CHAPTER 1 The Leray-Serre spectral sequence Our main tool from algebraic topology is the Leray-Serre spectral sequence. It relates the cohomology rings of the fibre and the base of a fibration with the cohomology of the total space. Although spectral sequences are standard devices in topology, they tend to be somewhat intimidating to non-specialists. The aim of this chapter is to give a basic introduction to the relevant notions and techniques. Let us consider a specific example, the Leray-Serre spectral sequence with field coefficients. Let K be a field, and let .F, B be topological spaces. The Kiinneth Theorem asserts that the K-cohomology of the product E = F x B is given by U.k(E K)^ 0 HX^ /0®HJ'(£ K). 2+7 = fc It is convenient to visualize the /^-modules E M = VLl{F\K) 0 H3(B K) as dis- tributed on the lattice Z2 CM2: attached to the point (i,j) is the vector space E u . Then ~H.k(E K) is obtained by adding up all vector spaces along the line i+j = k. Now suppose that E is not a product, but the total space of a (jFf-simple) fibre bundle F E B. Then the cohomology of E is 'smaller' than the cohomology of the trivial bundle F x B. The recipe to obtain the K-modules Hk(E K) is as follows. Start with the collection of if-modules E ^ = H%(F K) g W(B K) as before. There exists a collection of maps, the differentials, denoted c^ : E1^3 E%2 ,3 ~ . These differentials should be visualized as arrows going from (i, j) to (* + 2 , j - l ) . E ^ They satisfy the relation ^ 2 ° ^ 2 = 0, so we can take their cohomology (kernel mod image) at each point (i, j). Call the resulting K-module E^3. Note that this is a quotient of a submodule of E^'*7, so its dimension is smaller. Again, these K-modules should be viewed as distributed in the plane. There is another differential i3, this time from (i,j) to (i + 3, j 2). Now this process is iterated ad infinitum. The arrows ^2,^3,^4,. become longer, and their slope approaches 1. In the limit, one obtains a collection of K-modules denoted E ^ . Similarly as in the Kiinneth Theorem one has 2+7 = fc
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