Related papers: The mapping class group of connect sums of $S^2 \t…

We give alternative computations of the Schur multiplier of $Sp(2g,\mathbb Z/D\mathbb Z)$, when $D$ is divisible by 4 and $g\geq 4$: a first one using $K$-theory arguments based on the work of Barge and Lannes and a second one based on the…

This note has two related but independent parts. Firstly, we prove a generalisation of a recent result of Gay on the smooth mapping class group of $S^4$. Secondly, we give an alternative proof of a consequence of work of Saeki, namely that…

We record various properties of twisted Becker-Gottlieb transfer maps and study their multiplicative properties analogous to Becker-Gottlieb transfer. We show these twisted transfer maps factorise through Becker-Schultz-Mann-Miller-Miller…

For all $n \geq 1$, there is a notion of $n$-smooth group scheme over any $\mathbb{F}_p$-algebra $R$, which may be thought of as a ``Frobenius analogue" of $n$-truncated Barsotti-Tate groups over $R$. We show that the category of $n$-smooth…

In a seminal paper, Brin demonstrates that the outerautomorphism group of Thompson group $T$ is isomorphic to the cyclic group of order two. In this article, building on characterisation of automorphisms of the Higman-Thompson groups…

Wajnryb proved that the mapping class group of an orientable surface is generated by two elements. We prove that one of these generators can be taken as a Dehn twist. We also prove that the extended mapping class group is generated by two…

The extended mapping class group of a surface $\Sigma$ is defined to be the group of isotopy classes of (not necessarily orientation-preserving) homeomorphisms of $\Sigma$. We are able to show that the extended mapping class group of an…

The primary interest of this paper is to discuss the role of twisting cochains in the theory of characteristic classes. We begin with the homological description of monodromy map, associated with a connection on a trivial bundle over a…

For any $N \geq 1$, let $M_N$ denote the rational $4$-manifold $\mathbb{CP}^2 \# N \overline{\mathbb{CP}^2}$. In this paper we study the stabilizer $\text{Stab}(w)$ of a primitive, isotropic class $w\in H_2(M_N; \mathbb Z)$ of minimal genus…

In this paper we show that the normal closure of the mth power of a half-twist has infinite index in the mapping class group of a punctured sphere. Furthermore, in some cases we prove that the quotient of the mapping class group of the…

In this paper, we examine mapping class group relations of some symplectic manifolds. For each $n\geq 1$ and $k \geq 1$, we show that the $2n$-dimensional Weinstein domain $W = \{f=\delta\} \cap B^{2n+2}$, determined by the degree $k$…

As shown by H. Gluck in 1962, the diffeotopy group of S^1 \times S^2 is isomorphic to Z_2 + Z_2 + Z_2. Here an alternative proof of this result is given, relying on contact topology. We then discuss two applications to contact topology: (i)…

We consider finite-sheeted, regular, possibly branched covering spaces of compact surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides…

Over an algebraically closed base field $k$ of characteristic 2, the ring $R^G$ of invariants is studied, $G$ being the orthogonal group O(n) or the special orthogonal group SO(n) and acting naturally on the coordinate ring $R$ of the…

A well known quotient of the real Stiefel manifold is the projective Stiefel manifold. We introduce a new family of quotients of the real Stiefel manifold by cyclic group of order 2 whose action is induced by simultaneous pairwise flipping…

In this note we prove that the mapping class group of a compact topological manifold $M$ with boundary is of finite type, under assumptions on its dimension and connectivity.

We present a rigidity theorem for the action of the mapping class group $\pi_0(\mathrm{Diff}(M))$ on the space $\mathcal{R}^+(M)$ of metrics of positive scalar curvature for high dimensional manifolds $M$. This result is applicable to a…

We construct infinite rank summands isomorphic to $\mathbb{Z}^\infty$ in the higher homotopy and homology groups of the diffeomorphism groups of certain $4$-manifolds. These spherical families become trivial in the homotopy and homology…

For each integer q>0 there is a cohomology theory such that the zero cohomology group of a manifold N of dimension n is a certain group of cobordism classes of proper fold maps of manifolds of dimension n+q into N. We prove a splitting…

We construct smooth bundles with base and fiber products of two spheres whose total spaces have non-vanishing $\hat{A}$-genus. We then use these bundles to locate non-trivial rational homotopy groups of spaces of Riemannian metrics with…