nLab
mapping telescope

Context

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

Homotopy

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

Given a sequence

X =(X 0f 0X 1f 1X 2f 2) X_\bullet = \left( X_0 \overset{f_0}{\longrightarrow} X_1 \overset{f_1}{\longrightarrow} X_2 \overset{f_2}{\longrightarrow} \cdots \right)

of (pointed) topological spaces, then its mapping telescope is the result of forming the (reduced) mapping cylinder Cyl(f n)Cyl(f_n) for each nn and then attaching all these cylinders to each other in the canonical way.

At least if all the f nf_n are inclusions, this is the sequential attachment of ever “larger” cylinders, whence the name “telescope”.

The mapping telescope is a representation for the homotopy colimit over X X_\bullet. It is used for instance for discussion of lim^1 and Milnor sequences (and that’s maybe the origin of the concept?).

Definition

Definition

For

X =(X 0f 0X 1f 1X 2f 2) X_\bullet = \left( X_0 \overset{f_0}{\longrightarrow} X_1 \overset{f_1}{\longrightarrow} X_2 \overset{f_2}{\longrightarrow} \cdots \right)

a sequence in Top, its mapping telescope is the quotient topological space of the disjoint union of product topological spaces

Tel(X )(n(X n×[n,n+1]))/ Tel(X_\bullet) \coloneqq \left( \underset{n \in \mathbb{N}}{\sqcup} \left( X_n \times [n,n+1] \right) \right)/_\sim

where the equivalence relation quotiented out is

(x n,n)(f(x n),n+1) (x_n, n) \sim (f(x_n), n+1)

for all nn\in \mathbb{N} and x nX nx_n \in X_n.

Analogously for X X_\bullet a sequence of pointed topological spaces then use reduced cylinders to set

Tel(X )(n(X n[n,n+1] +))/ . Tel(X_\bullet) \coloneqq \left( \underset{n \in \mathbb{N}}{\sqcup} \left( X_n \wedge [n,n+1]_+ \right) \right)/_\sim \,.

Properties

For CW-complexes

Proposition

For X X_\bullet the sequence of stages of a (pointed) CW-complex X=lim nX nX = \underset{\longleftarrow}{\lim}_n X_n, then the canonical map

Tel(X )X Tel(X_\bullet) \longrightarrow X

from the mapping telescope, def. 1, is a weak homotopy equivalence.

Proof

Write in the following Tel(X)Tel(X) for Tel(X )Tel(X_\bullet) and write Tel(X n)Tel(X_n) for the mapping telescop of the substages of the finite stage X nX_n of XX. It is intuitively clear that each of the projections at finite stage

Tel(X n)X n Tel(X_n) \longrightarrow X_n

is a homotopy equivalence, hence a weak homotopy equivalence. A concrete construction of a homotopy inverse is given for instance in (Switzer 75, proof of prop. 7.53).

Moreover, since spheres are compact, so that elements of homotopy groups π q(Tel(X))\pi_q(Tel(X)) are represented at some finite stage π q(Tel(X n))\pi_q(Tel(X_n)) it follows that

lim nπ q(Tel(X n))π q(Tel(X)) \underset{\longrightarrow}{\lim}_n \pi_q(Tel(X_n)) \overset{\simeq}{\longrightarrow} \pi_q(Tel(X))

are isomorphisms for all qq\in \mathbb{N} and all choices of basepoints (not shown).

Together these two facts imply that in the following commuting square, three morphisms are isomorphisms, as shown.

lim nπ q(Tel(X n)) π q(Tel(X)) lim nπ q(X n) π q(X). \array{ \underset{\longleftarrow}{\lim}_n \pi_q(Tel(X_n)) &\overset{\simeq}{\longrightarrow}& \pi_q(Tel(X)) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow \\ \underset{\longleftarrow}{\lim}_n \pi_q(X_n) &\underset{\simeq}{\longrightarrow}& \pi_q(X) } \,.

Therefore also the remaining morphism is an isomorphism (two-out-of-three). Since this holds for all qq and all basepoints, it is a weak homotopy equivalence.

examples of universal constructions of topological spaces:

AAAA\phantom{AAAA}limitsAAAA\phantom{AAAA}colimits
\, point space\,\, empty space \,
\, product topological space \,\, disjoint union topological space \,
\, topological subspace \,\, quotient topological space \,
\, fiber space \,\, space attachment \,
\, mapping cocylinder, mapping cocone \,\, mapping cylinder, mapping cone, mapping telescope \,
\, cell complex, CW-complex \,

References

  • Robert Switzer, Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.
Revised on May 2, 2017 13:17:41 by Urs Schreiber (131.220.184.222)