Related papers: Manifold structures on highly connected Poincar\'e…

We establish a loop space decomposition for certain $CW$-complexes with a single top cell in the presence of a spherical pair, thereby generalizing several known decompositions of Poincar\'{e} duality complexes in which a loop of a product…

It is a well-known result of C.T.C. Wall's that one may decompose a simply connected 6-manifold as a connected sum of two simpler manifolds. Recent work of Beben and Theriault on decomposing based loop spaces of highly connected Poincar\'e…

Holomorphic vector bundles on $\mathbb C\times M$, $M$ a complex manifold, with meromorphic connections with poles of Poincar\'e rank 1 along $\{0\}\times M$ arise naturally in algebraic geometry. They are called $(TE)$-structures here.…

We prove that, if M is a compact oriented manifold of dimension 4k+3, where k>0, such that pi_1(M) is not torsion-free, then there are infinitely many manifolds that are homotopic equivalent to M but not homeomorphic to it. To show the…

It is proved that there exists an integer $L$ such that a framed manifold of dimension $2^l-2$, $l\le L$ has the trivial Kervaire Invariant.

We define the Bianchi-Massey tensor of a topological space X to be a linear map from a subquotient of the fourth tensor power of H*(X). We then prove that if M is a closed (n-1)-connected manifold of dimension at most 5n-3 (and n > 1) then…

Let p be a fibration over a finite simplicial complex, whose fibers have the homotopy type of finite simplicial complexes. Then p is equivalent to an approximate fibration whose total space is a compact ENR. The proof uses homotopy coherent…

We compute the topological mapping class group of every compact, simply connected, topological 4-manifold. This was previously only known in the closed case. If the 4-manifold is smooth, we deduce an analogous description of the stable…

Motivated by conjectures in Heegaard Floer homology, we introduce an invariant HC(Y) of the cohomology ring of a closed 3-manifold Y whose behavior mimics that of the Heegaard Floer homology HF^\infty(Y,s) for s a torsion spin-c structure.…

We find geometric conditions on a four-dimensional almost Hermitian manifold under which the almost complex structure is a harmonic map or a minimal isometric imbedding of the manifold into its twistor space.

We show that the homotopy type of a 4-manifold $M$ whose fundamental group is a finitely presentable $PD_3$-group $\pi$ and with $w_1(M)=w_1(\pi)$ is determined by $\pi$, $\pi_2(M)$, $k_1(M)$ and the equivariant intersection pairing…

Manifolds endowed with three foliations pairwise transversal are known as 3-webs. Equivalently, they can be algebraically defined as biparacomplex or complex product manifolds, i.e., manifolds endowed with three tensor fields of type…

A smooth map between manifolds is said to be \emph{image simple} if its restriction to its singular point set is a topological embedding. We study the parity of the number of connected components of the singular point set for image simple…

We establish various criteria for the inertness of the top cell attachments of Poincar\'{e} duality complexes through nonzero degree maps, algebraic intersection theory and various types of homotopy fibrations. Many examples are provided,…

We show that all homotopy $\mathbb{C}P^n$s, smooth closed manifolds with the oriented homotopy type of $\mathbb{C}P^n$, admit almost complex structures for $3 \leq n \leq 6$, and classify these structures by their Chern classes. Our methods…

We discuss conditions for the integrability of an almost complex structure defined on the total space of an induced Hopf S^3-bundle over a Sasakian manifold . As an application, we obtain an uncountable family of inequivalent complex…

The fine curve complex of a surface is a simplicial complex whose vertices are essential simple closed curves and whose $k$-simplices are collections of $k+1$ disjoint curves. We prove that the fine curve complex is homotopy equivalent to…

For each composite number $n\ne 2^k$, there does not exist a single connected closed $(n+1)$-manifold such that any smooth, simply-connected, closed $n$-manifold can be topologically flat embedded into it. There is a single connected closed…

We identify the space of left-invariant oriented complex structures on the complex Heisenberg group, and prove that it has the homotopy type of the disjoint union of a point and a 2-sphere.

We construct closed $(k-1)$-connected manifolds of dimensions $\ge 4k-1$ that possess non-trivial rational Massey triple products. We also construct examples of manifolds $M$ such that all the cup-products of elements of $H^k(M)$ vanish,…