nLab Borsuk's shape theory

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

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

If $X$ and $Y$ are compact subsets of $s$, then a fundamental sequence, $\underline{f} : X\to Y$, is defined to be a sequence of maps $f_n : Q\to Q$ such that for every neighbourhood $V$ of $Y$ in $Q$, there exists a neighbourhood $U$ of $X$ in $Q$ and an integer $n_0$ such that if $n, n^\prime \geq n_0$, the restrictions $f_n|_U$ and $f_{n^\prime}|_U$ are homotopic within $V$.

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

Two fundamental sequences, $\underline{f},\underline{f}^\prime : X\to Y$, are said to be homotopic, $\underline{f}\sim \underline{f}^\prime$ provided that for every neighbourhood $V$ of $Y$ in $Q$, there is a neighbourhood $U$ of $X$ in $Q$ and an integer $n_0$ such that if $n \geq n_0$, then $f_n|_U$ and $f^\prime_{n}|_U$ are homotopic within $V$.

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

Two compacta contained in $s$ are said to have the same shape if they are isomorphic in $Shape_B$. As an example, the Warsaw circle has the same shape as the circle.

Chapman complement theorem

Theorem (Chapman, 1972)

If $X$ and $Y$ are compacta in $s$, then $X$ and $Y$ have the same shape if and only if their complements $Q\setminus X$ and $Q\setminus Y$ are homeomorphic.

Chapman extended the association $X$ ‘goes to’ $Q\setminus X$ to a functor from the Borsuk shape category to the weak proper homotopy category of complements in $Q$ 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.

Related concepts

• shape theory

  • strong shape theory

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.