nLab Alexander-Čech duality

Idea

Let $X$ be a finite CW complex and $X \subseteq S^{n+1}$. Then there is a map

$$X \times (S^{n+1}\backslash X) \to S^n,\,\,\,\,\,\, (x,y) \mapsto \frac{x-y}{\|x - y\|}.$$

This element determines an element

$$\delta \in H^n( X \times (S^{n+1}\backslash X)).$$

In topology there is a slant product operation, sort of dividing, and in this case one can do slant product with $\delta$. This way one obtains a map

$$\delta_{/} : H_{i}(X)\to H^{n-i}(S^{n+1}\backslash{X}).$$

This map is an isomorphism and it is called the Alexander-Čech duality (or sometimes simply Alexander duality). It can be considered for infinite complexes as well, but in that case one has to change the flavour of (co)homology theories involved. $H_i$ is then the Steenrod-Sitnikov homology and $H^{n-i}$ has to be cohomology (?).

The Spanier-Whitehead duality is a generalization, where $S^{n+1}\backslash X$ is replaced by any space $D_n(X)$, together with an element $\delta$ such that the corresponding map

$$\delta_/ : H_i(X) \to H^{n-i}(\mathcal{D}_n X)$$

is an isomorphism. It follows that one may replace $\mathcal{D}_n X$ by its suspension and so on, hence the stable homotopy theory is a natural setup for this duality.

References

Named after James W. Alexander and Eduard Čech.