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Beginnings of Group Cohomology -RE-SONANCEI--~-P-Ffi-Mb-Er

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Beginnings of Group Cohomology -RE-SONANCEI--~-P-Ffi-Mb-Er GENERAL I ARTICLE Beginnings of Group Cohomology R Sridharan Cohomology theory is a powerful mathematical tool. This theory applied to topology is a part of algebraic topology, which associates algebraic invariants to topological spaces. In this article, we give a brief outline of group cohomology. 1. Introduction R Sridharan has been an Cohomology theory is a powerful mathematical tool. Adjunct Professor at the This theory applied to topology is a part of algebraic CMI, Chennai since topology, which associates algebraic invariants to topo­ retiring from TIFR, logical spaces. It can be applied equally fruitfully to Bombay in 2000 where he algebra to study the structure of algebraic objects like was a senior professor. He is also an INSA senior groups, associative algebras, etc. and this was initiated scientist now. His by Eilenberg-Mac Lane, Eckmann, Hochschild and oth­ scholarship permeates to ers. (It turns out that there is a unifying link between literature (English and these approaches, which was formalised for the first time Sanskrit) and philosophy as well. Many of the by H Cartan and S Eilenberg in their classic book H 0- algebraists in the country mological Algebra in 1956). Eilenberg and Mac Lane were his students. published in 1946, in the Annals of Mathematics, their first paper on the cohomology theory of groups. Al­ most at the same time, Eckmann also introduced the same theory. These authors were led to the cohomology Dedicated to my teacher theory of groups, with motivation from algebraic topol­ 5 Eilenberg ogy, an area in which they were specialists. Indeed, the first definition of cohomology of groups in the paper of Eilenberg and Mac Lane is based on the so-called 'ho­ mogeneous cochains' and this approach is really mod­ elled on the usual definitions in algebraic topology. On the other hand, there is another, equivalent approach to the cohomology theory of groups via the so-called 'non Keywords Homological algebra, factor homogeneous cochains', which was given by Eilenberg­ sets, group extensions, Hil­ Mac Lane and Eckmann and is motivated by the work bert's theorem 90, cocycles, of some earlier authors on group theory. crossed homomorphisms. -RE-S-O-N-A-N-C-E-I--~-p-ffi-m-b-er--2-0-05----------~~----------------------------~ GENERAL I ARTICLE The first definition of In this article, we give a brief outline of the work of cohomology of mathematicians like D Hilbert, E Noether, 0 Schreier, groups in the paper R Baer (and others) in group theory and note that their of Eilenberg and work already contained motivating material for defin­ Mac Lane is based ing and studying the cohomology theory of groups in general. For instance, the 90 for the on the so-called Hilbert Theorem class of 'cyclic Galois extensions' and its generalization 'homogeneous by Emmy Noether for 'all finite Galois extensions' is re­ cochains' and this ally to show that a certain first cohomology group, of approach is really the 'Galois group' of the extension is trivial. The work modelled on the of Schreier on group extensions with abelian kernels al­ usual definitions in ready contained the definition of the second cohomology algebraic topology. of groups. The work of R Baer on the construction of obstructions to the existence of group extensions with 'non abelian kernels' contained implicitly the definition of the third cohomology. We shall discuss some of these results in as elementary a manner as is possible, assum­ ing only rudimentary knowledge of group theory. 2. Group Actions and the First Cohomology The notion of a group has played a fundamental role in mathematics. For instance, Felix Klein in his famous "Erlangen Programme", emphasized the role of groups in geometry by remarking that geometry can be thought of as the study of properties of spaces invariant under the action of certain groups. A very fruitful notion in this direction is that of a group G acting on another group A. We start with a definition: We say that a group G acts on or operates on a group A, if there is a homomorphism of G in to the group Aut(A) of all automorphisms of A. (We recall that an automorphism of a group A is a bijective map of A onto A which preserves the group operation of A. The set of all automorphisms of a group A is again a group under the usual composition of maps as the group operation). -3S----------------------------VV\Afvv---------------------------­ RESONANCE I September 2005 GENERAL I ARTICLE Writing the definition of G action on A in long hand, Felix Klein in his this means that for each 9 E G, and a E A, there exists famous "Erlangen an element ga E A such that the following conditions Programme", are satisfied: emphasized the role For gl,g2 E G, and a E A, we have (glg2)a = gl(g2a). of groups in la = a, and g( a.b) = ga.gb for a, b E A. geometry by remarking that If, in particular, A is an abelian group, we shall denote geometry can be the group operation in A by '+' instead of '.', the iden­ thought of as the tity element by 0 and the inverse of a by -a. study of properties of If a group G acts on an abelian group A, we call A a spaces invariant G-module. We shall now assume that A is a G-module. under the action of It is natural to ask for elements of A which are fixed certain groups. A very points for this action. We recall that an element a of A fruitful notion in this is a fixed point if G fixes it or what is the same, g.a = a direction is that of a for all 9 E G. Now, 0 is clearly a fixed point. If a and b group G acting on are fixed points, then it is immediate from our definition another group A. that a-b is also fixed, since g.(a-b) = g.a-g.b = a-b. Hence the set of fixed points of A is a subgroup of A. As we have seen, 0 E A is a fixed point and we would like to know whether there are 'nontrivial fixed points'. We shall formulate the question in a slightly different way as follows: Let B be an abelian group on which a group G acts and let A be a subgroup of B which is stable under this action of G; that is for all a E A, and for all 9 E G, we have ga E A. It can then be seen immediately that G acts also on A which we call the action 'induced' from that of G on B. The group G then acts also on the quotient group B / A, if we set for any 9 E G and b E B/A, gb = gb, for any lift b of b E B. Suppose that there is a non zero fixed point of B / A for this action of G. We wish to find out the obstruction to 'lifting' this fixed point of B / A to a fixed point for the action of G on B. To do this, let us take a fixed -RE-S-O-NA-N-C-E-I--~-~-em-oo--r-2-0-05------~-~--------------------------3-9 GENERAL I ARTICLE point b E B / A and choose an arbitrary element b in B whose coset modulo A is b. Consider, for any 9 E G the element gb - b. We note that the coset of gb - b modulo A is the coset 0, since gb - b = gb - b = 0, so that gb - b E" A. Obviously, these elements are zero for all 9 E G if and only if b E B is a fixed point of G. Thus, the map G~A Ib:g~gb-b measures the obstruction to lifting b to a fixed point of G. We also note that the map Ib : G ~ A satisfies the equation Ib(g.g') = g.lb(g') + Ib(g) for g, g' E G. For any abelian group A on which G acts, a map I : G ~ A which satisfies I(g.g') = gl(g') + I(g), for g,g' E G is called, classically,;a crossed homomorphism or 1 - cocycle of G with values in A. Thus, given a fixed point for G in B/A, we get a 1-cocycle I : G ~ A. A 1- co cycle I : G ~ A is called a 1-coboundary if there exists an a E A such that I(g) = g.a - a, Vg E G. We note that any 1-coboundary is a 1-cocycle, since for , , ( , g,g E G,g.g a - a =g 9 a - a) + ga - a. It is easily seen that if I, I' : G ~ A are 1-cocycles then I + I' defined by (I + I')(g) = I(g) + I' (g) is again a 1-cocycle. The 1-cocycles from G ~ A form an abelian group (denoted by Zl ( G, A)) under this binary opera­ tion. The map 0 : G ~ A, defined by O(g) = 0 is the identity element and for any I E Zl ( G, A), - I defined by (-I) (g) = - I (g) is the inverse of I. It is immedi­ ately verified that the subset Bl(G, A) of 1-coboundaries of Zl ( G, A) is, in fact, a subgroup of Zl ( G, A). The quo­ tient group Hl(G, A) = Zl(G, A)/Bl(G, A) is called the first cohomology group of G with coefficients in A.
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