Connecting homomorphism question

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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 \ Im 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
 

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lavinia
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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 \ Im 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.
 

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