Related papers: The homotopy theory of function spaces: a survey

We prove that the mapping stack Map(Y,X) of topological stacks X and Y is again a topological stack if Y admits a compact groupoid presentation. If Y admits a locally compact groupoid presentation, we show that Map(Y,X) is a paratopological…

We bring a linkage from representation theory of Lie groups to homotopy theory for maps between flag manifolds. As applications we derive from representation theory abundant families of homotopy classes of maps between flag manifolds whose…

For complete metric spaces $X$ and $Y$, a description of linear biseparating maps between spaces of vector-valued Lipschitz functions defined on $X$ and $Y$ is provided. In particular it is proved that $X$ and $Y$ are bi-Lipschitz…

The notion of the \emph{homotopy type} of a topological stack has been around in the literature for some time. The basic idea is that an atlas $X \to \mathfrak{X}$ of a stack determines a topological groupoid $\mathbb{X}$ with object space…

This is a survey paper on spaces of automorphisms of manifolds and spaces of manifolds in a fixed homotopy type. It describes the main theorems of traditional surgery theory, but also the main theorems of pseudoisotopy theory, alias…

We discuss topological versions of the closed graph theorem, where continuity is inferred from near continuity in tandem with suitable conditions on source or target spaces. We seek internal characterizations of spaces satisfying a closed…

We show "free theorems" in the style of Wadler for polymorphic functions in homotopy type theory as consequences of the abstraction theorem. As an application, it follows that every space defined as a higher inductive type has the same…

Let w: Map(X,Y;f) -> Y denote a general evaluation fibration. Working in the setting of rational homotopy theory via differential graded Lie algebras, we identify the long exact sequence induced on rational homotopy groups by w in terms of…

This is the second in a series of papers. Here we develop here an intersection theory for manifolds equipped with an action of a finite group. As in our previous paper, our approach will be homotopy theoretic, enabling us to circumvent the…

Given a topological group G, its orbit category Orb_G has the transitive G-spaces G/H as objects and the G-equivariant maps between them as morphisms. A well known theorem of Elmendorf then states that the category of G-spaces and the…

We study the homotopy of the connected sum of a manifold with a projective space, viewed as a typical way to stabilize manifolds. In particular, we show a loop homotopy decomposition of a manifold after stabilization by a projective space,…

We prove that the inclusion of map(X,Y) into map(K(X),K(Y)) is continuous, where K(X) is the space of non-empty compact subsets of X (also known as the hyperspace of compact subsets of X), and both spaces of maps are endowed with the…

We establish certain conditions which imply that a map $f:X\to Y$ of topological spaces is null homotopic when the induced integral cohomology homomorphism is trivial; one of them is: $H^*(X)$ and $\pi_*(Y)$ have no torsion and $H^*(Y)$ is…

We propose a homology theory for locally compact spaces with ends in which the ends play a special role. The approach is motivated by results for graphs with ends, where it has been highly successful. But it was unclear how the original…

This paper centers around two basic problems of topological coincidence theory. First, try to measure (with help of Nielsen and minimum numbers) how far a given pair of maps is from being loose, i.e. from being homotopic to a pair of…

Associated to any manifold equipped with a closed form of degree >1 is an `L-infinity algebra of observables' which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group…

We interpret mathematically the pair (master equation, solution of master equation) up to equivalence, as the pair (a presentation of a free triangular dga T over a combination operad O, dga map of T into C, a dga over O) up to homotopy…

In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces, and maps of graphs embedded in the sphere, in homotopy type theory. This allows us to provide an elementary…

We investigate the problem of describing the homotopy classes $[X,Y]$ of continuous functions between $\omega$-bounded non metrizable manifolds $X,Y$. We define a family of surfaces $X$ built with the first octant $C$ in $L^2$ ($L$ is the…

In this paper, we show that for finite $CW$-complexes $X$ and two-stage space $Y$ (for example $n$-spheres $S^n$, homogeneous spaces and $F_0$-spaces), the rational homotopy type of $\map(X, Y)$ is determined by the cohomology algebra…