nLab
shrinkable map

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

shrinkable map \Rightarrow Dold fibration

Contents

Definition

A shrinkable map p:XYp:X \to Y in the category Top is a map with a section s:YXs:Y \to X such that sp:XXs\circ p:X \to X is vertically homotopic to id Xid_X (i.e. is homotopic to id X\id_X in the slice category Top/YTop/Y). In particular, a shrinkable map is a homotopy equivalence.

Properties

Every shrinkable map is a Dold fibration. This result follows from a theorem of Dold about locally homotopy trivial map?s being the same as Dold fibrations. It should be obvious that a shrinkable map is globally homotopy trivial, with trivial fibre.

Examples

Example:(Segal) Let U iYU_i \to Y be a numerable open cover. Then the geometric realization of the Cech nerve NCˇ(U i)N\check{C}(U_i) comes with a canonical map |NCˇ(U i)|Y|N\check{C}(U_i)| \to Y which is shrinkable.

There are extensions of this to other categories with a notion of homotopy.

References

This definition is due to Dold, in his 1963 Annals paper Partitions of unity in the theory of fibrations.

Revised on August 16, 2010 07:24:37 by David Roberts (203.24.207.134)