The notion of the homotopy type of a topological stack has been around in the literature for some time. The basic idea is that an atlas X → X of a stack determines a topological groupoid X with object space X. The homotopy type of X should be the classifying space BX. The choice of an atlas is not part of the data of a stack and hence it is not immediately clear why this construction of a homot...

2 Small steps towards a determination of the absolute homotopy type of X := P − {0, 1,∞} 10 2.1 The absolute homotopy type of X is not formal . . . . . . . . . . 10 2.2 Massey products of Steinberg type: a Lie algebra calculation . . 14 2.3 The first few stages of the minimal model of the absolute homotopy type of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 Detecting the tr...

A fundamental construction in the study of stable homotopy is the free infinite loop space generated by a space X. This is the colimit QX = lim −→ ΩΣX. The i homotopy group of QX is canonically isomorphic to the i stable homotopy group of X. Thus, one may obtain stable information about X by obtaining topological results about QX. One such result is the Kahn-Priddy theorem [7]. In another direc...

We prove that if M is a CW-complex, then the homotopy type of the skeletal filtration of M does not depend on the cell decomposition of M up to wedge products with n-disks D, when the later are given their natural CW-decomposition with unique cells of order 0, (n − 1) and n; a result resembling J.H.C. Whitehead’s work on simple homotopy types. From the Colimit Theorem for the Fundamental Crosse...

The homotopy type and homotopy groups of some spectra TAFGU of topological automorphic forms associated to a unitary similitude group GU of type (1, 1) are explicitly described in quasi-split cases. The spectrum TAFGU is shown to be closely related to the spectrum TMF in these cases, and homotopy groups of some of these spectra are explicitly computed.

Finite type nilpotent spaces are weakly equivalent if and only if their singular cochains are quasi-isomorphic as E∞ algebras. The cochain functor from the homotopy category of finite type nilpotent spaces to the homotopy category of E∞ algebras is faithful but not full.

A functor is constructed from the category of globular CW-complexes to that of flows. It allows the comparison of the S-homotopy equivalences (resp. the T-homotopy equivalences) of globular complexes with the S-homotopy equivalences (resp. the T-homotopy equivalences) of flows. Moreover, it is proved that this functor induces an equivalence of categories from the localization of the category of...

The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Geometry, Algebra, and Logic, which has recently come to light in the form of an interpretation of the constructive type theory of Per Martin-Löf into homotopy theory, resulting in new examples of certain algebraic structures which are important in topology. This connection was disc...

A flag complex can be defined as a simplicial complex whose simplices correspond to complete subgraphs of its 1-skeleton taken as a graph. In this article, by introducing the notion of s-dismantlability, we shall define the s-homotopy type of a graph and show in particular that two finite graphs have the same s-homotopy type if, and only if, the two flag complexes determined by these graphs hav...