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The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k , where k is the non-orientable genus.
It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves. The mapping class group can be defined for arbitrary manifolds (indeed, for arbitrary topological spaces) but the 2-dimensional setting is the most studied in group theory .
The curve has genus one (genus formula); in particular, it is not isomorphic to the projective line P 1, which has genus zero. Using genus to distinguish curves is very basic: in fact, the genus is the first invariant one uses to classify curves (see also the construction of moduli of algebraic curves).
This is because the surface N has a unique class of one-sided curves such that, when N is cut open along such a curve C, the resulting surface is a torus with a disk removed. As an unoriented surface, its mapping class group is GL ( 2 , Z ) {\displaystyle \operatorname {GL} (2,\mathbb {Z} )} .
The double cover X of the projective plane branched along a smooth sextic (degree 6) curve is a K3 surface of genus 2 (that is, degree 2g−2 = 2). (This terminology means that the inverse image in X of a general hyperplane in is a smooth curve of genus 2.)
1.2 Curves of genus one. 1.3 Curves with genus greater than one. 1.4 Curve families with variable genus. 2 Transcendental curves. Toggle Transcendental curves subsection.
The most common examples are the curves X(N), X 0 (N), and X 1 (N) associated with the subgroups Γ(N), Γ 0 (N), and Γ 1 (N). The modular curve X(5) has genus 0: it is the Riemann sphere with 12 cusps located at the vertices of a regular icosahedron. The covering X(5) → X(1) is realized by the action of the icosahedral group on the Riemann ...
The genus (sometimes called the demigenus or Euler genus) of a connected non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ , via the relationship χ = 2 − g , where g is the non-orientable ...