Wikipedia:Reference desk/Archives/Mathematics/2011 April 24

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April 24

Group Cohomology

I'm familiar with topological homology and cohomology, but I'm trying to figure out group cohomology and running into some trouble. In group cohomology there are several definitions given for Hⁿ(G,M) where M is a G-module. Then in the section Group_cohomology#Connections_with_topological_cohomology_theories, they only discuss Hⁿ(G;M). It seems like these are not the same thing, but I can't really tell. How is Hⁿ(G;M) defined and how does it relate to the definitions earlier for Hⁿ(G,M)? If it's at all helpful, I'm particularly trying to understand H¹(P_n;ℚ) where P_n is the pure braid group on n strands. Thanks. Rckrone (talk) 01:35, 24 April 2011 (UTC)

You can define the group cohomology as the derived functors of the taking invariants functor. Taking invariants is the functor $\mathrm {Hom} _{\mathbb {Z} [G]}(\mathbb {Z} ,-)$, so we get the derived functors $\mathrm {Ext} _{\mathbb {Z} [G]}^{n}(\mathbb {Z} ,-)$ which define the group cohomology. This is tailored so that a short exact sequence of G-modules yields a corresponding long exact sequence with the connecting homomorphisms. Now you can also define group cohomology as the topological cohomology of the classifying space BG. However, this is not entirely straightforward in the case that your module M has a nontrivial G action: indeed, you have to consider a local coefficient system. If you work through the definitions, a local coefficient system on BG is the same thing as a G-module, and trivial G-modules correspond to constant local coefficient systems. So instead of considering the usual, say singular, cohomology, you have to instead consider some form of sheaf cohomology. But you get the same answer.

The two notations $H^{n}(G,M)$ and $H^{n}(G;M)$ are the same thing; it's just that algebraic topologists tend to use a semicolon for the coefficients and reserve the comma for relative cohomology, a convention which is not observed by everyone.

Hope that answers your questions! --Sam^Talk 14:25, 24 April 2011 (UTC)

I was mostly thrown off by the difference between the semicolon and the comma since, like you mentioned, in the topology I've seen the two are used to distinguish different meanings. That, combined with the fact that ℚ as a P_n-module seemed really mysterious to me, led me to think maybe they meant something different when they used the semicolon. (Now I see the action is supposed to be trivial.) Thanks for clarifying that the notations mean the same thing. A lot of the rest of your explanation is somewhat over my head, but I got the part I need for now, and will probably come back to this topic later. Rckrone (talk) 21:45, 24 April 2011 (UTC)

Clarification: homework question

Hi everyone,

A question I'm working on says "Show that a poset is complete if and only if it is directed-complete and has joins for all its finite subsets". To the best of my knowledge, directed-complete, complete and poset are just the standard definitions here. However, since this is a question from a while back, I have no idea what a 'join' is: does anyone know what this could refer to? It was never defined in the course I took.

Staytime101 (talk) 03:52, 24 April 2011 (UTC)

Join includes the entry:

• Join (mathematics), a least upper bound in lattice theory

PrimeHunter (talk) 04:25, 24 April 2011 (UTC)

Homeomorphism

Please give me example of topological spaces which are homeomorphic and a homeomorphism between themMathematics2011 (talk) 08:24, 24 April 2011 (UTC)

There are some examples in our article on homeomorphism. Gandalf61 (talk) 09:52, 24 April 2011 (UTC)

The arctangent (times a constant) is a homeomorphism between the real numbers and the interval (-1,1). Actually it's a diffeomorphism. Staecker (talk) 11:41, 24 April 2011 (UTC)

Working out loan APR from monthly cost

What formula should I use to work out a loan annual percentage rate, when I know the balance and the monthly cost? It's a bank account with a flat daily charge for an overdraft, instead of a percentage based interest rate. I think it will be poor value, but need to know the APR so I can compare other options. Thanks. —Preceding unsigned comment added by 78.105.229.165 (talk) 12:31, 24 April 2011 (UTC)

It's going to depend on how long you are borrowing the money for and at what rate you are paying it off. If you are borrowing for a year and you aren't paying off anything and just let the fees increase the balance then, at the start you will be £x in debt and at the end you will be £(x+12*f) (where f is the monthly fee) in debt, so that's an annual interest rate of ${\frac {x+12f}{x}}-1={\frac {12f}{x}}$. However, I suggest you make sure you have understood the offer correctly. Overdraft facilities usually have a percentage rate in addition to the fixed fee. --Tango (talk) 13:43, 24 April 2011 (UTC)