View source for Leray spectral sequence

''spectral sequence of a continuous mapping''

A spectral sequence connecting the cohomology with values in a sheaf of Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l0581901.png" /> on a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l0581902.png" /> with the cohomology of its direct images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l0581903.png" /> under a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l0581904.png" />. More precisely, the second term of the Leray spectral sequence has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l0581905.png" /></td> </tr></table>

and its limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l0581906.png" /> is the bigraded group associated with a filtration of the graded group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l0581907.png" />. The construction of the Leray spectral sequence can be generalized to cohomology with support in specified families. In the case of locally compact spaces and cohomology with compact support, the Leray spectral sequence was constructed by J. Leray in 1946 (see [[#References|[1]]], [[#References|[2]]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l0581908.png" /> is the constant sheaf corresponding to an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l0581909.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l05819010.png" /> is the projection of the locally trivial fibre bundle with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l05819011.png" /> and the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l05819012.png" /> is locally contractible, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l05819013.png" /> are locally constant sheaves and the term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l05819014.png" /> takes a particularly simple form.

The condition of local contractibility can be replaced by other topological conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l05819015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l05819016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l05819017.png" /> (for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l05819018.png" /> is locally compact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058190/l05819019.png" /> is compact).

Using singular cohomology, for any Serre fibration with path-connected fibres one can construct an analogue of the Leray spectral sequence that has all the properties listed above of the Leray spectral sequence of a locally trivial fibre bundle (the Serre spectral sequence). There is an analogous spectral sequence in singular homology.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Leray,   "L'anneau spectral et l'anneau fibré d'homologie d'un espace localement compact et d'une application continue"  ''J. Math. Pures Appl.'' , '''29'''  (1950)  pp. 1–139</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Leray,   "L'homologie d'un espace fibré dont la fibre est connexe"  ''J. Math. Pures Appl.'' , '''29'''  (1950)  pp. 169–213</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Godement,   "Topologie algébrique et théorie des faisceaux" , Hermann  (1958)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.-T. Hu,   "Homotopy theory" , Acad. Press  (1959)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.W. Whitehead,   "Elements of homotopy theory" , Springer  (1978)  pp. 228</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.P. Serre,   "Homologie singulière des espaces fibrés. Applications"  ''Ann. of Math. (2)'' , '''54'''  (1951)  pp. 425–505</TD></TR></table>