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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Extending the concept of genus to dimension $ n$


Author: Carlo Gagliardi
Journal: Proc. Amer. Math. Soc. 81 (1981), 473-481
MSC: Primary 57M15; Secondary 05C10
MathSciNet review: 597666
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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0597666-0
PII: S 0002-9939(1981)0597666-0
Keywords: PL-manifold, genus, Heegaard genus, Heegaard diagram, multigraph, $ 2$-cell imbedding, line-colouring, $ (n + 1)$-coloured graph, regular imbedding, regular genus, crystallisation, pseudocomplex, contracted complex
Article copyright: © Copyright 1981 American Mathematical Society