Related papers: Multiplicity of continuous maps between manifolds

The purpose of this paper is to present, for all $n\ge 3$, very simple examples of continuous maps $f:M^{n-1} \to M^{n}$ from closed $(n-1)$-manifolds $M^{n-1}$ into closed $n$-manifold $M^n$ such that even though the singular set $S(f)$ of…

We call a multigraph {\em non-homotopic} if it can be drawn in the plane in such a way that no two edges connecting the same pair of vertices can be continuously transformed into each other without passing through a vertex, and no loop can…

We show that the category of affine bundles over a smooth manifold M is equivalent to the category of affine spaces modelled on projective finitely generated C^\infty(M)-modules. Using this equivalence of categories, we are able to give an…

We prove an analogue of the Madsen-Weiss theorem for high dimensional manifolds. For example, we explicitly describe the ring of characteristic classes of smooth fibre bundles whose fibres are connected sums of g copies of S^n x S^n, in the…

We study quotients of multi-graded bundles, including double vector bundles. Among other things, we show that any such quotient fits into a tower of affine bundles. Applications of the theory include a construction of normal bundles for…

We give a bound for the number of rational maps between algebraic varieties of general type under mild hypothesis on the canonical map. We use an idea inspired by Tanabe's work. Instead of attaching a morphism of Hodge structures to a…

In this article we study two "strong" topologies for spaces of smooth functions from a finite-dimensional manifold to a (possibly infinite-dimensional) manifold modeled on a locally convex space. Namely, we construct Whitney type topologies…

Let $l$ be a finite field of cardinality $q$ and let $n$ be in $\mathbb{Z}_{\geq 1}$. Let $f_1,\ldots,f_n \in l[x_1,\ldots,x_n]$ not all constant and consider the evaluation map $f=(f_1,\ldots,f_n) \colon l^n \to l^n$. Set…

In this revised form, the proof of the principal lemma has been simplified and the main theorem has been extended to all characteristics for those varieties which are smooth in codimension one. This principal theorem essentially says the…

Given a linear map $\Phi : M_n \rightarrow M_m$, its multiplicity maps are defined as the family of linear maps $\Phi \otimes \text{id}_k : M_n \otimes M_k \rightarrow M_m \otimes M_k$, where $\text{id}_k$ denotes the identity on $M_k$. Let…

Let $F$ be a totally real field, $\mathfrak{p}$ an unramified place of $F$ dividing $p$ and $\overline{r}: \mathrm{Gal}(\overline{F}/F)\rightarrow\mathrm{GL}_2(\overline{\mathbb{F}}_p)$ a continuous irreducible modular representation. The…

We give an intrinsic definition of (affine very) special real manifolds and realise any such manifold $M$ as a domain in affine space equipped with a metric which is the Hessian of a cubic polynomial. We prove that the tangent bundle $N=TM$…

In this paper we study the relationship between two different compactifications of the space of vector bundle quotients of an arbitrary vector bundle on a curve. One is Grothendieck's Quot scheme, while the other is a moduli space of stable…

We show that there exists a unique possible definition, with certain natural properties, of the multiple point space of a holomorphic map between complex manifolds. Our construction coincides with the double point space and the k-th…

Maps between manifolds $M^m\to N^{m+\ell}$ ($\ell>0$) have multiple points, and more generally, multisingularities. The closure of the set of points where the map has a particular multisingularity is called the multisingularity locus. There…

The definition of the intersection number of a map with a closed manifold can be extended to the case of a closed stratified set such that the difference between dimensions of its two biggest strata is greater than $1$. The set Sigma of…

We prove that 1) There exist infinitely many non-trivial codimension one "thick" knots in $\mathbb{R}^5$; 2) For each closed four-dimensional smooth manifold $M$ and for each sufficiently small positive $\epsilon$ the set of isometry…

Let $\mathfrak{R}$ and $\mathfrak{R}'$ be two associative rings (not necessarily with the identity elements). A bijective map $\varphi$ of $\mathfrak{R}$ onto $\mathfrak{R}'$ is called a \textit{$m$-multiplicative isomorphism} if {$\varphi…

Let M be a smooth manifold, A a local algebra, M^{A} the manifold of near points on M of kind A. We use the structure of C^{\infty}(M^{A},A)-module on the set X(M^{A}) of vector fields on M^{A} for to give the equivalence of parallelism of…

We study the space of oriented genus g subsurfaces of a fixed manifold M, and in particular its homological properties. We construct a "scanning map" which compares this space to the space of sections of a certain fibre bundle over M…