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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJun 1st 2011

In order to accompany the nCafe discussion I have started to add some content to the entry Euler characteristic

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJun 1st 2011

added the notion of “homotopical Euler characteristic” for “$\infty$-groupoid cardinality” and mentioned how Leinster cardinality interpolates between that and “homological” Euler characteristic.

Added stub sections for Euler char of enriched and higher categories and for Gauss-Bonnet theorem

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeJun 2nd 2011

I have added some more discussion of the symmetric-monoidal Euler characteristic, and rearranged a little bit. I think it would be better to group the various definitions by style, rather than by input data – the “homological” Euler characteristic of a topological space fits into the same picture as the Euler characteristic of a chain complex and the symmetric monoidal notion, whereas the “homotopical” one is a different sort of beast.

I’m also not sold on the phrase “homotopical Euler characteristic”. The notion of “Euler characteristic of a topological space” is ancient and well-established, and I don’t think we should try to bifurcate its meaning. Especially since we have a perfectly good alternate name: “$\infty$-groupoid cardinality”. I know that the “Euler characteristic of a category” reduces to both in special cases, but I would rather just say that “the Euler characteristic of a groupoid, considered as a category, is its groupoid cardinality”.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJun 2nd 2011

Sure, thanks. I have edited a bit more, accordingly. Also added a brief comment on Euler char of higher categories and added plenty of references, using John’s material.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeJun 2nd 2011

Looks great, thanks!

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeMay 20th 2012
• (edited May 20th 2012)

I added the reference by Efremovich and Rudyak which singles Euler characteristic of finite CW complexes as the only (up to multiplicative constant) additive homotopy invariant and the related 1966 reference by C.T.C. Wall on arithmetic invariants of finite simplicial complexes, with a free pdf link at Canadian J. Math.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMar 23rd 2019
• (edited Mar 23rd 2019)

added the following statement (here):

Let $X$ and $Y$ be closed manifolds (topological or differentiable) of even dimension. Then the Euler characteristic of their connected sum is the sum of their separate Euler characteristics, minus 2:

$\chi \big( X \sharp Y \big) \;=\; \chi \big(X\big) + \chi \big(Y\big) -2$

added this also to connected sum

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeOct 27th 2021

• Euler char of even dim spheres is 2 (here)

• Euler char of covering map is multiplication by degree (here)

Both without proofs for the moment, but with pointers to proofs. I just need to be able to refer to these statements from another entry…

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeOct 27th 2021
• (edited Oct 27th 2021)

I have touched the defining formula (here) via alternating sum of Betti numbers: it had $rk_{\mathbb{Z}}(H(-;R))$ with a comment on how $R$ could be either $\mathbb{Z}$ or $\mathbb{Q}$. I have made it read $dim_{\mathbb{Q}}(H(-;\mathbb{Q}))$ in the second case.

• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeMar 7th 2023