nLab
cell spectrum

Contents

Context

Stable Homotopy theory

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

Definition

A cell spectrum is a topological sequential spectrum XX realized as the colimit over a sequence of spectra *=X 0X 1X 2X 3\ast = X_0 \to X_1 \to X_2 \to X_3 \to \cdots such that there are morphisms

j n:(iI nΣ S q n)X n j_n \;\colon\; \left( \underset{i \in I_n}{\sqcup} \Sigma^\infty S^{q_n} \right) \longrightarrow X_n

with X n+1=Cone(j n)X_{n+1}= Cone(j_n) (the mapping cone).

A cell spectrum is a CW-spectrum if each attaching map Σ S q nX n\Sigma^\infty S^{q_n}\to X_n factors through a X kX nX_k \to X_n with k<qk \lt q.

(e.g. Lewis-May-Steinberger 86, def. 5.1, def. 52)

Definition

A CW-spectrum, def. , is called a finite spectrum (or countable spectrum, etc.) if it has finitely many cells (countably many cells) according to def. .

References

Discussion in the generality of equivariant spectra is in

  • L. Gaunce Lewis, Peter May, and M. Steinberger (with contributions by J.E. McClure), section I.5 of Equivariant stable homotopy theory Springer Lecture Notes in Mathematics Vol.1213. 1986 (pdf)

Last revised on May 18, 2016 at 05:07:39. See the history of this page for a list of all contributions to it.