shows that if M is an irreducible compact connected orientable 3-manifold with non-empty boundary, then the classifying space BDiff(M rel dM) of the space of diffeomorphisms of M which restrict to the identity map on…

A localisation of the category of n-manifolds is introduced by formally inverting the connected sum construction with a chosen n-manifold Y. On the level of automorphism groups, this leads to the stable diffeomorphism groups of n-manifolds.…

We establish a pseudoisotopy result for embedding spaces in the line of that of Weiss and Williams for diffeomorphism groups. In other words, for $P\subset M$ a codimension at least three embedding, we describe the difference in a range of…

We consider the groups $\operatorname{Diff}_{\mathcal B}(\mathbb R^n)$, $\operatorname{Diff}_{H^\infty}(\mathbb R^n)$, and $\operatorname{Diff}_{\mathcal S}(\mathbb R^n)$ of smooth diffeomorphisms on $\mathbb R^n$ which differ from the…