The homotopy excision theorem.

A-homotopy groups, excision, and solvable quotients

We study some properties of A-homotopy groups: geometric interpretations of connectivity, excision results, and a re-interpretation of geometric quotients by solvable groups in terms of covering spaces in the sense of A-homotopy theory. These concepts and results are well-suited to the study of certain quotients via geometric invariant theory. As a case study in geometry of solvable group quoti...

The Seifert–van Kampen Theorem in Homotopy

Homotopy type theory is a recent research area connecting type theory with homotopy theory by interpreting types as spaces. In particular, one can prove and mechanize type-theoretic analogues of homotopy-theoretic theorems, yielding “synthetic homotopy theory”. Here we consider the Seifert–van Kampen theorem, which characterizes the loop structure of spaces obtained by gluing. This is useful in...

A Borsuk Theorem on Homotopy Types

We present a Borsuk’s theorem published first in [3] (compare also [4, pages 119–120]). It is slightly generalized, the assumption of metrizability is omitted. We introduce concepts needed for the formulation and the proof of theorems on upper semi-continuous decompositions, retracts, strong deformation retract. However, only those facts that are necessary in the proof have been proved.

A Homotopy Theorem for Matroids, Ii

A Torus Theorem for Homotopy Nilpotent Groups

Nilpotency for discrete groups can be defined in terms of central extensions. In this paper, the analogous definition for spaces is stated in terms of principal fibrations having infinite loop spaces as fibers, yielding a new invariant we compare with classical cocategory, but also with the more recent notion of homotopy nilpotency introduced by Biedermann and Dwyer. This allows us to character...