Editing Euler characteristic (section)

===Arbitrary dimensions===
[[File:Euler_characteristic_hypercube_simplex.svg|thumb|250px|Comparison of Euler characteristics of [[hypercube]]s and [[simplex|simplices]] of dimensions&nbsp;1 to 4]]
{| class="wikitable floatright" style="text-align:right;"
|+ <small>Euler characteristics of the six [[regular 4-polytope|4&nbsp;dimensional analogues of the regular polyhedra]]</small>
! Regular<br/>4&nbsp;polytope
! [[Vertex (geometry)|{{mvar|V}}]]<br/>{{mvar|k}}{{sub|0}}
!   [[Edge (geometry)|{{mvar|E}}]]<br/>{{mvar|k}}{{sub|1}}
!   [[Face (geometry)|{{mvar|F}}]]<br/>{{mvar|k}}{{sub|2}}
!   [[Cell (geometry)|{{mvar|C}}]]<br/>{{mvar|k}}{{sub|3}}
! {{right| <math> \chi = V - E\ +</math><br/><math> +\ F - C</math>}}
|-
| [[5-cell|5&nbsp;cell]] || 5 || 10 || 10 || 5 || {{center|'''0'''}}
|-
| [[8-cell|8&nbsp;cell]] || 16 || 32 || 24 || 8 || {{center|'''0'''}}
|-
| [[16-cell|16&nbsp;cell]] || 8 || 24 || 32 || 16 || {{center|'''0'''}}
|-
| [[24-cell|24&nbsp;cell]] || 24 || 96 || 96 || 24 || {{center|'''0'''}}
|-
| [[120-cell|120&nbsp;cell]] || 600 || 1200 || 720 || 120 || {{center|'''0'''}}
|-
| [[600-cell|600&nbsp;cell]] || 120 || 720 || 1200 || 600 || {{center|'''0'''}}
|}
The {{mvar|n}}&nbsp;dimensional sphere has singular [[homology group]]s equal to
:<math>H_k(\mathrm{S}^n) = \begin{cases} \mathbb{Z} ~& k = 0 ~~ \mathsf{ or } ~~ k = n \\ \{0\} & \mathsf{otherwise}\ , \end{cases}</math>

hence has Betti number&nbsp;1 in dimensions&nbsp;0 and {{mvar|n}}, and all other Betti numbers are&nbsp;0. Its Euler characteristic is then {{nobr|{{math|&chi; {{=}} 1 + (−1)}}{{sup| {{mvar|n}} }} ;}} that is, either&nbsp;0 if {{mvar|n}} is [[odd number|odd]], or&nbsp;2 if {{mvar|n}} is [[even number|even]].

The {{mvar|n}}&nbsp;dimensional real [[projective space]] is the quotient of the {{mvar|n}}&nbsp;sphere by the [[antipodal map]]. It follows that its Euler characteristic is exactly half that of the corresponding sphere &ndash; either&nbsp;0 or&nbsp;1.

The {{mvar|n}}&nbsp;dimensional torus is the product space of {{mvar|n}}&nbsp;circles. Its Euler characteristic is&nbsp;0, by the product property. More generally, any compact [[parallelizable manifold]], including any compact [[Lie group]], has Euler characteristic&nbsp;0.<ref>{{cite book |author1-link=John Milnor |last1=Milnor |first1=J.W. |name-list-style=amp |last2=Stasheff |first2=James D. |year=1974 |title=Characteristic Classes |publisher=Princeton University Press}}</ref>

The Euler characteristic of any [[closed manifold|closed]] odd-dimensional manifold is also&nbsp;0.<ref>{{harvp|Richeson|2008|p=261}}</ref> The case for [[Orientability|orientable]] examples is a corollary of [[Poincaré duality]]. This property applies more generally to any [[compact space|compact]] [[topologically stratified space|stratified space]] all of whose strata have odd dimension. It also applies to closed odd-dimensional non-orientable manifolds, via the two-to-one [[Orientability#Orientable double cover|orientable double cover]].