The genus $g$ of a closed, orientable surface is the maximum number of disjoint simple closed curves that can be drawn on the surface without disconnecting it. In terms of the Euler characteristic, $\chi = 2 - 2g$.

For closed nonorientable surfaces, Wikipedia defines its nonorientable genus, demigenus or Euler genus $k$ as the number of cross-caps attached to a sphere. In terms of the Euler characteristic, $\chi = 2 - k$.

Here we have four different "numbers" for any given surface:

  1. The maximum number of disjoint simple closed curves that can be drawn on the surface without disconnecting it
  2. $g = (2 - \chi)/2$
  3. The number of cross-caps attached to a sphere
  4. $k = 2 - \chi$

My question: What is the genus of a nonorientable surface?

The first paragraph of page 3 of this article defines the genus of a nonorientable surface using $(1)$. This is compatible with Wikipedia which avoids defining the genus of a nonorientable surface.

Wolfram Mathworld defines the genus of a nonorientable surface using $(4)$. The Manifold Atlas Project defines the genus of a nonorientable surfaces using $(3)$.

Assuming that all of the above are consistent, the only way I can resolve this conflict is:

Possible answer: Could it be that for orientable surfaces $(1) = (2)$, but for nonorientable surfaces $(1) = (3) = (4)$, so that definition $(1)$ works for everything?

Is there even a consensus among mathematicians? If there is, could I have a reference for this consensus?

  • 1
    Your "possible answer" is correct. Do you want to see a proof? Definition 3 is indeed the most common one and its source (the "atlas") is the most professional one of all you link. I do not believe there is a reference for a consensus on anything in mathematics, including genus. – Moishe Cohen Apr 21 '14 at 18:15
up vote 2 down vote accepted
+50

Here is a proof of equivalence of 1, 3 and 4. Below, $S$ is a compact connected surface without boundary.

Definition. A maximal cut in $S$ is a 1-dimensional submanifold $L\subset S$ such that $S\setminus L$ is connected and contains no nonseparating simple loops. The latter condition just means that $S\setminus L$ is the 2-dimensional sphere with $q=q(S,L)$ points removed. (To see the latter, you can use the classification of compact surfaces.) In particular, $\chi(S_L)=2-q$.

Let $L$ be a maximal cut in $S$ and let $S_L$ denote the surface with such that $S_L\setminus \partial S_L$ is homeomorphic to $S\setminus L$. Then $S$ can be reconstructed from $S_L$ as follows: represent $\partial S_L$ as a disjoint union of circles: $$ C_1\cup C_1' \sqcup ... \sqcup C_{n}\sqcup C_n' \cup A_{1} \sqcup ... A_m, $$ $q=2n+m$. Identify each $C_i$ to $C_i'$, $i\le n$ via a homeomorhism and identify each $A_i$ to itself via an antipodal map of the circle. The space $S_L/\sim$ equipped with the quotient topology is homeomorphic to $S$.

Using this, we can compute the Euler characteristics: $$ \chi(S)= \chi(S_L)= 2-q. $$ To see the first identity, triangulate $S_L$ so that the identification maps of the boundary are simplicial and just count (with the usual sign) the simplices in $S$ using simplices in $S_L$.

In particular, we see that $q$ is independent of the maximal cut. Moreover, this shows that 1 is equivalent to 4 (orientation is irrelevant). To prove equivalence of 1 and 3 in the non-orientable case, note that we can obtain $S$ from the surface $S_L$ (which is the sphere with $q$ holes) by adding a cross-cap to each hole ($q$ cross-cups in total). In the more modern terminology, we represent $S$ as the connected sum of $q$ projective planes. Equivalently, $S=S_L/\sim$, where we set $n=0, m=q$ holes with my notation above.

As for references, you can use Massey's "A basic course in Algebraic Topology". Another thing I am sure is this: There is no document confirming a consensus of topologists on this matter.

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