For a simply-connected closed manifold X of $\dim X \neq 4$, the mapping class group $\pi _0(\mathrm {Diff}(X))$ is known to be finitely generated. We prove that analogous finite generation fails in dimension 4. Namely, we show that there exist simply-connected closed smooth 4-manifolds whose mapping class groups are not finitely generated. More generally, for each $k>0$, we prove that there are simply-connected closed smooth 4-manifolds X for which $H_k(B\mathrm {Diff}(X);\mathbb {Z})$ are not finitely generated. The infinitely generated subgroup of $H_k(B\mathrm {Diff}(X);\mathbb {Z})$ which we detect are topologically trivial, and unstable under the connected sum of $S^2 \times S^2$. These results are proven by constructing and computing an infinite family of characteristic classes using Seiberg–Witten theory.

Let G be a compact connected Lie group, and let $\operatorname {Hom}({\mathbb {Z}}^m,G)$ be the space of pairwise commuting m-tuples in G. We study the problem of which primes $p \operatorname {Hom}({\mathbb {Z}}^m,G)_1$, the connected component of $\operatorname {Hom}({\mathbb {Z}}^m,G)$ containing the element $(1,\ldots ,1)$, has p-torsion in homology. We will prove that $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ for $m\ge 2$ has p-torsion in homology if and only if p divides the order of the Weyl group of G for $G=SU(n)$ and some exceptional groups. We will also compute the top homology of $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ and show that $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ always has 2-torsion in homology whenever G is simply-connected and simple. Our computation is based on a new homotopy decomposition of $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$, which is of independent interest and enables us to connect torsion in homology to the combinatorics of the Weyl group.

We prove two theorems about the Malcev Lie algebra associated to the Torelli group of a surface of genus g: Stably, it is Koszul and the kernel of the Johnson homomorphism consists only of trivial $\mathrm {Sp}_{2g}(\mathbb {Z})$ -representations lying in the centre.

We study generators of the fundamental group of the group of symplectomorphisms $\operatorname {\mathrm{Symp}} (\mathbb C\mathbb P^2\#\,5\overline { \mathbb C\mathbb P}\,\!^2, \omega )$ for some particular symplectic forms. It was observed by Kȩdra (2009, Archivum Mathematicum 45) that there are many symplectic 4-manifolds $(M, \omega )$, where M is neither rational nor ruled, that admit no circle action and $\pi _1 (\operatorname {\mathrm {Ham}} (M,\omega ))$ is nontrivial. On the other hand, it follows from Abreu and McDuff (2000, Journal of the American Mathematical Society 13, 971–1009), Anjos and Eden (2019, Michigan Mathematical Journal 68, 71–126), Anjos and Pinsonnault (2013, Mathematische Zeitschrift 275, 245–292), and Pinsonnault (2008, Compositio Mathematica 144, 787–810) that the fundamental group of the group $ \operatorname {\mathrm{Symp}}_h(\mathbb C\mathbb P^2\#\,k\overline { \mathbb C\mathbb P}\,\!^2,\omega )$, of symplectomorphisms that act trivially on homology, with $k \leq 4$, is generated by circle actions on the manifold. We show that, for some particular symplectic forms $\omega $, the set of all Hamiltonian circle actions generates a proper subgroup in $\pi _1(\operatorname {\mathrm{Symp}}_{h}(\mathbb C\mathbb P^2\#\,5\overline { \mathbb C\mathbb P}\,\!^2,\omega )).$ Our work depends on Delzant classification of toric symplectic manifolds, Karshon’s classification of Hamiltonian $S^1$-spaces, and the computation of Seidel elements of some circle actions.

The homotopy theory of gauge groups has received considerable attention in recent decades. In this work, we study the homotopy theory of gauge groups over some high-dimensional manifolds. To be more specific, we study gauge groups of bundles over (n − 1)-connected closed 2n-manifolds, the classification of which was determined by Wall and Freedman in the combinatorial category. We also investigate the gauge groups of the total manifolds of sphere bundles based on the classical work of James and Whitehead. Furthermore, other types of 2n-manifolds are also considered. In all the cases, we show various homotopy decompositions of gauge groups. The methods are combinations of manifold topology and various techniques in homotopy theory.

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 $2n\geqslant 6$. Our calculation is also valid for $2n=2$ assuming that the rational cohomology groups of these Torelli groups are finite-dimensional in a stable range.

We study quotients of mapping class groups ${\Gamma _{g,1}}$ of oriented surfaces with one boundary component by the subgroups ${{\cal I}_{g,1}}(k)$ in the Johnson filtrations, and we show that the stable classifying spaces ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(k))^ + }$ after plus-construction are infinite loop spaces, fitting into a tower of infinite loop space maps that interpolates between the infinite loop spaces ${\mathbb {Z}} \times B\Gamma _\infty ^ + $ and ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(1))^ + } \simeq {\mathbb {Z}} \times B{\rm{Sp}}{({\mathbb {Z}})^ + }$ . We also show that for each level k of the Johnson filtration, the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity.

The group of ${\mathcal{C}}^{1}$ -diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson’s groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin’s higher-dimensional generalizations $nV$ of Thompson’s group $V$ arise when we consider products of central ternary Cantor sets. We derive that the ${\mathcal{C}}^{2}$ -smooth mapping class group of a sparse Cantor sphere pair is a discrete countable group and produce this way versions of the braided Thompson groups.

Let Mμ0 denote S2×S2 endowed with a split symplectic form normalized so that μ≥1 and σ(S2)=1. Given a symplectic embedding of the standard ball of capacity c∈(0,1) into Mμ0, consider the corresponding symplectic blow-up . In this paper, we study the homotopy type of the symplectomorphism group and that of the space of unparametrized symplectic embeddings of Bc into Mμ0. Writing ℓ for the largest integer strictly smaller than μ, and λ∈(0,1] for the difference μ−ℓ, we show that the symplectomorphism group of a blow-up of ‘small’ capacity c<λ is homotopically equivalent to the stabilizer of a point in Symp(Mμ0), while that of a blow-up of ‘large’ capacity c≥λ is homotopically equivalent to the stabilizer of a point in the symplectomorphism group of a non-trivial bundle obtained by blowing down . It follows that, for c<λ, the space is homotopy equivalent to S2 ×S2, while, for c≥λ, it is not homotopy equivalent to any finite CW-complex. A similar result holds for symplectic ruled manifolds diffeomorphic to . By contrast, we show that the embedding spaces and , if non-empty, are always homotopy equivalent to the spaces of ordered configurations and . Our method relies on the theory of pseudo-holomorphic curves in 4 -manifolds, on the computation of Gromov invariants in rational 4 -manifolds, and on the inflation technique of Lalonde and McDuff.

It is shown that the group of compactly supported, measure-preserving homeomorphisms of a connected, second countable manifold is locally contractible in the direct limit topology. Furthermore, this group is weakly homotopically equivalent to the more general group of compactly supported homeomorphisms.

In this paper we primarily consider two natural subgroups of the autohomeomorphism group of the real line $\mathbb{R}$, endowed with the compact-open topology. First, we prove that the subgroup of homeomorphisms that map the set of rational numbers $\mathbb{Q}$ onto itself is homeomorphic to the infinite power of $\mathbb{Q}$ with the product topology. Secondly, the group consisting of homeomorphisms that map the pseudoboundary onto itself is shown to be homeomorphic to the hyperspace of nonempty compact subsets of $\mathbb{Q}$ with the Vietoris topology. We obtain similar results for the Cantor set but we also prove that these results do not extend to ${{\mathbb{R}}^{n}}$ for $n\ge 2$, by linking the groups in question with Erdős space.

It is known that the group of homeomorphisms of a non-compact, connected, metrizable manifold acts transitively on certain spaces of measures defined, among other properties, by their behaviour at the manifold's set of ‘ends’. As a generalization of previous work of Fathi in the realm of compact manifolds, it is shown that the given action admits a continuous section when restricted to ‘biregular’ homeomorphisms and measures. This result exhibits the isotropy groups of measure-preserving homeomorphisms as deformation retracts of the corresponding acting groups.