Leray Spectral Sequence - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T00:21:38Z http://mathoverflow.net/feeds/question/112618 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112618/leray-spectral-sequence Leray Spectral Sequence Ru 2012-11-16T22:48:07Z 2012-11-17T02:27:44Z <p>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 $F$.<br> 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.<br> 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.</p> http://mathoverflow.net/questions/112618/leray-spectral-sequence/112621#112621 Answer by algori for Leray Spectral Sequence algori 2012-11-16T23:13:20Z 2012-11-17T02:27:44Z <p>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 <code>$$E_2^{p,q}=H^p(Y,R^q f_*F)$$</code> where <code>$R^q f_*F$</code> 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:</p> <ol> <li><p>If $f$ is a locally trivial fibration and $F$ is constant then all <code>$R^q f_*F$</code> 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 $Y$.</p></li> <li><p>It may happen that all fibers $f^{-1}(y),y\in Y$ are homeomorphic but some or all <code>$R^q f_*F$</code> are non-constant; take e.g. <code>$X=(\mathbb{R}\setminus \{ 0\})\sqcup \{ 0\}, Y=\mathbb{R},f$</code> the identity map.</p></li> <li><p>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 <code>$R^q f_*F$</code> is constant with stalk $H^q(G,\mathbb{Q})$ (resp., $H^q(G,\mathbb{R})$ and $H^q(G,\mathbb{C})$).</p></li> </ol> <p>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</p>