arXiv research

Classifies 1-connected 8-manifolds with specific homology.

Study classifies certain 7-manifolds with specific cohomology and proves positive Ricci curvature.

Classifies 1-connected Lorentzian manifolds with essential conformal groups.

In this note we classify the diffeomorphism classes rel. boundary of smooth h-cobordisms between two fixed 1-connected 4-manifolds in terms of isometries between the intersection forms.

Study flat GL(1|1) connections using fatgraphs and coordinates.

The paper shows a 3D shape can't fit into certain compact spaces.

The paper identifies manifolds with free circle actions.

Trivalent $2$-stratifolds are a generalization of $2$-manifolds in that there are disjoint simple closed curves where three sheets meet. We obtain a classification of $1$-connected $2$-stratifolds in terms of their associated labeled graphs and develop operations that will construct from a single vertex all graphs that…

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 its rational homotopy type is determined by its cohomology algebra and Bianchi-M…

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, while the group $H^{3k-1}(M;\Q)$ is generated by Massey products: such examples ar…

Our goal in this paper is to compute the integral free loop space homology of $(n-1)$-connected $2n$-manifolds $M$, $n\geq 2$. We do this when $n\neq 2,4,8$, or when $n\neq 2$ and $\tilde H^*(M)$ has trivial cup product squares, though the techniques used here should extend to a much wider range of manifolds. We also g…

We consider the moduli space of flat $SO(2n+1)$-connections (up to gauge transformations) on a Riemann surface, with fixed holonomy around a marked point. There are natural line bundles over this moduli space; we construct geometric representatives for the Chern classes of these line bundles, and prove that the ring ge…

Study harmonic measures and rigidity in Seifert 3-manifolds using $S^1$-connections.

The paper characterizes cohomotopy sets of specific manifolds.

If M is an oriented 3-manifold, let S(M) denote the Homflypt skein module of M. We show that S(M_1 connect sum M_2) is isomorphic to S(M_1) tensor S(M_2) modulo torsion. In fact, we show that S(M_1 connect sum M_2) is isomorphic to S(M_1) tensot S(M_2) if we are working over a certain localized ring. We show the simila…

The paper computes inertia groups of certain high-dimensional manifolds.

We generalize a result of the author about the classification of 1-connected 7-manifolds and demonstrate its use by two concrete applications, one to 2-connected 7-manifolds (a new proof -- and slightly different formulation -- of an up to now unpublished Theorem by Crowley and Nordstroem and one to simply connected 7-…

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

Every 1-connected topological 4-manifold M admits a $S^{1}$-covering by $#_{r-1}S^{2}\times S^{3}$, where $r=$rank$H^{2}(M;\QTR{Bbb}{Z})$.

The paper explores the formality of low-dimensional manifolds using algebraic structures.

We classify all closed 1-connected manifolds $M$ which look like projective planes, i.e. with integral homology $H_*(M)=Z^3$. Furthermore, we give an explicit construction of these manifolds as Thom spaces of open disk bundles.

Let $X$ be a Stein manifold of complex dimension at least two, $F : X \rightarrow \mathbb{C}^n$ a local biholomorphism, and $q \in F(X)$. In this paper we formulate sufficient conditions involving only objects naturally associated to $q$, in order for the fiber over $q$ to be finite. Assume that $F^{-1}(l)$ is 1-connec…

We show that, for any $k\geq 1$, there exist non-formal compact orientable $(k-1)$-connected $n$-manifolds with $k$-th Betti number $b_k=b\geq 0$ if and only if $n\geq \max \{4k-1, 4k+3-2b\}$.

Method counts connected 2D stratifolds with singular curves and components.

Thom-Pontrjagin constructions are used to give a computable necessary and sufficient condition when a homomorphism $φ: H^n(L;Z)\to H^n(M;Z)$ can be realized by a map $f:M\to L$ of degree $k$ for closed $(n-1)$-connected $2n$-manifolds $M$ and $L$, $n>1$. A corollary is that each $(n-1)$-connected $2n$-manifold admits s…

The work of Ray and Singer which introduced analytic torsion, a kind of determinant of the Laplacian operator in topological and holomorphic settings, is naturally generalized in both settings. The couplings are extended in a direct way in the topological setting to general flat bundles and in the holomorphic setting t…

A Banach symmetric space in the sense of O. Loos is a smooth Banach manifold $M$ endowed with a multiplication map $μ\colon M \times M \to M$ such that each left multiplication map $μ_x := μ(x,\cdot)$ (with $x \in M$) is an involutive automorphism of $(M,μ)$ with the isolated fixed point $x$. We show that morphisms of …

Improved bounds on embedding $k$-skeletons into $2k$-manifolds.

We prove that the stable moduli space of $(n-1)$-connected, $n$-parallelizable, $(2n+1)$-dimensional manifolds is homology equivalent to an infinite loopspace for $n \geq 4, n \neq 7$. The main novel ingredient is a version of the cobordism category incorporating surgery data in the form of Lagrangian subspaces.

Let $M^{2n}$ denote a closed $(n-1)$-connected smoothable topological $2n$-manifold. We show that the group $\mathcal{C}(M^{2n})$ of concordance classes of smoothings of $M^{2n}$ is isomorphic to the group of smooth homotopy spheres $\overlineΘ_{2n}$ for $n=4$ or $5$, the concordance inertia group $I_c(M^{2n})=0$ for $…

We show that if S is a finite type orientable surface of genus g and p punctures where 3g+p > 4, then EL(S) is (n-1)-connected and (n-1)-locally connected where dim(PML(S))=2n+1=6g+2p-7. Furthermore, if g=0, then EL(S) is homeomorphic to the p-4 dimensional Nobeling space.

We present a necessary condition for $(\ell-1)$-connected combinatorial $(2\ell +1)$-manifolds to be tight. As a corollary, we show that there is no tight combinatorial three-manifold with Betti number at most two other than the boundary of the four-simplex and the nine-vertex triangulation of the three-dimensional Kle…

Trivalent $2$-stratifolds are a generalization of $2$-manifolds in that there are disjoint simple closed curves where three sheets meet. We develop operations on their associated labeled graphs that will effectively construct from a single vertex all graphs that represent $1$-connected $2$-stratifolds. We describe an i…

We determine loop space decompositions of simply-connected four-manifolds, $(n-1)$-connected $2n$-dimensional manifolds provided $n\notin\{4,8\}$, and connected sums of products of two spheres. These are obtained as special cases of a more general loop space decomposition of certain torsion-free $CW$-complexes with wel…

New invariants define the rational and real homotopy types of closed manifolds.

Research proves the connectedness of classifying spaces for Haefliger structures.

A closed, orientable, splitting surface in an oriented $3$-manifold is a topologically minimal surface of index $n$ if its associated disk complex is $(n-2)$-connected but not $(n-1)$-connected. A critical surface is a topologically minimal surface of index $2$. In this paper, we use an equivalent combinatorial definit…

Surgery method proves category inequality for specific manifolds.

Extends foliation results to singular cases.

Completes classification of high-dimensional manifolds up to exotic sums.

New technique connects graph matching complexes to Morse theory for better topology understanding.

Proves Rudyak's conjecture for low-dimensional simply connected spin manifolds.

In this article, we discuss the local rigidity of Clifford-Klein forms of homogeneous spaces of 1-connected completely solvable Lie groups. In fact, we introduce a splitting of the local rigidity: vertical rigidity and horizontal rigidity. By using this splitting, we refine some existing results about the local rigidit…

We classify nonnegatively curved simply connected 4-manifolds with circle symmetry up to equivariant diffeomorphisms. The main problem is rule out knotted curves in the singular set of the orbit space. As an extension of this work we classify all knots in S^3 which can be realized as an extremal set with respect to an …

We show that the path construction integration of Lie algebroids by Lie groupoids is an actual equivalence from the category of integrable Lie algebroids and complete Lie algebroid comorphisms to the category of source 1-connected Lie groupoids and Lie groupoid comorphisms. This allows us to construct an actual symplec…

Recently R. Cohen and V. Godin have proved that the homology of the free loop space of a closed oriented manifold with coefficients in a field has the structure of a Frobenius algebra without counit. In this short note we prove that when the characteristic of the field is zero and when the manifold is 1-connected the a…

We consider the unique Hermitian connection with totally skew-symmetric torsion on a Hermitian manifold. We prove that if the torsion is parallel and the holonomy is Sp(n)U(1), considered as a subgroup of U(2n) x U(1), then the manifold is locally isomorphic to the twistor space of a quaternionic Kaehler manifold with …

We establish a 1:1 correspondence between Poisson-Lie group actions on integrable Poisson manifolds and twisted multiplicative hamiltonian actions on source 1-connected symplectic groupoids. For an action of a Poisson-Lie group $G$ on a Poisson manifold $M$, we find an explicit description of the lifted hamiltonian act…