Hoffmann-Zeller theorem

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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. $B \times A$ and those of the spaces $B$ and $A$ . The theorem first appeared in a 1949 paper published by the American Mathematical Monthly.

[edit] Theorem statement

The theorem can be formulated as follows. Suppose $B$ and $A$ are topological spaces, followed by the three chain complexes $C * (B)$ , $C * (A)$ , and $C_*(B \times A)$ . (The argument applies equally to the simplicial or cellular chain complexes.) We then have the tensor equation complex $C_*(B) \otimes C_*(A)$ , it follows that the differential is, by definition,

$\delta( \sigma \otimes \tau) = \delta_B \sigma \otimes \tau + (-1)^p \sigma \otimes \delta_A \tau$

for $\sigma \in C_p(B)$ and $δ B$ , $δ A$ the differentials on $C * (B)$ , $C * (A)$ .

The theorem then states that we have a chain maps

$F: C_*(B \times A) \rightarrow C_*(B) \otimes C_*(A), \quad G: C_*(B) \otimes C_*(A) \rightarrow C_*(B \times A)$

therefore $F G$ is the identity and $G F$ is chain-homotopic to the identity. Moreover, the maps are natural in $B$ and $A$ . Consequently the two products must have the same root homology:

$H_*(C_*(B \times A)) \cong H_*(C_*(B) \otimes C_*(A))$ .

The chain-homotopic would not apply if the product outcome were greater than the initial homology.

[edit] Importance

The Hoffmann-Zeller theorem is a key factor in establishing the principal link between the cellular and simplicial homologicals.

[edit] References

Hoffmann, Jan & Zeller, M. H. (1949), “Homology, Principles and Products”, American Mathematical Monthly. 476 (55): 189–196 .
Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0 .