Editing Euler characteristic (section)

===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>