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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The genus of a map


Author: Sara Hurvitz
Journal: Trans. Amer. Math. Soc. 264 (1981), 1-28
MSC: Primary 55P60
DOI: https://doi.org/10.1090/S0002-9947-1981-0597863-9
MathSciNet review: 597863
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Abstract: The elements $ [f'](f':X' \to Y')$ of the genus $ - G(f)$ of a map $ f:X \to Y$ are equivalence classes of homotopy classes of maps $ f'$ which satisfy: For every prime $ p$ there exist homotopy equivalences $ {h_p}:{X'_p} \to {X_p}$ and $ {k_p}:{Y'_p} \to {Y_p}$ so that $ {f_p}{h_p} \sim {k_p}{f'_p}$. The genus of $ f$ under $ X - {G^X}(f)$ and the genus of $ f$ over $ Y - {G_Y}(f)$ are defined similarly.

In this paper we prove that under certain conditions on $ f$, the sets $ G(f)$, $ {G^X}(f)$ and $ {G_Y}(f)$ are finite and admit an abelian group structure. We also compare the genus of $ f$ to those of $ X$ and $ Y$, calculate it for some principal fibrations of the form $ K(G,n - 1) \to X \to Y$, and deal with the noncancellation phenomenon.


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DOI: https://doi.org/10.1090/S0002-9947-1981-0597863-9
Article copyright: © Copyright 1981 American Mathematical Society