Testwiki:Reference desk/Archives/Mathematics/2022 May 7

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May 7

two or three questions about 2-manifolds

Can a manifold (ADD: a connected manifold) have Euler characteristic χ>2? Intuitively, higher χ means more simply connected, and you can't get simpler than simple; but is there a proof, or a counterexample?

Related: the sphere is a double cover for the real projective plane, and the torus is a double cover for the Klein bottle (a property that I have put to practical use). Does this generalize? Is every orientable 2-manifold with n handles a double cover of a manifold with n+1 cross-caps?

Are there manifold mappings with more multiplicity? —Tamfang (talk) 04:45, 7 May 2022 (UTC)

First question: if $\chi (M)$ and $\chi (N)$ are the Euler characteristics of two manifolds $M$ and $N$, the Euler characteristic of their disjoint union is given by $\chi (M\bigsqcup N)=\chi (M)+\chi (N).$ See Template:Section link.  --Lambiam 07:17, 7 May 2022 (UTC)

Covers with any multiplicity are possible with 1-manifolds, namely simple close curves, specifically circles. The Cartesian product with another 1-manifold gives covers of any multiplicity for 2-manifolds, or manifolds of any dimension. If X is a k-sheet covering of Y, then χ(X) = k⋅χ(Y) for "sufficiently nice" X and Y. (For example Theorem 2.3 here states this holds when Y is a finite CW-complex. See also this article at Topospaces.) This greatly restricts which spaces can be covers of other spaces, but it does leave open some interesting possibilities. --RDBury (talk) 10:34, 7 May 2022 (UTC)