How to compute relative homology group

Asked Jan 12 '18 at 12:33

Active Jan 12 '18 at 12:33

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How to compute relative homology group $H_n(SO(n),SO(n)-E_n)$

I try to use the long exact sequence as follows $$ ...H_n(SO(n)-E_n)\rightarrow H_n(SO(n))\rightarrow H_n(SO(n),SO(n)-E_n) \rightarrow H_{n-1}(SO(n)-E_n)...$$
I find difficulty both in compute $H_n(SO(n))$ and $H_n(SO(n)-E_n)$

asked Jan 12 '18 at 12:33

Bruce

1

What is $E_n$ ? – Ahr Jan 12 '18 at 12:36
the Unitary matrix – Bruce Jan 12 '18 at 12:39
You mean the identity? – Ahr Jan 12 '18 at 12:40
yes,you are right – Bruce Jan 12 '18 at 12:40
Maybe you can try by excising? – Ahr Jan 12 '18 at 12:43
could you show me more – Bruce Jan 12 '18 at 12:43
$SO(n)$ is a manifold, the identity has a neighborhhod isomorphic to a ball in $\mathbb{R}^d$, where $d$ is the dimension of $SO(n)$ (easy to compute). Let $B$ be the complement of this neighborhood. Apply excision to the pairs $(SO(n), SO(n)-E_n)$ and $(SO(n)-B, (SO(n)-E_n)-B)\simeq(\mathbb{B}^d, \mathbb{B}^d-{center})$ – Ahr Jan 12 '18 at 12:50
could you help me show why $(SO(n)-B, (SO(n)-E_n)-B)\simeq(\mathbb{B}^d, \mathbb{B}^d-{center})$ – Bruce Jan 12 '18 at 13:05
It's simply how we chose $B$. It is the complement of neighborhood isomorphic to a ball. – Ahr Jan 12 '18 at 13:08

How to compute relative homology group

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