Editing Euler characteristic (section)

==Properties==

The Euler characteristic behaves well with respect to many basic operations on topological spaces, as follows.

===Homotopy invariance===
Homology is a topological invariant, and moreover a [[homotopy invariant]]: Two topological spaces that are [[homotopy equivalent]] have [[group isomorphism|isomorphic]] homology groups. It follows that the Euler characteristic is also a homotopy invariant.

For example, any [[contractible]] space (that is, one homotopy equivalent to a point) has trivial homology, meaning that the 0th Betti number is 1 and the others 0. Therefore, its Euler characteristic is 1. This case includes [[Euclidean space]] <math>\mathbb{R}^n</math> of any dimension, as well as the solid unit ball in any Euclidean space &mdash; the one-dimensional interval, the two-dimensional disk, the three-dimensional ball, etc.

For another example, any convex polyhedron is homeomorphic to the three-dimensional [[ball (mathematics)|ball]], so its surface is homeomorphic (hence homotopy equivalent) to the two-dimensional [[sphere]], which has Euler characteristic&nbsp;2. This explains why the surface of a convex polyhedron has Euler characteristic 2.

===Inclusion–exclusion principle===
If ''M'' and ''N'' are any two topological spaces, then the Euler characteristic of their [[disjoint union]] is the sum of their Euler characteristics, since homology is additive under disjoint union:

:<math>\chi(M \sqcup N) = \chi(M) + \chi(N).</math>

More generally, if ''M'' and ''N'' are subspaces of a larger space ''X'', then so are their union and intersection. In some cases, the Euler characteristic obeys a version of the [[inclusion–exclusion principle]]:

:<math>\chi(M \cup N) = \chi(M) + \chi(N) - \chi(M \cap N).</math>

This is true in the following cases:

*if ''M'' and ''N'' are an [[excisive couple]]. In particular, if the [[interior (topology)|interiors]] of ''M'' and ''N'' inside the union still cover the union.<ref>Edwin Spanier: Algebraic Topology, Springer 1966, p. 205.</ref>
*if ''X'' is a [[locally compact space]], and one uses Euler characteristics with [[compact space|compact]] [[support (mathematics)|supports]], no assumptions on ''M'' or ''N'' are needed.
*if ''X'' is a [[topologically stratified space|stratified space]] all of whose strata are even-dimensional, the inclusion–exclusion principle holds if ''M'' and ''N'' are unions of strata.  This applies in particular if ''M'' and ''N'' are subvarieties of a [[complex number|complex]] [[algebraic variety]].<ref>William Fulton: Introduction to toric varieties, 1993, Princeton University Press, p. 141.</ref>

In general, the inclusion–exclusion principle is false. A [[counterexample]] is given by taking ''X'' to be the [[real line]], ''M'' a [[subset]] consisting of one point and ''N'' the [[complement (set theory)|complement]] of ''M''.

===Connected sum===

For two connected closed n-manifolds <math>M, N</math> one can obtain a new connected manifold <math>M \# N</math> via the [[connected sum]] operation. The Euler characteristic is related by the formula <ref>{{cite web
 | url = http://topospaces.subwiki.org/wiki/Homology_of_connected_sum
 | title = Homology of connected sum
 | access-date = 2016-07-13
}}</ref>

:<math> \chi(M \# N) = \chi(M) + \chi(N) - \chi(S^n).</math>

===Product property===
Also, the Euler characteristic of any [[product space]] ''M'' &times; ''N'' is

:<math>\chi(M \times N) = \chi(M) \cdot \chi(N).</math>

These addition and multiplication properties are also enjoyed by [[cardinality]] of [[set (mathematics)|set]]s. In this way, the Euler characteristic can be viewed as a generalisation of cardinality; see [http://math.ucr.edu/home/baez/counting/].

===Covering spaces===
{{details|Riemann–Hurwitz formula}}
Similarly, for a ''k''-sheeted [[covering space]] <math>\tilde{M} \to M,</math> one has
:<math>\chi(\tilde{M}) = k \cdot \chi(M).</math>
More generally, for a [[ramified covering space]], the Euler characteristic of the cover can be computed from the above, with a correction factor for the ramification points, which yields the [[Riemann–Hurwitz formula]].

===Fibration property===
The product property holds much more generally, for [[fibrations]] with certain conditions.

If <math>p\colon E \to B</math> is a fibration with fiber ''F,'' with the base ''B'' [[path-connected]], and the fibration is orientable over a field ''K,'' then the Euler characteristic with coefficients in the field ''K'' satisfies the product property:<ref>{{citation
|title=Algebraic Topology
|first=Edwin Henry
|last=Spanier
|author-link=Edwin Spanier
|publisher=Springer
|year=1982
|isbn=978-0-387-94426-5
|url=https://books.google.com/books?id=h-wc3TnZMCcC
}}, [https://books.google.com/books?id=h-wc3TnZMCcC&pg=PA481 Applications of the homology spectral sequence, p. 481]</ref>
:<math>\chi(E) = \chi(F)\cdot \chi(B).</math>
This includes product spaces and covering spaces as special cases,
and can be proven by the [[Serre spectral sequence]] on homology of a fibration.

For fiber bundles, this can also be understood in terms of a [[transfer map]] <math>\tau\colon H_*(B) \to H_*(E)</math> – note that this is a lifting and goes "the wrong way" – whose composition with the projection map <math>p_*\colon H_*(E) \to H_*(B)</math> is multiplication by the [[Euler class]] of the fiber:<ref>{{citation
|title=Fibre bundles and the Euler characteristic
|first=Daniel Henry
|last=Gottlieb
|journal=Journal of Differential Geometry
|volume=10
|issue=1
|year=1975
|pages=39–48
|doi=10.4310/jdg/1214432674
|s2cid=118905134
|url=http://www.math.purdue.edu/~gottlieb/Bibliography/17FibreBundlesAndtheEulerCharacteristic.pdf
}}</ref>
:<math>p_* \circ \tau = \chi(F) \cdot 1.</math>