Connecting homomorphism question

[panzervlad]

For the cohomology of a n-sphere, I am having difficulty explicitly showing that the connecting homomorphism from (n-1) deRham group of U∩V which is isomorphic to (n - 1) sphere (which is simply R) to the n-deRham group of n-sphere is surjective!

On the exact long sequence, I have showed that i*-j* leading up to (n-1) deRham group of U∩V from the trivial deRham group of the direct sum U,V open covers of the sphere iso. to Rn is injective.

By definition of long sequence, (n-1) deRham group of U∩V is exact.

Now I am missing to explicitly show connecting homomorphism is surjective that way I can use the isomorphism theorems to show that

n-deRham group of n-sphere = R \ I am i*-j* where this image is trivial leaving us that this group is isomorphic to R.

I solved this problem by just explicitly making a homomorphism from each group which has an inverse, both right and left, hence it is isomorphic. But I'm having trouble using the isomorphism theorems, zigzag lemma to prove the isomorphism. Thanks

 

[lavinia]

For the cohomology of a n-sphere, I am having difficulty explicitly showing that the connecting homomorphism from (n-1) deRham group of U∩V which is isomorphic to (n - 1) sphere (which is simply R) to the n-deRham group of n-sphere is surjective!

On the exact long sequence, I have showed that i*-j* leading up to (n-1) deRham group of U∩V from the trivial deRham group of the direct sum U,V open covers of the sphere iso. to Rn is injective.

By definition of long sequence, (n-1) deRham group of U∩V is exact.

Now I am missing to explicitly show connecting homomorphism is surjective that way I can use the isomorphism theorems to show that

n-deRham group of n-sphere = R \ I am i*-j* where this image is trivial leaving us that this group is isomorphic to R.

I solved this problem by just explicitly making a homomorphism from each group which has an inverse, both right and left, hence it is isomorphic. But I'm having trouble using the isomorphism theorems, zigzag lemma to prove the isomorphism. Thanks

The cohomology of U and V are both zero except in dimension zero. Thus the connecting homomorphism is surjective.

 

[Greg Bernhardt]

Thank you for sharing your difficulty with the connecting homomorphism in the cohomology of an n-sphere. It can be a tricky concept to grasp, but I will try my best to explain it in a way that may help you.

First, let's review the definition of the connecting homomorphism. In the long exact sequence of cohomology groups, the connecting homomorphism is the map that connects the cohomology group of U∩V to the cohomology group of V. In other words, it takes an element in the (n-1) deRham group of U∩V and maps it to an element in the n-deRham group of V.

Now, in order to show that the connecting homomorphism is surjective, we need to show that for any element in the n-deRham group of V, there exists an element in the (n-1) deRham group of U∩V that maps to it under the connecting homomorphism. This means that every element in the n-deRham group of V has a preimage in the (n-1) deRham group of U∩V.

One way to do this is by constructing an explicit map from the n-deRham group of V to the (n-1) deRham group of U∩V. This is what you have done by creating a homomorphism with an inverse. However, as you mentioned, we can also use the isomorphism theorems and the zigzag lemma to show this.

The zigzag lemma states that if we have two exact sequences that are connected by a series of homomorphisms, then the induced maps between the corresponding cohomology groups are also exact. In our case, we have the exact sequence of the (n-1) deRham group of U∩V, the connecting homomorphism, and the n-deRham group of V. This is connected to the trivial deRham group of the direct sum of U and V, which is isomorphic to Rn, by the maps i* and j*. So, by the zigzag lemma, we know that the induced map between the (n-1) deRham group of U∩V and the n-deRham group of V is also exact.

Now, since we have already shown that the (n-1)