Mapping degrees between spherical $3$-manifolds

D. Gonçalves ; P. Wong ; X. Zhao

Sbornik. Mathematics, Tome 208 (2017) no. 10, pp. 1449-1472

Résumé

Let $D(M,N)$ be the set of integers that can be realized as the degree of a map between two closed connected orientable manifolds $M$ and $N$ of the same dimension. For closed $3$-manifolds $M$ and $N$ with $S^3$-geometry, every such degree $\operatorname{deg} f\equiv \overline {\operatorname{deg}}\psi \mod |\pi_1(N)|$ where $0\le \overline {\operatorname{deg}}\psi |\pi_1(N)|$ and $\overline {\operatorname{deg}}\psi$ only depends on the induced homomorphism $\psi=f_{\pi}$ on the fundamental group. In this paper, we calculate the set $\{\overline{\operatorname{deg}}\psi\}$ explicitly when $\psi$ is surjective and then we show how to determine $\overline{\operatorname{deg}}(\psi)$ for arbitrary homomorphisms. This leads to the determination of the set $D(M,N)$. Bibliography: 22 titles.

Détail

Keywords: $3$-manifolds, mapping degrees.