Group Cohomology: Borel's Finite & Lie Group Cases

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In summary, Dijkgraaf and Witten's paper "Topological Gauge Theory and Group Cohomology" discusses the properties of cohomology for compact Lie groups and finite groups. For compact Lie groups, it is proven that all odd cohomology vanishes, while for finite groups, all cohomology is finite. This is due to the fact that for a discrete group, the Eilenberg-Maclane space K(G,1) can be used to determine information about the cohomology of the group. This can be shown using the Leray-Serre spectral sequence, which reveals that the cohomology of BG is a polynomial algebra with generators of even degree.
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electroweak
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In Dijkgraaf and Witten's paper "Topological Gauge Theory and Group Cohomology" it is claimed that...

For a compact Lie group we have the very useful property, due to Borel, that with real coefficients all odd cohomology vanishes: H^odd(BG; R) = 0. So the odd cohomology (and homology) consists completely of torsion. For finite groups an even stronger result holds: all cohomology is finite: H(BG; R) = 0.

Why are either of these statements (the Lie group case or the finite case) true?
 
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Note: for discrete G, BG is the Eilenberg-Maclane space K(G,1). Perhaps this will help with the finite case.
 
  • #3
One argument that works: Notice there is a fibration G→EG→BG so using the Leray-Serre spectral sequence given information about H*(G;Q) and H*(EG;Q) one can hopefully determine something about H*(BG;Q). Since EG is contractible this gives us one piece of the puzzle and since G has the homotopy type of a finite CW-complex with some difficulty one can actually show H*(G;Q) is an exterior algebra with generators of odd degree. Using our spectral sequence it then turns out H*(BG;Q) is a polynomial algebra with generators of even degree and the desired result follows. This might be overkill, but it works at least!

Edit: I wrote the above for coefficients in Q, but the same argument should work for R. Essentially the important fact is that over Q one can ignore torsion and the same obviously holds for R.
 

FAQ: Group Cohomology: Borel's Finite & Lie Group Cases

1. What is group cohomology?

Group cohomology is a branch of mathematics that studies the algebraic structure of groups and their associated cohomology. It involves investigating the properties of cohomology groups, which are mathematical objects that describe the ways in which a group can act on other mathematical objects.

2. Who is Borel and what is his contribution to group cohomology?

Armand Borel was a Swiss mathematician who made significant contributions to the study of group cohomology. In particular, he developed a method for computing the cohomology of finite groups and Lie groups, which are important types of groups in mathematics.

3. What are the finite and Lie group cases in Borel's work on group cohomology?

The finite group case refers to Borel's work on computing the cohomology of finite groups, which are groups with a finite number of elements. The Lie group case, on the other hand, deals with computing the cohomology of Lie groups, which are groups that are continuously varying and play a key role in many areas of mathematics and physics.

4. What are some applications of group cohomology?

Group cohomology has many applications in mathematics and beyond. In algebraic topology, it is used to study topological spaces and their symmetries. In algebraic geometry, it helps to understand the geometry of algebraic varieties. In number theory, it has connections to the study of class field theory. Furthermore, group cohomology has also found applications in physics, particularly in the study of gauge theories and topological phases of matter.

5. What are some open problems in group cohomology?

Despite the significant progress made in the study of group cohomology, there are still many open problems that researchers are actively working on. These include finding explicit formulas for the cohomology of finite and Lie groups, developing new techniques for computing cohomology groups, and understanding the connections between group cohomology and other areas of mathematics. Additionally, there are ongoing efforts to generalize the theory of group cohomology to more general types of groups and spaces.

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