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We classify, up to homeomorphism, all closed manifolds having the homotopy type of a connected sum of two copies of real projective n-space.

A quasitoric manifold is a $2n$-dimensional compact smooth manifold with a locally standard action of an $n$-dimensional torus whose orbit space is a simple polytope. In this article, we classify quasitoric manifolds with the second Betti…

We show that the homotopy type of a finite oriented Poincar\'{e} 4-complex is determined by its quadratic 2-type provided its fundamental group is finite and has a dihedral Sylow 2-subgroup. By combining with results of Hambleton-Kreck and…

We construct the stable (representable) homotopy category of finite orbispectra, whose objects are formal desuspensions of finite orbi-CW-pairs by vector bundles and whose morphisms are stable homotopy classes of (representable) relative…

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…

The primary algebraic model of a ring spectrum is the ring of homotopy groups. We introduce the secondary model which has the structure of a secondary analogue of a ring. This new algebraic model determines Massey products and cup-one…

To a bicomplex one can associate two natural filtrations, the column and row filtrations, and then two associated spectral sequences. This can be generalized to $N$-multicomplexes. We present a family of model category structures on the…

We give a criterion on a group $\pi$ and a homomorphism $w \colon \pi \to C_2$ under which closed $4$-manifolds with fundamental group $\pi$ and orientation character $w$ are classified up to homotopy equivalence by their quadratic…

We describe of the topology of the geometric quotients of 2n dimensional compact connected symplectic manifolds with n-1 dimensional torus actions. When the isotropy weights at each fixed point are in general position, the quotient is…

Suppose that $N_1$ and $N_2$ are closed smooth manifolds of dimension $n$ that are homeomorphic. We prove that the spaces of smooth knots $Emb(S^1, N_1)$ and $Emb(S^1, N_2)$ have the same homotopy $(2n-7)$-type. In the 4-dimensional case…

We show that for an oriented 4-dimensional Poincar\'e complex with finite fundamental group, whose 2-Sylow subgroup is abelian with at most 2 generators, the homotopy type is determined by its quadratic 2-type.

We define 2-calibrated structures, which are analogs of symplectic structures in odd dimensions. We show the existence of differential topological constructions compatible with the structure.

Closed oriented 4-manifolds with the same geometrically 2-dimensional fundamental group (satisfying certain properties) are classified up to $s$-cobordism by their $w_2$-type, equivariant intersection form and the Kirby-Siebenmann…

In this paper, we prove that certain spherical fibrations over certain CW-complexes are stably fibre homotopy equivalent to $\mm{TOP}$-spherical fibrations (see Definition 1,1). Applying this result, we get a sufficient condition for…

We answer a weaker version of the classification problem for the homotopy types of $(n-2)$-connected closed orientable $(2n-1)$-manifolds. Let $n\geq 6$ be an even integer, and $X$ be a $(n-2)$-connected finite orientable Poincar\'e…

Let $S$ be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class of the space of all curves on $S$ which start and end at given points in given directions and whose curvatures are constrained to lie in a…

Let M be a closed simply connected 2n-dimensional manifold. The present paper is concerned with the cohomology of classifying spaces of connected groups of homeomorphisms of M.

For every relation $R$ between Stiefel-Whitney numbers of closed $(n+1)$-manifolds we consider an associated invariant $\varkappa_R$ of null-cobordant $n$-manifolds with a certain additional structure. For $n=2k-1$ and $R = w_{n+1}+v_k^2$…

A manifold $M^n$ inherits a labeled $n$-dimensional graph $\widetilde{M}[G^L]$ structure consisting of its charts. This structure enables one to characterize fundamental groups of manifolds, classify those of locally compact manifolds with…

In this paper, we prove that a two-dimensional self-shrinker, homeomorphic to the sphere, immersed in the three dimensional Euclidean space is a round sphere, provided its mean curvature and the norm of its position vector have an upper…