19032
Lecture
WiSe 13/14: Group Cohomology
Pavle Blagojevic
Comments
BMS Advanced Course: Cohomology of groups
The cohomology of groups is a specialized topic that share flavors of algebra, geometry and topology. It first appeared in the work of Schur related to a question of group extensions while in the work of Hurewicz it comes in connection with aspherical spaces.
In this course we will introduce and develop both algebraic and topological view points on the group cohomology. While developing the general framework of homological algebra related to Tor and Ext functors we will use direct definitions, examples and explicit calculations to shed light to the definition and basic notions related to group cohomology. The equivalence of algebraic and topological approach will be discussed and used in study of group cohomology of explicit examples, product structures in group cohomology as well as relationships of group cohomology between group and its subgroups.
The highlight of the course should be the study the cohomology of wreath products and use of the developed material on the following applications from equivariant topology and discrete geometry.
1. An application of group cohomology on Borsuk--Ulam type theorems
The famous theorem of Borsuk and Ulam from 1933 states that a Z/2-equivariant map between free Z/2 spheres exists if and only if dimension of the domain sphere is less then or equal to the dimension of the codomain sphere. The variety of applications of Borsuk--Ulam theorem, like Ham Sandwich theorem, Kneser's theorem, Van Kampen--Flores theorem and many others, motivate search for similar statement that would apply to spaces beyond spheres and groups more complex then Z/2.
A classical tool that provides necessary conditions for the existence of G-equivariant maps between G-spaces is the Fadell--Husseini index. Let G be a finite group, X be a G-space and R a commutative ring with unit. Consider the G-eqivariant projection map p: X--->pt. The Fadell--Husseini index of the space X is the kernel ideal of the homomorphism induced by p in the equivariant cohomology Since the G-equivariant cohomology of the point coincides with the cohomology of the group G, the Fadell--Husseini index of a G-space is an ideal in the cohomology of group G.
The main property of the Fadell--Huseini index, that provides a necessary condition for the existence of an equivariant map, is that: if there is a G-equivariant map X--->Y, then the index of X has to contain the index of Y.
2. An application of group cohomology to discrete geometry
An isometry of a d-dimensional Euclidean space is distance preserving self map. Given a group of such isometries together with its action on a Euclidean space, one can associate a fundamental domain with it. This is just a simple connected set that contains exactly one point from each orbit. It tiles a Euclidean space in a very symmetric manner via the group action.
Particularly interesting are groups of isometries with a bounded fundamental domain. These groups are called space groups. Bieberbach proved that every space group G contains a subgroup isometric to Z^d, and that the quotient group P:=G/Z^d is finite. In other words, there exists a short exact sequence of groups:
0---> Z---> G---> P---> 1.
Amazingly, the converse is true as well: Giving a finite group P, every group G that fits in a short exact sequence of the above type is isomorphic to a space group. But such groups are so-called extensions of P by Z^d which are fully captured by the second cohomology group of P with coefficients in Z^d. close
The cohomology of groups is a specialized topic that share flavors of algebra, geometry and topology. It first appeared in the work of Schur related to a question of group extensions while in the work of Hurewicz it comes in connection with aspherical spaces.
In this course we will introduce and develop both algebraic and topological view points on the group cohomology. While developing the general framework of homological algebra related to Tor and Ext functors we will use direct definitions, examples and explicit calculations to shed light to the definition and basic notions related to group cohomology. The equivalence of algebraic and topological approach will be discussed and used in study of group cohomology of explicit examples, product structures in group cohomology as well as relationships of group cohomology between group and its subgroups.
The highlight of the course should be the study the cohomology of wreath products and use of the developed material on the following applications from equivariant topology and discrete geometry.
1. An application of group cohomology on Borsuk--Ulam type theorems
The famous theorem of Borsuk and Ulam from 1933 states that a Z/2-equivariant map between free Z/2 spheres exists if and only if dimension of the domain sphere is less then or equal to the dimension of the codomain sphere. The variety of applications of Borsuk--Ulam theorem, like Ham Sandwich theorem, Kneser's theorem, Van Kampen--Flores theorem and many others, motivate search for similar statement that would apply to spaces beyond spheres and groups more complex then Z/2.
A classical tool that provides necessary conditions for the existence of G-equivariant maps between G-spaces is the Fadell--Husseini index. Let G be a finite group, X be a G-space and R a commutative ring with unit. Consider the G-eqivariant projection map p: X--->pt. The Fadell--Husseini index of the space X is the kernel ideal of the homomorphism induced by p in the equivariant cohomology Since the G-equivariant cohomology of the point coincides with the cohomology of the group G, the Fadell--Husseini index of a G-space is an ideal in the cohomology of group G.
The main property of the Fadell--Huseini index, that provides a necessary condition for the existence of an equivariant map, is that: if there is a G-equivariant map X--->Y, then the index of X has to contain the index of Y.
2. An application of group cohomology to discrete geometry
An isometry of a d-dimensional Euclidean space is distance preserving self map. Given a group of such isometries together with its action on a Euclidean space, one can associate a fundamental domain with it. This is just a simple connected set that contains exactly one point from each orbit. It tiles a Euclidean space in a very symmetric manner via the group action.
Particularly interesting are groups of isometries with a bounded fundamental domain. These groups are called space groups. Bieberbach proved that every space group G contains a subgroup isometric to Z^d, and that the quotient group P:=G/Z^d is finite. In other words, there exists a short exact sequence of groups:
0---> Z---> G---> P---> 1.
Amazingly, the converse is true as well: Giving a finite group P, every group G that fits in a short exact sequence of the above type is isomorphic to a space group. But such groups are so-called extensions of P by Z^d which are fully captured by the second cohomology group of P with coefficients in Z^d. close
Suggested reading
Literature:
(A) A. Adem, R. J. Milgram, Cohomology of Finite Groups, Second Edition, Grundlehren der mathematischen Wissenschaften 309, Springer-Verlag, Berlin, 2004.
(B) K. S. Brown, Cohomology of Groups, Graduate Texts in Math. 87, Springer-Verlag, New York, Berlin, 1982.
(C) L. Evens, The Cohomology of Groups, Oxford Mathematical Monographs, Oxford University Press, New York, 1991. xii+159 pp.
(D) M. S. Osborne, Basic homological algebra, Graduate Texts in Mathematics, 196. Springer-Verlag, New York, 2000. x+395 pp. close
(A) A. Adem, R. J. Milgram, Cohomology of Finite Groups, Second Edition, Grundlehren der mathematischen Wissenschaften 309, Springer-Verlag, Berlin, 2004.
(B) K. S. Brown, Cohomology of Groups, Graduate Texts in Math. 87, Springer-Verlag, New York, Berlin, 1982.
(C) L. Evens, The Cohomology of Groups, Oxford Mathematical Monographs, Oxford University Press, New York, 1991. xii+159 pp.
(D) M. S. Osborne, Basic homological algebra, Graduate Texts in Mathematics, 196. Springer-Verlag, New York, 2000. x+395 pp. close
16 Class schedule
Regular appointments
Thu, 2013-10-17 10:00 - 12:00
Thu, 2013-10-24 10:00 - 12:00
Thu, 2013-10-31 10:00 - 12:00
Thu, 2013-11-07 10:00 - 12:00
Thu, 2013-11-14 10:00 - 12:00
Thu, 2013-11-21 10:00 - 12:00
Thu, 2013-11-28 10:00 - 12:00
Thu, 2013-12-05 10:00 - 12:00
Thu, 2013-12-12 10:00 - 12:00
Thu, 2013-12-19 10:00 - 12:00
Thu, 2014-01-09 10:00 - 12:00
Thu, 2014-01-16 10:00 - 12:00
Thu, 2014-01-23 10:00 - 12:00
Thu, 2014-01-30 10:00 - 12:00
Thu, 2014-02-06 10:00 - 12:00
Thu, 2014-02-13 10:00 - 12:00