Tag: Euler

Let $X$ be a differentiable manifold and $G$ a finite group acting
differentiably on $X$. The following formula for the Euler number $text{e}(X/G)$ of the orbit space $X/G$ appears to be well-known:
begin{equation*}tag{1}
text{e}(X/G) = dfrac{1}{lvert G rvert} sum_{g in G} text{e}(X^g)
end{equation*}
where, for each $g in G$, $X^g$ is the subspace of elements fixed by $g$. Instances of this formula can be found in (3) and (5), but without proof or
link to such. A quite elaborate search on the web did not produce one, either.

Following the allusion to Lefschetz numbers in (3), a proof might be given, I suppose, along the following lines. Consider the canonical projection $pi : X rightarrow X/G$ and its induced map $pi^* : text{H}^*(X/G;Bbb{Q}) rightarrow text{H}^*(X;Bbb{Q})$ on rational cohomology. According to a celebrated result of Grothendieck in (4), this map is injective and maps $text{H}^*(X/G;Bbb{Q})$ isomorphically onto the $G$-invariants $text{H}^*(X;Bbb{Q})^G$ (this follows from the Leray SS for the map $pi$, which collapses). Now for any representation of $G$ on a finite dimensional vector space $V$ over a field of characteristic 0, one has that
begin{equation*}
P := dfrac{1}{lvert G rvert} sum_{g in G} g : V rightarrow V
end{equation*}
is the projector onto the $G$-invariants $V^G$ and so, by taking traces,
begin{equation*}
dim V^G = text{tr}, P =
dfrac{1}{lvert G rvert} sum_{g in G} text{tr}, g.
end{equation*}
Therefore, for each $i$,
begin{equation*}
dim text{H}^i(X/G;Bbb{Q}) = dim text{H}^i(X;Bbb{Q})^G =
dfrac{1}{lvert G rvert} sum_{g in G} text{tr}, g^i,
end{equation*}
where $g^i$ is the endomorphism induced by $g$ on $text{H}^i(X;Bbb{Q})$. Taking the alternating sum and interchanging the summation order leaves us with
begin{equation*}
begin{aligned}
text{e}(X/G) &= sum_i (-1)^i dim text{H}^i(X/G;Bbb{Q}) \
&= dfrac{1}{lvert G rvert} sum_{g in G} sum_i (-1)^i text{tr}, g^i
end{aligned}
end{equation*}
and so
begin{equation*}tag{2}
text{e}(X/G) = dfrac{1}{lvert G rvert} sum_{g in G} L(g),
end{equation*}
where
begin{equation*}
L(g) := sum_i (-1)^i text{tr}, g^i
end{equation*}
is the Lefschetz number of the endomorphism $g : X rightarrow X$.

But now we are in business; (1) boils down to the sixty-four dollar question
begin{equation*}tag{3}
forall g in G: L(g) = text{e}(X^g).
end{equation*}
A high-brow reference for this is is the general Lefschetz fixed point theorem
Theorem (2.12) of (2) (which is a synthesis of the main theorem Theorem (6.7) of (1) and the Localization Theorem (1.1) of (2)), which, after plodding through the quite intricate intricacies of the $K$-theory apparatus surrounding the Index Theorem, should grind out the desired answer (3), which is, in a sense a localized version of the Gauss-Bonnet Theorem.
What makes me unhappy with this is that the derivation of (1) up to stage
(3) is maybe on the level of a first year graduate course in Algebraic Topology,
wherea the application of the full force of the Index Theorem in (3) appears
like the proverbial cracking of a peanut with a sledgehammer. Now it is known that the Gauss-Bonnet Theorem is much more elementary than the Index Theorem
(which just illustrates it but is not needed for its proof), and also localization results come in various flavours, so my question is whether there exists a much more elementary proof of (1) somewhere in the world.

(1) Atiyah, M.F. & Singer, I.M.,
The Index of Elliptic Operators: I, Ann.of Math, Sec.Ser., Vol. 87, No. 3 (May, 1968), pp. 484-530.

(2) Atiyah, M.F. & Segal, G.B.,
The Index of Elliptic Operators: II, Ann.of Math, Sec.Ser., Vol. 87, No. 3 (May, 1968), pp. 531-545.

(3) Bryan, J. & Fulman, J.,
Orbifold Euler characteristics and the number of commuting n-tuples in symmetric groups,
(math/9712248) on $mathtt{arxiv.org}$

(4) Grothendieck, A.,
Sur quelques points d’algèbre homologique, II, Tohoku Math. J. (2)
Volume 9, Number 3 (1957), 119-221.

(5) Hirzebruch, F. & Höfer, T.,
On the Euler number of an orbifold, Math. Ann. 286, 255-260 (1990)