Leray spectral sequence

spectral sequence of a continuous mapping

A spectral sequence connecting the cohomology with values in a sheaf of Abelian groups $ {\mathcal F} $ on a topological space $ H $ with the cohomology of its direct images $ f _ {q} ( {\mathcal F} ) $ under a continuous mapping $ f : X \rightarrow Y $. More precisely, the second term of the Leray spectral sequence has the form

$$ E _ {2} ^ {p , q } = H ^ {p} ( Y , f _ {q} ( {\mathcal F} ) ) , $$

and its limit $ E _ \infty $ is the bigraded group associated with a filtration of the graded group $ H ^ {*} ( X , {\mathcal F} ) $. 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 [1], [2]).

If $ {\mathcal F} = A $ is the constant sheaf corresponding to an Abelian group $ A $, $ f $ is the projection of the locally trivial fibre bundle with fibre $ F $ and the space $ Y $ is locally contractible, then the $ f _ {q} ( {\mathcal F} ) $ are locally constant sheaves and the term $ E _ {2} $ takes a particularly simple form.

The condition of local contractibility can be replaced by other topological conditions on $ X $, $ Y $, $ F $( for example, $ Y $ is locally compact, $ F $ 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

[1]	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
[2]	J. Leray, "L'homologie d'un espace fibré dont la fibre est connexe" J. Math. Pures Appl. , 29 (1950) pp. 169–213
[3]	R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958)
[4]	S.-T. Hu, "Homotopy theory" , Acad. Press (1959)

References

[a1]	G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) pp. 228
[a2]	J.P. Serre, "Homologie singulière des espaces fibrés. Applications" Ann. of Math. (2) , 54 (1951) pp. 425–505