Related papers: Obstruction theory for coincidences of multiple ma…

Let $X$ be a complete intersection inside a variety $M$ with finite dimensional motive and for which the Lefschetz-type conjecture $B(M)$ holds. We show how conditions on the niveau filtration on the homology of $X$ influence directly the…

Klein and Williams developed an obstruction theory for the homotopical equivariant fixed point problem, which asks whether an equivariant map can be deformed, through an equivariant homotopy, into another map with no fixed points…

Let $E,F$ be two topological spaces and $u:E\rightarrow F$ be a map. \ If $F$ is Haudorff and $u$ is continuous, then its graph is closed. \ \ The Closed Graph Theorem establishes the converse when $E$ and $F$ are suitable objects of…

A Frechet algebra endowed with a multiplicatively convex topology has two types of invariants: homotopy invariants (topological K-theory and periodic cyclic homology) and secondary invariants (multiplicative K-theory and the non-periodic…

A covariant functor from the category of mapping tori to a category of AF-algebras is constructed; the functor takes continuous maps between such manifolds to stable homomorphisms between the corresponding AF-algebras. We use this functor…

The Decomposition Problem in the class $LIP(\mathbb{S}^2)$ is to decompose any bi-Lipschitz map $f:\mathbb{S}^2 \to \mathbb{S}^2$ as a composition of finitely many maps of arbitrarily small isometric distortion. In this paper, we construct…

Let $W$ be a nonorientable $4$-dimensional handlebody without $3$- and $4$-handles. We show that $W$ admits a Lefschetz fibration over the $2$-disk, whose regular fiber is a nonorientable surface with nonempty boundary. This is an analogue…

Thom-Pontrjagin constructions are used to give a computable necessary and sufficient condition when a homomorphism $\phi : H^n(L;Z)\to H^n(M;Z)$ can be realized by a map $f:M\to L$ of degree $k$ for closed $(n-1)$-connected $2n$-manifolds…

We approach the problem of finding obstructions to curvature distinguished Riemannian metrics by considering Lorentzian metrics to which they are dual in a suitable sense. Obstructions to the latter then yield obstructions to the former.…

We survey work by the author and Ralf Meyer on equivariant KK-theory. Duality plays a key role in our approach. We organize the survey around the objective of computing a certain homotopy-invariant of a space equipped with a proper action…

Coincidences of maps between smooth manifolds are studied via a geometric approach which involves (nonstabilized) normal bordism theory and pathspaces.

For a holomorphic one-form $\mathbf{\xi}$ on a weakly 1-complete manifold $X$ with certain properties, we discussed the connectivity of the pair $(\hat{X}, F^{-1}(z))$, where $\pi : \hat{X} \to X$ is a covering map and…

Caffarelli's contraction theorem states that probability measures with uniformly logconcave densities on R d can be realized as the image of a standard Gaussian measure by a globally Lipschitz transport map. We discuss some counterexamples…

For M and N closed oriented connected smooth manifolds of the same dimension, we consider the mapping space Map(M,N;f) of continuous maps homotopic to f:M--> N.We show that the evaluation map from the space of maps to the manifold N induces…

A wide and natural class of closed currents - which are differences of positive closed currents - can be constructed by pulling back smooth closed forms using rational maps. These currents are very singular in general, and hence defining…

With any (open or closed) cover of a space T we associate certain homotopy classes of maps T into n-spheres. These homotopy invariants can be considered as obstructions for extensions of covers of a subspace A to a space X. We using these…

We consider a continuous map $f :M\to N$ between two manifolds and try to estimate its multiplicity from below, i.e. find a $q$-tuple of pairwise distinct points $x_1,..., x_q\in M$ such that $f(x_1) = f(x_2) = ... = f(x_q)$. We show that…

Using higher topos theory, we explore the obstruction to the \v{C}ech-de Rham map being an isomorphism in each degree for diffeological spaces. In degree 1, we obtain an exact sequence which interprets Iglesias-Zemmour's construction from…

This paper explores the Bernstein problem of smooth maps $f:\mathbb{R}^4 \to \mathbb{R}^3$ whose graphs form coassociative submanifolds in $\mathbb{R}^7$. We establish a condition, expressed in terms of the second elementary symmetric…

We prove that a pseudoholomorphic diffeomorphism between two almost complex manifolds with boundaries satisfying some pseudoconvexity type condition cannot map a pseudoholomorphic disc in the boundary to a single point. This can be viewed…