The genus of a map

Hurvitz, Sara

doi:10.1090/S0002-9947-1981-0597863-9

The genus of a map
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by Sara Hurvitz PDF

Trans. Amer. Math. Soc. 264 (1981), 1-28 Request permission

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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