nLab
Alexander-Čech duality

Idea

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

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

This element determines an element

δH n(X×(S n+1\X)).\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 δ\delta. This way one obtains a map

δ /:H i(X)H ni(S n+1\X). \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 iH_i is then the Steenrod-Sitnikov homology and H niH^{n-i} has to be cohomology (?).

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

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

is an isomorphism. It follows that one may replace 𝒟 nX\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.

Last revised on December 11, 2018 at 06:59:46. See the history of this page for a list of all contributions to it.