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.
-dimensional CW complex
-connected space
, the well-defined map: with a certain cohomology class
ι ∈
is an isomorphism.
The Hurewicz theorem claims that the well-defined map
with a fundamental class
is an isomorphism and that
, which implies
Ext
{\displaystyle \operatorname {Ext} _{\mathbb {Z} }^{1}(H_{n-1}(Y,\mathbb {Z} ),\pi _{n}(Y))\cong 1}
for the Ext functor.
The Universal coefficient theorem then simplifies and claims:
ι ∈
is then the cohomology class corresponding to the identity
id ∈
{\displaystyle \operatorname {id} \in \operatorname {End} _{\mathbb {Z} }(\pi _{n}(Y))}
In the Postnikov tower removing homotopy groups from above, the space
only has a single nontrivial homotopy group
(up to weak homotopy equivalence), which classifies singular cohomology.
Combined with the canonical map
, the map from the Hopf–Whitney theorem can alternatively be expressed as a postcomposition: For homotopy groups, cohomotopy sets or cohomology, the Hopf–Whitney theorem reproduces known results but weaker: