Hopf–Whitney theorem
In mathematics, especially algebraic topology and homotopy theory, the Hopf–Whitney theorem is a result relating the homotopy classes between a CW complex and a multiply connected space with singular cohomology classes of the former with coefficients in the first nontrivial homotopy group of the latter. It can for example be used to calculate cohomotopy as spheres are multiply connected.
Statement
For a -dimensional CW complex and a -connected space, the well-defined map:with a certain cohomology class is an isomorphism.
The Hurewicz theorem claims that the well-defined map with a fundamental class is an isomorphism and that, which implies for the Ext functor. The Universal coefficient theorem then simplifies and claims:
is then the cohomology class corresponding to the identity.
In the Postnikov tower removing homotopy groups from above, the space only has a single nontrivial homotopy group and hence is an Eilenberg–MacLane space, which classifies singular cohomology. Combined with the canonical map, the map from the Hopf–Whitney theorem can alternatively be expressed as a postcomposition:
Examples
For homotopy groups, cohomotopy sets or cohomology, the Hopf–Whitney theorem reproduces known results but weaker:- For every -connected space one has:
- For a -dimensional CW complex one has:
- For a topological group and a natural number, the Eilenberg–MacLane space is -connected by construction, hence for every -dimensional CW-complex one has: