The “first” interesting compact Lie group: the special unitary group

This is a (Lie) group under matrix multiplication.

Let . Then , and . Equating these, we find the conditions

So we can write as

We find that . So .

linearly, so there is an action , lines in . This action is transitive, with stabiliser of any point . The coset space is . This is another way to construct the Hopf fibration.

If is a topological group, there exists a space which

with ; and

. These spaces are unique up to homotopy.

The Serre spectral sequence tells us that for the bundle

we have

Here, we have , and therefore

We have on the page:

We have:

The conclusion is that

Show that where . Read Hatcher’s unpublished chapter on spectral sequences for how the Serre spectral sequence interacts with products in cohomology.

Let

be a short exact sequence in an abelian category (e.g. ) with enough injectives (for all there exists injective with ). Then there exists a diagram

where:

The LemmaLemma implies this Proposition (Horseshoe)Proposition (Horseshoe) because if we have

then we can iteratively apply the LemmaLemma to to get

and so on.

Let

be a short exact sequence in an abelian category with enough injectives. Then there exists a diagram

where:

We can embed and into injective objects and respectively:

By injectivity of , there exists a map :

There is also a map via the composition . Take .

The vertical sequence

is exact by construction. is injective by the Snake Lemma. One can check that the diagram commutes. We now take , , to be the cokernels

The rows are now exact by construction.

We now do a diagram chase or use the Snake Lemma:

Let the diagram below have exact rows:

Then the dashed arrows form an exact sequence. If further is injective then is injective, and if is surjective, then is surjective.

Let

be a short exact sequence in an abelian category with enough injectives. Let be a left exact functor. Then there is a long exact sequence

We apply to the projective resolutions

to get

Here, we lose exactness on the left for all the rows, but the columns are still exact because all the objects are injective:

A short exact sequence of chain complexes induces a long exact sequence in (co)homology.

Where at , we get a long exact sequence