# Leray Spectral Sequence

Let $f:X\to Y$ be a smooth map between paracompact differential manifolds $X$ and $Y$. Let $U$ be an open and dense subset of $Y$. For any $y\in U$, let $f^{-1}(y)=F$ be a generic fiber that is a submanifold of $F$.
Assume the singular fibers are $F/\Gamma_t$, where for each $t\in Y\setminus U$, $\Gamma_t$ is a finite subgroup (depending on $t$) of the automorphism group of $F$ that is acting properly discontinuously on $F/\Gamma_t$, i.e., the latter is also a smooth manifold.
If $\Gamma_t$ is the identity for all $t$, and $f$ is a fibration, then there is a Leray spectral sequence relating the homology of $X$ to that of $F$ and $Y$. Is there some spectral sequence for the case when $\Gamma_t$ is not always the identity, and if so what? A reference for this would be appreciated too.

• As Algori mentions, the answer is yes, and it's also the Leray spectral sequence. What reference are you using? – Ryan Budney Nov 16 '12 at 23:37

Ru -- the Leray spectral sequence exists for any map $f:X\to Y$ of arbitrary topological spaces and any sheaf $F$ on $X$ and its second term is $$E_2^{p,q}=H^p(Y,R^q f_*F)$$ where $R^q f_*F$ are the sheaves on $Y$ that are obtained by sheafifying the presheaves $U\mapsto H^q(f^{-1}(U),F)$. Here are some remarks that might help:

1. If $f$ is a locally trivial fibration and $F$ is constant then all $R^q f_*F$ are locally constant; if in addition $Y$ is simply-connected then the sheaves are constant and we can express $E_2$ in terms of the constant cohomology of $Y$.

2. It may happen that all fibers $f^{-1}(y),y\in Y$ are homeomorphic but some or all $R^q f_*F$ are non-constant; take e.g. $X=(\mathbb{R}\setminus \{ 0\})\sqcup \{ 0\}, Y=\mathbb{R},f$ the identity map.

3. Nevertheless, if $f:X\to X/G$ where $G$ is a connected Lie group that acts nicely on $X$ (say so that the quotient is Hausdorff) with finite stabilizers, and $F$ is a constant sheaf with stalk $\mathbb{Q}$ (or $\mathbb{R}$ or $\mathbb{C}$) then any sheaf $R^q f_*F$ is constant with stalk $H^q(G,\mathbb{Q})$ (resp., $H^q(G,\mathbb{R})$ and $H^q(G,\mathbb{C})$).

Two possible references (which means, to be honest, that there may be better references but that's where I first learned this from) are Godement, Topologie alg\'ebrique et th\'eorie des faisceaux, the very end of chapter 4, and Griffiths-Harris, the very end of vol.1

• Thanks Algori, As you mentioned spectral sequence argument seems to work for cohomology - how about homology? – user13559 Nov 18 '12 at 20:48
• Ru -- welcome. Re how about homology: it depends: for locally trivial fibrations everyng works fine in a similar way; for more general maps the homological version exists but is quite a bit more complicated (the strategy basically consists in reducing everything to the cohomological case via the Verdier duality). – algori Nov 18 '12 at 22:49
• Thanks again. Do you know of any reference about homological case? – user13559 Nov 19 '12 at 18:34
• Ru -- I've never seen it done in detail in the homological case but if I had to guess I would define, following Borel-Moore, Homology theory for locally compact spaces, Michigan Math. J. 7, 2, 1960, thm 3.8 and \S 5, $H_i(X,F)=\mathbb{H}_c^{-i}(X,DF)$ where $\mathbb{H}_c$ stands for compactly supported hypercohomology and $D$ for the Verdier dual, and then see how it goes. For a careful introduction to constructible sheaves and related things see e.g. Borel, Intersection cohomology. Also, there was an Asterisque volume called "Etale homology" by Deligne et al, which may also be relevant. – algori Nov 20 '12 at 2:12