Related papers: Smooth structures on $\mathbb{C}P^{m}$ for $5\leq …

Let M be a closed simply connected n-manifold of positive sectional curvature. We determine its homeomorphism or homotopic type if M also admits an isometric elementary p-group action of large rank. Our main results are: There exists a…

It is well known that isoperimetric regions in a smooth compact $(n+1)$-manifold are smooth, up to a closed set of codimension at most $6$. In this note, we first construct an $8$-dimensional compact smooth manifold whose unique…

We study smooth proper embeddings of compact orientable surfaces in compact orientable $4$-manifolds and elements in the mapping class group of that surface which are induced by diffeomorphisms of the ambient $4$-manifolds. We call such…

We study closed orientable manifolds whose topological complexity is at most 3 and determine their cohomology rings. For some of admissible cohomology rings we are also able to identify corresponding manifolds up to homeomorphism.

We consider the general problem of constructing the structure of a smooth manifold on a given space of loops in a smooth finite dimensional manifold. By generalising the standard construction for smooth loops, we derive a list of conditions…

For each n greater than 7 we explicitly construct a sequence of Stein manifolds diffeomorphic to complex affine space of dimension n so that there is no algorithm to tell us in general whether a given such Stein manifold is symplectomorphic…

We give a complete diffeomorphism classification of 1-connected manifolds (of dimension different from 4) whose integral homology is H(M)=Z+Z+Z.

We construct stable minimal hypersurfaces with simple topology in certain compact $4$-manifolds $X$ with boundary, where $X$ embeds into a smooth manifold homeomorphic to $S^4$. For example, if $X$ is equipped with a Riemannian metric $g$…

We construct the first examples of non-smoothable self-homeomorphisms of smooth $4$-manifolds with boundary that fix the boundary and act trivially on homology. As a corollary, we construct self-diffeomorphisms of $4$-manifolds with…

Every smooth homotopy 4-sphere is diffeomorphic to the 4-sphere.

In this paper, we study the existence of high-dimensional, closed, smooth manifolds whose rational homotopy type resembles that of a projective plane. Applying rational surgery, the problem can be reduced to finding possible Pontryagin…

We introduce Smale A-homeomorphisms that includes regular, semi-chaotic, chaotic, and super chaotic homeomorphisms of topo\-lo\-gi\-cal $n$-manifold $M^n$, $n\geq 2$. Smale A-homeo\-mor\-p\-hisms contain A-diffeomorphisms (in particular,…

The Arnold conjecture states that a Hamiltonian diffeomorphism of a closed and connected symplectic manifold must have at least as many fixed points as the minimal number of critical points of a smooth function on the manifold. It is well…

We construct smooth 4-manifolds homeomorphic but not diffeomorphic to CP^2+6CP^2-bar.

We consider closed topological 4-manifolds $M$ with universal cover ${S^2\times{S^2}}$ and Euler characteristic $\chi(M) = 1$. All such manifolds with $\pi=\pi_1(M)\cong {\mathbb Z}/4$ are homotopy equivalent. In this case, we show that…

One approach to produce a pair of homeomorphic-but-not-diffeomophic closed 4-manifolds is to find a knot which is smoothly slice in one but not the other. This approach has never been run successfully. We give the first examples of a pair…

For a closed smooth manifold $M$ admitting a symplectic structure, we define a smooth topological invariant $Z(M)$ using almost-K\"ahler metrics, i.e. Riemannian metrics compatible with symplectic structures. We also introduce $Z(M,…

We show that any eight-dimensional oriented manifold $M$ possessing smooth circle action with exactly three fixed points has the same weight system as some circle action on $\mathbb HP^2$. It follows that Pontryagin numbers and equivariant…

Given a closed, smooth, connected, orientable $4$-manifold $M$, whose integral homology groups can have $2$-torsion, we determine the homotopy decomposition of the double suspension $\Sigma^2M$ as wedge sums of some elementary…

In 2006 Masuda and Suh asked if two compact non-singular toric varieties having isomorphic cohomology rings are homeomorphic. In the first part of this paper we discuss this question for topological generalizations of toric varieties,…