Definition
Let be a continuous map (assume ). Then we can form the cell complex
where is a -dimensional disc attached to via . The cellular chain groups are just freely generated on the -cells in degree, so they are in degree 0, and and zero everywhere else. Cellular (co-)homology is the (co-)homology of this chain complex, and since all boundary homomorphisms must be zero (recall that ), the cohomology is
Denote the generators of the cohomology groups by
- and
For dimensional reasons, all cup-products between those classes must be trivial apart from . Thus, as a ring, the cohomology is
The integer is the Hopf invariant of the map .
Read more about this topic: Hopf Invariant
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