Extending the concept of genus to dimension 𝑛

Gagliardi, Carlo

doi:10.1090/S0002-9939-1981-0597666-0

Extending the concept of genus to dimension $n$
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Proc. Amer. Math. Soc. 81 (1981), 473-481 Request permission

Abstract:

Some graph-theoretical tools are used to introduce the concept of "regular genus" $\mathcal {G}({M_n})$, for every closed $n$-dimensional PL-manifold ${M_n}$. Then it is proved that the regular genus of every surface equals its genus, and that the regular genus of every $3$-manifold ${M_3}$ equals its Heegaard genus, if ${M_3}$ is orientable, and twice its Heegaard genus, if ${M_3}$ is nonorientable. A geometric approach, and some applications in dimension four are exhibited.

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Lehrbuch der Topologie

Graphs, groups, and surfaces