I would like to know something more than what is written on wikipedia http://en.wikipedia.org/wiki/Euler_characteristic

What would be some large (largest?) class of topological spaces for which $\chi$ is defined, so that all standard properties hold, for example that $\chi(X)=\chi(Y)+\chi(Z)$ if $X=Y \cup Z$, ($Y\cap Z=0$).

ADDED. The answer of Algori indicates that a reasonably large class of spaces for which Euler characteristics can be defined are locally compact spaces $X$, whose one point compactification $\bar X$ is a CW complex. Then we can define $\chi(X)=\chi(\bar X)-1$. For example, the Euler characteristics of an open interval according to this definition is $-1$. This definition rases a second (maybe obvious) question.

Question 2. Suppose $X$ is a locally compact space whose 1 point compactification is a $CW$ complex, and $Y$ is a subspace of $X$ such that both $Y$ and $X\setminus Y$ have this property. Is it ture that $\chi(X)=\chi(Y)+\chi(X\setminus Y)$?

Also, I was thinking, that Euler characteristics is more fundamental then homology.But can it be defined for spaces, where homology is not defined?

Finally, Quiaochu pointed out below that a very similar question was already discussed previously on mathoverflow.

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    $\begingroup$ Much like divergent series, there is more than one extension of the Euler characteristic to spaces other than finite CW-complexes. E.g. there is the rational-cohomology Euler characteristic or the Morava K-theory Euler characteristic K(n), which assign 1 and p^n to the classifying space of a cyclic group of order p respectively, or one can apply divergent series techniques to form the alternating sum of dimensions. $\endgroup$ – Tyler Lawson May 29 '10 at 19:40
  • $\begingroup$ The open interval is a(n infinite) CW-complex. $\endgroup$ – algori May 29 '10 at 23:03
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    $\begingroup$ There is a notion of "combinatorial Euler characteristic," but it's not homotopy invariant; see mathoverflow.net/questions/1184/… . $\endgroup$ – Qiaochu Yuan May 30 '10 at 0:02

The answer to the question as it is stated is that there is probably no "largest" class of spaces for which the Euler characteristic makes sense.

The answer also depends on where you would like the Euler characteristic to take values. Here is the tautological answer (admittedly not a very exciting one): if you have a category $C$ of spaces closed under taking cones and cylinders, then there is the universal Euler characteristic for that category: just take the free abelian group $K(C)$ that has a generator $[X]$ for each $X\in C$ and quotient it by the span of $[X]+[Cone(f)]-[Y]$ for all $X,Y\in C$ and any morphism $f:X\to Y$ in $C$. The Euler characteristic of any $X$ in $C$ is set to be $[X]$. (There may be variations and/or generalizations of this approach.)

The group $K(C)$ is complicated in general but for some choices of $C$ it has interesting quotients. This can happen e.g. when $C$ admits a good "cohomology-like" functor. For example if $C$ is the category of spaces with finitely generated integral homology groups then $K(C)$ maps to $\mathbf{Z}$ and this gives the usual Euler characteristic. If one takes $C$ to be formed by spaces that admit a finite cover with finitely generated integral homology groups (typical examples are the classifying spaces of $SL_2(\mathbf{Z})$ and more generally of mapping class groups), then $K(C)$ does not map to $\mathbf{Z}$ any more, but it maps to $\mathbf{Q}$. This gives the rational Euler characteristic.

Finally, let me address the last remark by Dmitri. For some categories the group $K(C)$ maps to $\mathbf{Z}$ in several different ways. Let us take e.g. $C$ to be the category formed by spaces whose one-point compactification is a finite CW-complex (with proper maps as morphisms). Then there are (at least) two characteristics; one is obtained using the ordinary cohomology and another one comes from the Borel-Moore homology. On complex algebraic varieties both agree. But the Borel-Moore Euler characteristic of an open $n$-ball is $(-1)^n$.

Here is the answer to the second question: suppose $Y$ is a locally closed subspace of a locally compact space $X$ such that $X,Y,\bar Y,\bar Y\setminus Y, X\setminus\bar Y$ and $X\setminus Y$ are of the form "a finite CW-complex minus a point". Then $\chi(Y)+\chi(X\setminus Y)=\chi(X)$ where $\chi$ is the Euler characteristic computed using the Borel-Moore homology.

The case when $Y$ is closed follows from the Borel-Moore homology long exact sequence. In general we can write $\chi(X)=\chi(X\setminus\bar Y)+\chi(\bar Y)=\chi(X\setminus\bar Y)+\chi(\bar Y\setminus Y)+\chi(Y)$. In the last sum the sum of the first two terms gives $\chi(X\setminus Y)$ since $X\setminus\bar Y$ is open in $X\setminus Y$.

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    $\begingroup$ Algori, thanks a lot! Are there some readable references on what you have stated (for example about K(C) and Borel-Moore)? $\endgroup$ – Dmitri Panov May 29 '10 at 23:30
  • $\begingroup$ Dmitri -- unfortunately I don't have a reference. But to make up for it here are some comments: 1. the definition of $K(C)$ is a slight variation of the definition of the motivic measure, which in turn goes back to other similar definitions (e.g. the Grothendieck group for coherent sheaves etc) 2. the Borel-Moore homology of a space $X$ (with constant coefficients) is the homology of the one point compactification modulo the added point. $\endgroup$ – algori May 30 '10 at 1:32

The Euler Characteristic can be defined for a larger class of spaces. For example, Euler Characteristic is defined for definable sets in an o-minimal system such as: semi-linear sets, semi-algebraic sets, algebraic varieties, compact smooth manifolds... and not all of them are locally compact.

However, when they are not locally compact, then it is not necessarily true that the different definitions of the characteristic give the same result. If you are interested in this point of view you can read:

  • L. Van den Dries. Tame topology and o-minimal structures. Vol. 248. London Mathematical Society Lecture Notes Series. Cambridge University Press, 1998.

  • J. Curry, R. Ghrist y M. Robinson. “Euler calculus with applications to signals and sensing”. Proceedings of Symposia in Applied Mathematics. Vol. 70. 2012, pages: 75-146. (they have a pdf freely and legally available online here)


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