Stable homotopy, $1$-dimensional NCCW complexes, and Property (H)

Complexes and Homotopy Chains

is the boundary chain of the oriented cell &J, the p% being the incidence numbers of the cells a\, a j ~ \ These chains and boundary matrices change by subdivision of the complex A, and the homology groups are the invariants with regard to this process. The chains and boundary matrices which I introduce are defined for complexes U with an adjoined group G of mappings 7 of U in itself, tha t is,...

On the Second Homotopy Module of Two-dimensional Cw Complexes

Let A' be a connected 2-dimensional CW complex. This note reproves from a very simple point of view two classical theorems of H. Hopf relating the homology of the fundamental group it = irx(X) of X and the Hurewicz map on ir2(X). This point of view also allows the dual theorems to be proved. If w is a finite group, a new interpretation is given for tf,(7r; Z) (/' = 2, 3) in terms of v2(X).

1 Leray Spectral Sequence Theorem for Nccw

We prove in this paper a noncommutative version of Leray Theorem and then Leray-Serre Spectral Theorem for noncommutative Serre fibrations: for NC Serre fibration there are converging spectral sequences with E 2 terms as E 2 p,q = HP p (A; HP q (B, A)) =⇒ HP p+q (B) and E 2 p,q = HP p (A; Kq(B, A)) =⇒ Kp+q(B).

Chapter 0. Some Underlying Geometric Notions..... 1 Homotopy and Homotopy Type 1. Cell Complexes 5. Operations on Spaces 8. Two Criteria for Homotopy Equivalence 11. The Homotopy Extension Property 14.

Stable A1-homotopy Theory

Definition 1. A T -prespectrum E is a sequence En of pointed spaces together with structure morphisms σn : En ∧ T → En+1 for all n. A morphism λ : E → F of T -prespectra is a sequence of maps λn : En → Fn commuting with the structure morphisms. We will denote the category of T -prespectra by PSpT (k). Example 1. For any based space X , we have the suspension prespectrum Σ∞(X ) of X given by Σ∞(...