The Haefliger–Thurston conjecture predicts that Haefliger's classifying space for $C^r$-foliations of codimension $n$ whose normal bundles are trivial is $2n$-connected. In this paper, we confirm this conjecture for piecewise linear (PL) foliations of codimension $2$. Using this, we use a version of the Mather–Thurston theorem for PL homeomorphisms due to the author to derive new homological properties for PL surface homeomorphisms. In particular, we answer the question of Epstein in dimension $2$ and prove the simplicity of the identity component of PL surface homeomorphisms.

We determine when the integral homology of the classifying space of a $PU(n)$-gauge group over the sphere $S^2$ has torsion.

We identify the cohomology of the stable classifying space of homotopy automorphisms (relative to an embedded disk) of connected sums of ${\mathrm {S}^{k}} \times {\mathrm {S}^{l}}$, where $3 \le k < l \le 2k - 2$. The result is expressed in terms of Lie graph complex homology.

Let $G$ be a compact connected simple Lie group of type $(n_{1},\,\ldots,\,n_{l})$, where $n_{1}<\cdots < n_{l}$. Let $\mathcal {G}_k$ be the gauge group of the principal $G$-bundle over $S^{4}$ corresponding to $k\in \pi _3(G)\cong \mathbb {Z}$. We calculate the mod-$p$ homology of the classifying space $B\mathcal {G}_k$ provided that $n_{l}< p-1$.

We study moduli spaces of d-dimensional manifolds with embedded particles and discs, which we refer to as decorations. These spaces admit a model in which points are unparametrised d-dimensional manifolds in $\mathbb{R}^\infty$ with particles and discs constrained to it. We compare this to the space of d-dimensional manifolds in $\mathbb{R}^\infty$ with particles and discs that are no longer constrained, i.e. the decorations are decoupled. We show that under certain conditions these spaces cannot be distinguished by homology groups within a range. This generalises work by Bödigheimer–Tillmann for oriented surfaces to different tangential structures and also to higher dimensional manifolds. We also extend this result to moduli spaces with more general submanifolds as decorations and specialise in the case of decorations being embedded circles.

In the early 1980s, Johnson defined a homomorphism $\mathcal {I}_{g}^1\to \bigwedge ^3 H_1\left (S_{g},\mathbb {Z}\right )$ , where $\mathcal {I}_{g}^1$ is the Torelli group of a closed, connected, and oriented surface of genus g with a boundary component and $S_g$ is the corresponding surface without a boundary component. This is known as the Johnson homomorphism.

We study the map induced by the Johnson homomorphism on rational homology groups and apply it to abelian cycles determined by disjoint bounding-pair maps, in order to compute a large quotient of $H_n\left (\mathcal {I}_{g}^1,\mathbb {Q}\right )$ in the stable range. This also implies an analogous result for the stable rational homology of the Torelli group $\mathcal {I}_{g,1}$ of a surface with a marked point instead of a boundary component. Further, we investigate how much of the image of this map is generated by images of such cycles and use this to prove that in the pointed case, they generate a proper subrepresentation of $H_n\left (\mathcal {I}_{g,1}\right )$ for $n\ge 2$ and g large enough.

We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

We consider the Birman–Hilden inclusion $\phi\colon\Br_{2g+1}\to\Gamma_{g,1}$ of the braid group into the mapping class group of an orientable surface with boundary, and prove that $\phi$ is stably trivial in homology with twisted coefficients in the symplectic representation $H_1(\Sigma_{g,1})$ of the mapping class group; this generalises a result of Song and Tillmann regarding homology with constant coefficients. Furthermore we show that the stable homology of the braid group with coefficients in $\phi^*(H_1(\Sigma_{g,1}))$ has only 4-torsion.

We construct a ring homomorphism comparing the tautological ring, fixing a point, of a closed smooth manifold with that of its stabilisation by S2a×S2b.

We construct a Baum–Connes assembly map localised at the unit element of a discrete group $\Gamma$. This morphism, called $\mu _\tau$, is defined in $KK$-theory with coefficients in $\mathbb {R}$ by means of the action of the idempotent $[\tau ]\in KK_{\mathbin {{\mathbb {R}}}}^\Gamma (\mathbb {C},\mathbb {C})$ canonically associated to the group trace of $\Gamma$. We show that the corresponding $\tau$-Baum–Connes conjecture is weaker than the classical version, but still implies the strong Novikov conjecture. The right-hand side of $\mu _\tau$ is functorial with respect to the group $\Gamma$.

Generalising the classical work of Atiyah and Hirzebruch on non-algebraic classes, recently Quick proved the existence of torsion non-algebraic elements in the Brown–Peterson tower. We construct non-torsion non-algebraic elements in the Brown–Peterson tower for the prime number 2.

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 construct a scheme $B(r; {\mathbb {A}}^n)$ such that a map $X \to B(r; {\mathbb {A}}^n)$ corresponds to a degree-n étale algebra on X equipped with r generating global sections. We then show that when $n=2$ , i.e., in the quadratic étale case, the singular cohomology of $B(r; {\mathbb {A}}^n)({\mathbb {R}})$ can be used to reconstruct a famous example of S. Chase and to extend its application to showing that there is a smooth affine $r-1$ -dimensional ${\mathbb {R}}$ -variety on which there are étale algebras ${\mathcal {A}}_n$ of arbitrary degrees n that cannot be generated by fewer than r elements. This shows that in the étale algebra case, a bound established by U. First and Z. Reichstein in [6] is sharp.

We construct a cycle in higher Hochschild homology associated to the two-dimensional torus which represents 2-holonomy of a nonabelian gerbe in the same way as the ordinary holonomy of a principal G-bundle gives rise to a cycle in ordinary Hochschild homology. This is done using the connection 1-form of Baez–Schreiber. A crucial ingredient in our work is the possibility to arrange that in the structure crossed module $\unicode[STIX]{x1D707}:\mathfrak{h}\rightarrow \mathfrak{g}$ of the principal 2-bundle, the Lie algebra $\mathfrak{h}$ is abelian, up to equivalence of crossed modules.

For each of the groups G = O(2), SU(2), U(2), we compute the integral and $\mathbb{F}_2$-cohomology rings of BcomG (the classifying space for commutativity of G), the action of the Steenrod algebra on the mod 2 cohomology, the homotopy type of EcomG (the homotopy fiber of the inclusion BcomG → BG), and some low-dimensional homotopy groups of BcomG.

We study tautological rings for high-dimensional manifolds, that is, for each smooth manifold $M$ the ring $R^{\ast }(M)$ of those characteristic classes of smooth fibre bundles with fibre $M$ which is generated by generalised Miller–Morita–Mumford classes. We completely describe these rings modulo nilpotent elements, when $M$ is a connected sum of copies of $S^{n}\times S^{n}$ for $n$ odd.

B. Schuster [19] proved that the mod 2 Morava K-theory K(s)*(BG) is evenly generated for all groups G of order 32. For the four groups G of order 32 with the numbers 38, 39, 40 and 41 in the Hall-Senior list [11], the ring K(2)*(BG) has been shown to be generated as a K(2)*-module by transferred Euler classes. In this paper, we show this for arbitrary s and compute the ring structure of K(s)*(BG). Namely, we show that K(s)*(BG) is the quotient of a polynomial ring in 6 variables over K(s)*(pt) by an ideal for which we list explicit generators.

The integral cohomology rings of the configuration spaces of $n$-tuples of distinct points on arbitrary surfaces (not necessarily orientable, not necessarily compact and possibly with boundary) are studied. It is shown that for punctured surfaces the cohomology rings stabilize as the number of points tends to infinity, similarly to the case of configuration spaces on the plane studied by Arnold, and the Goryunov splitting formula relating the cohomology groups of configuration spaces on the plane and punctured plane to arbitrary punctured surfaces is generalized. Moreover, on the basis of explicit cellular decompositions generalizing the construction of Fuchs and Vainshtein, the first cohomology groups for surfaces of low genus are given.

This paper is concerned with mod p Morita-Mumford classes of the mapping class group Γg of a closed oriented surface of genus g ≥ 2, especially triviality and nontriviality of them. It is proved that is nilpotent if n ≡ − 1 (mod p − 1), while the stable mod p Morita-Mumford class is proved to be nontrivial and not nilpotent if n ≢ −1 (mod p − 1). With these results in mind, we conjecture that vanishes whenever n ≡ − 1 (mod p − 1), and obtain a few pieces of supporting evidence.

Let K be a field of characteristic different from two. Let L be a finite separable extension of K. If is the separable closure of K, we have a continuous homomorphism π : Ga(/K) → ∑n(n - [L : K]). We give a very short proof of Serre's formula which evaluates the Hasse-Witt invariant of a symmetric bilinear form, transferred from L, in terms of the topological Stiefel-Whitney classes of IT.