Gender (area)

In topology, the gender of a compact, orientable surface is the number of “holes” (or “handles”) of the surface. The name and definition go back to Alfred Clebsch .

Gender is a topological invariant . The classification theorem for surfaces states that closed orientable surfaces are classified by their gender, except for homeomorphism .

definition

The gender of a surface is defined as the maximum number of possible cuts along disjoint, simply closed curves , so that the surface is still connected after the cutting process, i.e. after all cuts made .

term

Bernhard Riemann dealt with "holes" in surfaces as early as 1857 . He called this size class number. The term gender was introduced in 1864 by Alfred Clebsch .

Examples

This double torus has gender 2.

The spherical surface has gender 0, because it has no holes, or every section divides it into two non-connected parts. ${\ displaystyle S ^ {2}}$

The torus surface has genus 1.

Relationships with other sizes

The Euler characteristic and gender are related as follows for orientable, closed surfaces ${\ displaystyle \ chi}$ ${\ displaystyle g}$

{\ displaystyle \ chi = 2-2g}

.

literature

Wladimir G. Boltjanski , Vadim A. Efremovitsch: Illustrative combinatorial topology . With a foreword by SP Novikov . Translated from Russian and with a foreword by Detlef Seese and Martin Weese (= Mathematical Student Library . Volume 129 ). Deutscher Verlag der Wissenschaften, Berlin 1986, ISBN 3-326-00008-1 .

Individual evidence

^ DD Bleecker, B. Booss: Topology and Analysis: The Atiyah-Singer Index Formula and Gauge-Theoretic Physics . Springer Science & Business Media, 2012, ISBN 978-1-4684-0627-6 , p. 288 ( google.com ).

[BleeckerBooss2012-1] DD Bleecker, B. Booss: Topology and Analysis: The Atiyah-Singer Index Formula and Gauge-Theoretic Physics . Springer Science & Business Media, 2012, ISBN 978-1-4684-0627-6 , p. 288 ( google.com ).