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.

The paper identifies manifolds with free circle actions.

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

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…

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…

The paper characterizes cohomotopy sets of specific manifolds.

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

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applications, one to 2-connected 7-manifolds (a new proof -- and slightly different formulation -- of an up to now unpublished Theorem by Crowley and Nordstroem and one to simply connected 7-…

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

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.

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…

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.

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 $n\geq \max \{4k-1, 4k+3-2b\}$.

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…

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

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

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

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 …

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.

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…

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…

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

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 $…

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

Surgery method proves category inequality for specific manifolds.

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 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 …

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

This paper examines limits of Riemannian 2-manifolds with bounded curvature.

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…

Special Riemannian manifolds are characterized by their curvature.

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 …

An immersion of a compact manifold is tight if it admits the minimal total absolute curvature over all immersions of the manifold. A prominent result in the study of minimal total absolute curvature immersions is the theorem of Chern and Lashof, which characterizes minimal total absolute curvature immersions, and tight…

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…

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…

The paper studies Dirac operators on large spectral three-manifolds.

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…

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…

Study shows bounds on knot groups and nonembeddability of certain open manifolds.

Using the Wodzicki residue, we build Wodzicki-Chern-Simons (WCS) classes in $H^{2k-1}(LM)$ associated to the residue Chern character on the loop space $LM$ of a Riemannian manifold $M^{2k-1}$. These WCS classes are associated to the $L^2$ connection and the Sobolev $s=1$ connections on $LM.$ The WCS classes detect seve…

New invariant P helps classify simply-connected 8-manifolds.