nLab
Borsuk's shape theory

Contents

Reprise of the Idea

The basic idea behind Borsuk’s shape theory is explained in the entry on shape theory, so will not be repeated here, except to say that it considers compact metric spaces embedded in the Hilbert cube, then uses the open neighbourhoods of the space as a ‘net’ of approximations of the space. The space is, of course, the intersection of all these open neighbourhoods.

Any compact metric space can be embedded in the Hilbert cube, so it is sufficient to consider just compact subspaces of that space.

Some details

Let s= n=1 (1n,1n)s= \prod_{n=1}^\infty (\frac{1}{n},\frac{1}{n}) be the pseudo-interior of the Hilbert cube, Q= n=1 [1n,1n]Q= \prod_{n=1}^\infty [\frac{1}{n},\frac{1}{n}].

We will define (a category equivalent to) the Borsuk Shape category, Shape BShape_B, to have compact subsets of ss as objects and some morphisms that need a bit of explaining.

If XX and YY are compact subsets of ss, then a fundamental sequence, f̲:XY\underline{f} : X\to Y, is defined to be a sequence of maps f n:QQf_n : Q\to Q such that for every neighbourhood VV of YY in QQ, there exists a neighbourhood UU of XX in QQ and an integer n 0n_0 such that if n,n n 0n, n^\prime \geq n_0, the restrictions f n Uf_n|_U and f n Uf_{n^\prime}|_U are homotopic within VV.

Note that the f n(X)f_n(X) do not have to be contained in YY, they only have to be ‘near’ YY.

Two fundamental sequences, f̲,f̲ :XY\underline{f},\underline{f}^\prime : X\to Y, are said to be homotopic, f̲f̲ \underline{f}\sim \underline{f}^\prime provided that for every neighbourhood VV of YY in QQ, there is a neighbourhood UU of XX in QQ and an integer n 0n_0 such that if nn 0n \geq n_0, then f n Uf_n|_U and f n Uf^\prime_{n}|_U are homotopic within VV.

The morphisms of Shape BShape_B and taken to be the homotopy classes of fundamental sequences between the corresponding spaces.

Two compacta contained in ss are said to have the same shape if they are isomorphic in Shape BShape_B. As an example, the Warsaw circle has the same shape as the circle.

Chapman complement theorem

Theorem (Chapman, 1972)

If XX and YY are compacta in ss, then XX and YY have the same shape if and only if their complements QXQ\setminus X and QYQ\setminus Y are homeomorphic.

Chapman extended the association XX ‘goes to’ QXQ\setminus X to a functor from the Borsuk shape category to the weak proper homotopy category of complements in QQ of compacta. This was the basis for Edwards-Hastings formulation of strong shape theory, on replacing the weak form of proper homotopy by a strong form.

References

  • K. Borsuk, Concerning homotopy properties of compacta, Fund Math. 62 (1968) 223-254
  • K. Borsuk, Theory of Shape, Monografie Matematyczne Tom 59,Warszawa 1975.
  • T.A.Chapman, On Some Applications of Infinite Dimensional Manifolds to the Theory of Shape, Fund. Math. 6 (1972), 181 - 193.
  • D.A. Edwards and H. M. Hastings, (1976), Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Maths. 542, Springer-Verlag.
Revised on November 9, 2011 22:32:36 by Urs Schreiber (131.174.41.104)