Math 2.0

Manifolds and cell complexes Definition

Simple homotopy equivalence

Builds on CW complexn-skeletonHomotopy equivalenceClosed ballFundamental group

Definition

Let XX be a finite CW complex and let n≥1n \ge 1. Let DnD^n be the closed unit ball of ℝn\R^n. Its boundary sphere is the union of the closed lower hemisphere D−n−1D^{n-1}_- and the closed upper hemisphere D+n−1D^{n-1}_+. The two hemispheres meet in Sn−2S^{n-2}. Let g ⁣:D−n−1→Xg \colon D^{n-1}_- \to X be a continuous map with g(Sn−2)⊆X(n−2)g(S^{n-2}) \subseteq X^{(n-2)} and g(D−n−1)⊆X(n−1)g(D^{n-1}_-) \subseteq X^{(n-1)}, where X(k)X^{(k)} is the kk-skeleton. Let Y=X∪gDnY = X \cup_g D^n be the space obtained by gluing DnD^n to XX along gg. It is a CW complex with two new cells: the interior of D+n−1D^{n-1}_+ and the interior of DnD^n.

  1. The inclusion X↪YX \hookrightarrow Y is an elementary expansion, and YY is said to collapse elementarily onto XX.
  2. A formal deformation from XX to ZZ is a finite sequence X=X0,X1,…,Xm=ZX = X_0, X_1, \ldots, X_m = Z of finite CW complexes. Each step is an elementary expansion or an elementary collapse.
  3. A cellular map f ⁣:X→Zf \colon X \to Z is a simple homotopy equivalence if it is homotopic to the composite along a formal deformation from XX to ZZ. In this composite, an expansion contributes its inclusion and a collapse contributes a deformation retraction.

Remarks

Let XX be a point and n=1n = 1: gluing D1=[−1,1]D^1 = [-1, 1] to XX along its lower end point gives Y=[−1,1]Y = [-1, 1], with cells {−1}\set{-1}, {1}\set{1} and (−1,1)(-1, 1). So the inclusion of a point into an interval is an elementary expansion, and the constant map from the interval to the point is a simple homotopy equivalence.

The lens spaces L(7,1)L(7, 1) and L(7,2)L(7, 2) are homotopy equivalent, but no homotopy equivalence between them is simple. Their Reidemeister torsions differ, and Reidemeister torsion is invariant under formal deformations.

Every simple homotopy equivalence is a homotopy equivalence. Let f ⁣:X→Zf \colon X \to Z be a homotopy equivalence of finite CW complexes. Then ff is simple if and only if its Whitehead torsion τ(f)\tau(f) vanishes in the Whitehead group of the fundamental group of ZZ. In particular, if ZZ is simply connected, every homotopy equivalence X→ZX \to Z is simple, since the Whitehead group of the trivial group is zero. By a theorem of Thomas Chapman, every homeomorphism between finite CW complexes is a simple homotopy equivalence.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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CW complex, concept, no evidence levelCW complexconceptClosed ball, concept, no evidence levelClosed ballconceptFundamental group, concept, no evidence levelFundamental groupconceptHomotopy equivalence, concept, no evidence levelHomotopy equivalenceconceptn-skeleton, concept, no evidence leveln-skeletonconceptSimple homotopy equivalence, definition, no evidence levelSimple homotopy equiv…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. CW complex concept
  2. Closed ball concept
  3. Fundamental group concept
  4. Homotopy equivalence concept
  5. n-skeleton concept
  6. Simple homotopy equivalence definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Simple_homotopy_theory
About this page
TagF01D990B61
URIm2://F01D990B61
Kinddefinition
WritingW0 stub
MSC57Q10, 55P10
DefinesSimple homotopy equivalence
UsesCW complex, n-skeleton, Homotopy equivalence, Closed ball, Fundamental group
Changed2 November 2026

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