In mathematics, Spanier–Whitehead duality is a duality theory in homotopy theory, based on a geometrical idea that a topological space X may be considered as dual to its complement in the n-sphere, where n is large enough.
It is named for Edwin Spanier and J. H. C. Whitehead, who developed it in papers from 1955.
The basic point is that sphere complements determine the homology, but not the homotopy type, in general.
Thus Spanier–Whitehead duality fits into stable homotopy theory.
are dual objects in the category of pointed spectra with the smash product as a monoidal structure.
are reduced and unreduced suspensions respectively.
Taking homology and cohomology with respect to an Eilenberg–MacLane spectrum recovers Alexander duality formally.