The algebra is kept fairly simple here, and most of the discussion is about computing the homology of a double complex, and constructing the. Spectral sequence of lyndonhochschildserre for a group extension 1 h. We usually draw the rth stage of a spectral sequence in a tabular format with p increasing. The adams spectral sequence lecture 8 april 27, 2010 recall that our goal this week is to prove the following result. If a group g operates on a topological space x, then one can define equivariant homology and cohomology groups, which can be thought of heuristically as a mixture of hg and hx. Some problems in algebraic topology and the theory of fibre bundles, ann. Finally we use the technique of spectral sequence to compute the cohomology of some classes of eilenbergmaclane spaces, and apply it to the calculation 5s3. There is a map from hochschild to quillen cohomology, and a spectral sequence.
Chapter 1 introduction one of the main problems, if not \the problem of topology, is to understand when two. Short exact sequence and cohomology group stack exchange. Computing group cohomology rings from the lyndonhochschildserre spectral sequence graham ellis mathematics department, national university of ireland, galway, ireland paul smith1 mathematics department, national university of ireland, galway, ireland abstract we describe a method for computing presentations of cohomology rings of small. The aim of the book is to present a precise and comprehensive introduction to the basic theory of derived functors, with an emphasis on sheaf cohomology and spectral sequences. For example, in the serre spectral sequence for homology the first few pages. In a footnote, he says relative homology groups of groups have been considered by k. In mathematics, especially in the fields of group cohomology, homological algebra and number theory, the lyndon spectral sequence or hochschildserre spectral sequence is a spectral sequence relating the group cohomology of a normal subgroup n and the quotient group gn to the cohomology of the total group g. Brauer group as the brauer group of kvector spaces. In mathematics more specifically, in homological algebra, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. It keeps the treatment as simple as possible, aiming at the same time to provide a number of examples, mainly from sheaf.
The spectral sequences produced in this paper are those associated to the rst quadrant double complex 1. Introduction a space x having only one nontrivial homotopy group nx. Similarly, the mayervietoris sequence for a decomposition xab generalizes to a spectral sequence associated to a cover of xby any number of sets. The spectral sequence is named after roger lyndon, gerhard hochschild, and jean. The spectral sequence which we have been discussing for the group g. For example, stefan produces his spectral sequence as a grothendieck spectral sequence, whereas those of guccione and guccione are derived from ltrations on a certain rather large complex. The forgetful functor from amodules to graded vector spaces has a left.
Spectral sequences via examples the graduate journal of. The rst author generalized this theorem to nontrivial coe cient systems, in the form of fimodules over a eld, though one now obtains periodicity of the cohomology. Group cohomology lecture notes university of washington. Spectral sequence an overview sciencedirect topics. The serre spectral sequence is a way to compute the cohomology of y. In particular we show how a spectral sequence induced by a filtration on the complex provides a very natural and useful method to study recursively. Cohomology of topological groups with applications to the. A spectral sequence binsun abstract this is the second paper in a series of three papers aiming to study cohomology of group theoretic dehn. We say that a spectral sequence is first quadrant if all abelian groups.
When ais zngraded for some n2n and j is homogeneous. The cohomology spectral sequence additional topics that it would be nice to include. Guichardet shows that the hochschild cohomology of a crossed product can be. This formal group law is classi ed by an isomorphism of commutative rings l mu. Under the previous assumptions we have the following spectral sequence. Although it seems that we should calculate it by induction, i dont know how to continue. How do i get the correct long exact sequence for relative. Introduction we have been introduced to the idea of homology, which derives from a chain complex of. Moreover, someone could use any other method as long as it work. Any of the standard texts on homological algebra or cohomology of groups contains an account of the theory of spectral. X 1 x with x s ix i, and then there are associated homology and cohomology spectral sequences. It is in this way, for example, that quillen proved his theorem.
Josh swanson may 30, 2014 summary the serre spectral sequence e ectively computes cohomology rings for numerous classical spaces. We would like to understand the group cohomology by understanding the group cohomology of simpler groups. Solutions to exercises from kenneth browns cohomology of. Group cohomology, ss 2019 fakultat fur mathematik uni. This equivariant theory provides a powerful tool for extracting homological information about g from the action of g on x. Then, we try to use spectral sequence and lerays theorem to calculate. Analogous to group representations, group cohomology looks at the group actions of a group g in an associated gmodule m to elucidate the properties of the group. Derived functors and sheaf cohomology contemporary. The 1st goal of this paper is to build a formalism to construct several spectral sequences that, on the one hand, recover and extend the mayervietoris spectral sequence see also 37, theorem 2.
The associated mod 2 cohomology spectral sequence collapses to give the formula. The adamsnovikov spectral sequence and the homotopy. A generalization of the lyndonhochschildserre spectral sequence for polynomial cohomology a dissertation submitted to the faculty of purdue university by bobby william ramsey, jr. Cohomology of topological groups with applications to the weil group m. One of the reasons is that the rst homotopy group is generally nonabelian, so harder to study. A primer on spectral sequences university of chicago. Short exact sequence of chain complexes one concept that we will use repeatedly when discussing cohomology will be the idea that. In 1904 schur studied a group isomorphic to h2g,z, and this group. In 1904 schur studied a group isomorphic to h2g,z, and this group is known as the schur multiplier of g. The spectral sequence relating hg, z and rg now follows from a spectral sequence hx, zkx for any finite cwcomplex x i. Given a group gthere exists a connected cw complex xwhich is aspherical with. The universal complex orientation of the complex bordism spectrum mu determines a formal group law over. There are two standard approaches to spectral sequences. A gentle introduction to homology, cohomology, and sheaf cohomology.
With this great increase in generality comes, not surprisingly, a corresponding increase in complexity. A gentle introduction to homology, cohomology, and sheaf. As a second year graduate textbook, cohomology of groups introduces students to cohomology theory involving a rich interplay between algebra and topology with a minimum of prerequisites. A spectral sequence is a piece of terminol ogy used by algebraists to intimidate other mathematicians.
In group cohomology, this sequence is called the inflationrestriction sequence. As a sample application, we prove the gysin sequence and use it to compute h cpn. However, this sequence can be extended only for a couple more terms, but. Ken browns book, cohomology of groups also gives a fairly readable introduction to spectral sequences. A spectral sequence is a sequence of complexes which, by taking successive cohomologies converges to the cohomology of a given complex or to some graded version thereof. The spectral sequence whose existence is asserted in the above theorem is an example of a.