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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The only genus zero $n$-manifold is $S^{n}$


Authors: Massimo Ferri and Carlo Gagliardi
Journal: Proc. Amer. Math. Soc. 85 (1982), 638-642
MSC: Primary 57N15; Secondary 05C10, 05C15, 57Q15, 57Q99
DOI: https://doi.org/10.1090/S0002-9939-1982-0660620-5
MathSciNet review: 660620
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Abstract: All $n$-manifolds of regular genus zero, i.e. admitting a crystallization which regularly imbeds into ${{\mathbf {S}}^2}$, are proved to be homeomorphic to ${{\mathbf {S}}^n}$. A conjecture implying the Poincaré Conjecture in dimension four is also formulated. Si dimostra che tutte le $n$-varietà di genere regolare zero, cioè aventi una cristallizzazione che si immerge regularmente in ${{\mathbf {S}}^2}$, sono omeomorfe a ${{\mathbf {S}}^n}$. Si formula anche una congettura che implica quella di Poincaré in dimensione quattro.


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Keywords: <!– MATH ${\text {PL}}$ –> <IMG WIDTH="32" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img21.gif" ALT="${\text {PL}}$">-manifold, genus, Heegaard genus, multigraph, <IMG WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$2$">-cell imbedding, regular genus, crystallization, pseudocomplex, generalized Poincar&#233; Conjecture
Article copyright: © Copyright 1982 American Mathematical Society