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

= May 7 =

two or three questions about 2-manifolds
Can a manifold (ADD: a connected manifold) have Euler characteristic &chi;&gt;2? Intuitively, higher &chi; 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 $$\chi(M\sqcup N)=\chi(M)+\chi(N).$$ See .  --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)