Related papers: Mapping class groups of highly connected $(4k+2)$-…

For every integer $k\geq 2$, we construct infinite families of mutually nondiffeomorphic irreducible smooth structures on the topological $4$-manifolds $(2k-1)(S^2\times S^2)$ and $(2k-1)(\CP#\CPb)$, the connected sums of $2k-1$ copies of…

Let P be a connected smooth p-manifold. We describe the group of all cobordism classes of smooth maps of n-manifolds to P with singularities of a given $cal K$-invariant class in terms of certain stable homotopy groups by applying the…

The Torelli group $\mathcal T(X)$ of a closed smooth manifold $X$ is the subgroup of the mapping class group $\pi_0(\mathrm{Diff}^+(X))$ consisting of elements which act trivially on the integral cohomology of $X$. In this note we give…

For a connected based space $X$, let $[X,X]$ be the set of all based homotopy classes of base point preserving self map of $X$ and let $\E(X)$ be the group of self-homotopy equivalences of $X$. We denote by $\A_{\sharp}^k(X)$ the set of…

To every $Q$-irreducible representation $r$ of a finite group $H$, there corresponds a simple factor $A$ of $Q[H]$ with an involution $\tau$. To this pair $(A,\tau)$, we associate an arithmetic group $\Omega$ consisting of all $(2g-2)\times…

The mapping class group of a Heegaard splitting is the group of automorphisms of the ambient 3-manifold that take the surface onto itself, modulo isotopies that keep the surface on itself. We characterize the mapping classes that restrict…

We develop the theory of equivariant harmonic self-maps of compact cohomogeneity one manifolds and construct new harmonic self-maps of the compact Lie groups SO(4L+2), L >= 1, with degree -3, of SO(8), SO(14) and SO(26) with degree -5 each,…

For a smooth manifold $N$ denote by $E^m(N)$ the set of smooth isotopy classes of smooth embeddings $N\to\mathbb R^m$. A description of the set $E^m(S^p\times S^q)$ was known only for $p=q=0$ or for $p=0$, $m\ne q+2$ or for $2m\ge…

We prove several positive results regarding representation of homotopy classes of spheres and algebraic groups by regular mappings. Most importantly we show that every mapping from a sphere to an orthogonal or a unitary group is homotopic…

We describe a procedure to construct infinite sets of pairwise smoothly inequivalent 2-spheres in simply connected 4-manifolds, which are topologically isotopic and whose complement has a prescribed fundamental group that satisfies some…

In this paper we give a formula for the homotopy groups of $(n-1)$-connected $2n$-manifolds as a direct sum of homotopy groups of spheres in the case the $n^{th}$ Betti number is larger than $1$. We demonstrate that when the $n^{th}$ Betti…

These are the lecture notes for my course at the 2011 Park City Mathematics Graduate Summer School. The first two lectures covered the basics of the Torelli group and the Johnson homomorphism, and the third and fourth lectures discussed the…

In this article we show how to calculate the group of automorphisms of flat K\"ahler manifolds. Moreover we are interested in the problem of classification of such manifolds up to biholomorphism. We consider these problems from two points…

We show that the set of harmonic maps from the 2-dimensional stratified spheres with uniformly bounded energies contains only finitely many homotopy classes. We apply this result to construct infinitely many harmonic map flows and mean…

We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play…

In this paper we compute the homotopy groups of the symplectomorphism groups of the 3-, 4- and 5-point blow-ups of the projective plane (considered as monotone symplectic Del Pezzo surfaces). Along the way, we need to compute the homotopy…

For M_r = #_r(S^p \times S^p), p=3, 7, we calculate the group of isotopy classes of orientation preserving diffeomorphisms of $M_r$ modulo isotopy classes with representatives which are the identity outside a 2p-disc and also the group of…

For any natural numbers $k \leq n$, the rational cohomology ring of the space of continuous maps $S^{2k-1} \to S^{2n-1}$ (respectively, $S^{4k-1} \to S^{4n-1}$) equivariant under the Hopf action of the circle (respectively, of the group…

We show that mapping class groups associated to all types of real algebraic curves are virtual duality groups. We also deduce some results about the orbifold homotopy groups of the moduli spaces of real algebraic curves. We achieve these…

We study the Picard groups of moduli spaces of smooth complex projective curves that have a group of automorphisms with a prescribed topological action. One of our main tools is the theory of symmetric mapping class groups. In the first…