Hoffmann-Zeller theorem

The Hoffmann-Zeller theorem is a mathematical theorem in the field of algebraic topology. The theorem describes the connection between the simplicial homology products of one equation with the product of a cellular homology equation. <math>B \times A</math> and those of the spaces <math>B</math> and <math>A</math>. The theorem first appeared in a 1949 paper published by the American Mathematical Monthly.
Theorem statement
The theorem can be formulated as follows. Suppose <math>B</math> and <math>A</math> are topological spaces, followed by the three chain complexes <math>C_*(B)</math>, <math>C_*(A)</math>, and <math>C_*(B \times A) </math>. (The argument applies equally to the simplicial or cellular chain complexes.) We then have the tensor equation complex <math>C_*(B) \otimes C_*(A)</math>, it follows that the differential is, by definition,
:<math>\delta( \sigma \otimes \tau) = \delta_B \sigma \otimes \tau + (-1)^p \sigma \otimes \delta_A \tau</math>
for <math>\sigma \in C_p(B)</math> and <math>\delta_B</math>, <math>\delta_A</math> the differentials on <math>C_*(B)</math>,<math>C_*(A)</math>.
The theorem then states that we have a chain maps
:<math>F: C_*(B \times A) \rightarrow C_*(B) \otimes C_*(A), \quad G: C_*(B) \otimes C_*(A) \rightarrow C_*(B \times A)</math>
therefore <math>FG</math> is the identity and <math>GF</math> is chain-homotopic to the identity. Moreover, the maps are natural in <math>B</math> and <math>A</math>. Consequently the two products must have the same root homology:
:<math>H_*(C_*(B \times A)) \cong H_*(C_*(B) \otimes C_*(A))</math>.
The chain-homotopic would not apply if the product outcome were greater than the initial homology.
Importance
The Hoffmann-Zeller theorem is a key factor in establishing the principal link between the cellular and simplicial homologicals.

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