Related papers: Higher homotopy wild sets

We prove injectivity of the canonical map from singular homology to measure homology for certain ``mildly wild" spaces, that is, certain spaces not having the homotopy type of a CW-complex, but having countable fundamental groups.

Let $k$ be a field, $f \colon X \to Y$ a birational morphism of integral connected schemes proper over $k$ with $Y$ normal, $x \in X(k)$ lying over $y \in Y(k)$. For Tannakian categories $\cC_X \subset \Vect(X)$ and $\cC_Y \subset…

We show that Chen-Ruan cohomology is a homotopy invariant in certain cases. We introduce the notion of a T-representation homotopy, which is a stringent form of homotopy under which Chen-Ruan cohomology is invariant. We show that while…

We consider the homotopy types of $PD_4$-complexes $X$ with fundamental group $\pi$ such that $c.d.\pi=2$ and $\pi$ has one end. Let $\beta=\beta_2(\pi;F_2)$ and $w=w_1(X)$. Our main result is that (modulo two technical conditions on…

If $X$ is a (topological) space, the $n$th finite subset space of $X$, denoted by $X(n)$, consists of $n$-point subsets of $X$ (i.e., nonempty subsets of cardinality at most $n$) with the quotient topology induced by the unordering map…

An elementary notion of homotopy can be introduced between arrows in a cartesian closed category $E$. The input is a finite-product-preserving endofunctor $\Pi_0$ with a natural transformation $p$ from the identity which is surjective on…

In this paper we prove results on the existence and homotopy classification of holomorphic submersions from Stein manifolds to other complex manifolds. We say that a complex manifold Y satisfies Property S_n for some integer n bigger or…

An isovariant map is an equivariant map between $G$-spaces which strictly preserves isotropy groups. We consider an isovariant analogue of Klein--Williams equivariant intersection theory for a finite group $G$. We prove that under certain…

The space of all finite non-empty subsets of a topological space $X$, also known as the Ran space of $X$, is weakly contractible for $X$ path connected. We consider subspaces $\mathrm{Ran}_{\leqslant n}(X)$ of the Ran space given by all…

Given a manifold $M$ and a proper sub-bundle $\Delta\subset TM$, we study homotopy properties of the horizontal base-point free loop space $\Lambda$, i.e. the space of absolutely continuous maps $\gamma:S^1\to M$ whose velocities are…

Let $M$ be a smooth manifold. We use Chern-Weil theory to study the characteristic classes of principal $G$-bundles built from continuous families of $\pi_{1}(M)$-representations, where $G$ is a compact Lie group. We then relate these…

A spatial refinement of Bar-Natan homology is given, that is, for any link diagram $D$ we construct a CW-spectrum $\mathcal{X}_{\mathit{BN}}(D)$ whose reduced cellular cochain complex gives the Bar-Natan complex of $D$. The stable homotopy…

For a real $r\geq 0,$ we consider the notion of $r$-homotopy equivalence in the category quasimetric spaces, which includes metric spaces and directed graphs. We show that for a finite quasimetric space $X$ there is a unique (up to…

Classical persistent homology is a powerful mathematical tool for shape comparison. Unfortunately, it is not tailored to study the action of transformation groups that are different from the group Homeo(X) of all self-homeomorphisms of a…

Building on the work of Grinberg and Stanley, we begin a systematic study of permutations with a prescribed $X$-descent set. In particular, for a set $X \subseteq \mathbb{N}^2$, and $I \subseteq [n-1]$, we study the permutations $\pi \in…

It is known that shape injectivity implies homotopical Hausdorff and that the converse does not hold, even if the space is required to be a Peano continuum. This paper gives an alternative definition of homotopical Hausdorff inspired by a…

For a connected based space $X$, let $[X,X]$ be the set of all based homotopy classes of base point preserving self map of $X$ and let $\E(X)$ be the group of self-homotopy equivalences of $X$. We denote by $\A_{\sharp}^k(X)$ the set of…

Given two real algebraic varieties X and Y, we denote by R(X,Y) the set of all regular maps from X to Y. The set R(X,Y) is regarded as a topological subspace of the space C(X,Y) of all continuous maps from X to Y endowed with the…

The topological fundamental group $\pi_{1}^{top}$ is a topological invariant that assigns to each space a quasi-topological group and is discrete on spaces which are well behaved locally. For a totally path-disconnected, Hausdorff, unbased…

Let $\Hol_{x_0}^{{\bf n}} (\C\P^1, X)$ be the space of based holomorphic maps of degree ${\bf n}$ from $\C\P^1$ into a simply connected algebraic variety $X$. Under some condition we prove that the map $\map \Hol_{x_0}^{{\bf n}} (\C\P^1,…