Compatibility of orientation classes

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Let $U$ be an oriented topological manifold of pure dimension $n$ and $K$ a compact of $U$. There is an orientation class $$or_U,K in H_n(U, U backslash K).$$ Let $V$ an open subset of $U$ such that $K subset V subset U.$ There is as well an orientation class $or_V,K in H_n(V,Vbackslash K).$ Now let us consider the canonical map $$i : H_n(V,Vbackslash K) to H_n(U,Ubackslash K).$$ Do we have $$i(or_V,K) = or_U,K?$$

algebraic-topology homology-cohomology orientation

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asked Jul 17 at 12:52

C. Dubussy

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Let $U$ be an oriented topological manifold of pure dimension $n$ and $K$ a compact of $U$. There is an orientation class $$or_U,K in H_n(U, U backslash K).$$ Let $V$ an open subset of $U$ such that $K subset V subset U.$ There is as well an orientation class $or_V,K in H_n(V,Vbackslash K).$ Now let us consider the canonical map $$i : H_n(V,Vbackslash K) to H_n(U,Ubackslash K).$$ Do we have $$i(or_V,K) = or_U,K?$$

algebraic-topology homology-cohomology orientation

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asked Jul 17 at 12:52

C. Dubussy

7,59631338

add a comment | 

up vote
1
down vote

favorite

up vote
1
down vote

favorite

Let $U$ be an oriented topological manifold of pure dimension $n$ and $K$ a compact of $U$. There is an orientation class $$or_U,K in H_n(U, U backslash K).$$ Let $V$ an open subset of $U$ such that $K subset V subset U.$ There is as well an orientation class $or_V,K in H_n(V,Vbackslash K).$ Now let us consider the canonical map $$i : H_n(V,Vbackslash K) to H_n(U,Ubackslash K).$$ Do we have $$i(or_V,K) = or_U,K?$$

algebraic-topology homology-cohomology orientation

share|cite|improve this question

asked Jul 17 at 12:52

C. Dubussy

7,59631338

Let $U$ be an oriented topological manifold of pure dimension $n$ and $K$ a compact of $U$. There is an orientation class $$or_U,K in H_n(U, U backslash K).$$ Let $V$ an open subset of $U$ such that $K subset V subset U.$ There is as well an orientation class $or_V,K in H_n(V,Vbackslash K).$ Now let us consider the canonical map $$i : H_n(V,Vbackslash K) to H_n(U,Ubackslash K).$$ Do we have $$i(or_V,K) = or_U,K?$$

algebraic-topology homology-cohomology orientation

share|cite|improve this question

asked Jul 17 at 12:52

C. Dubussy

7,59631338

share|cite|improve this question

share|cite|improve this question

asked Jul 17 at 12:52

C. Dubussy

7,59631338

asked Jul 17 at 12:52

C. Dubussy

7,59631338

asked Jul 17 at 12:52

C. Dubussy

7,59631338

7,59631338

add a comment | 

add a comment | 

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I believe the orientation class is characterized by the fact that it’s image under inclusion for each $pin U$,
$$H_n(U,U-K)rightarrow H_n(U,U-p)$$ is sent to the chosen orientation. If they disagreed this couldn’t happen as the fact that manifolds are normal and excision implies that $$H_n(V,V-p)rightarrow H_n(U,U-p)$$ is an isomorphism.

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answered Jul 18 at 13:16

Charlie Frohman

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
0
down vote

I believe the orientation class is characterized by the fact that it’s image under inclusion for each $pin U$,
$$H_n(U,U-K)rightarrow H_n(U,U-p)$$ is sent to the chosen orientation. If they disagreed this couldn’t happen as the fact that manifolds are normal and excision implies that $$H_n(V,V-p)rightarrow H_n(U,U-p)$$ is an isomorphism.

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answered Jul 18 at 13:16

Charlie Frohman

1,239611

add a comment | 

up vote
0
down vote

I believe the orientation class is characterized by the fact that it’s image under inclusion for each $pin U$,
$$H_n(U,U-K)rightarrow H_n(U,U-p)$$ is sent to the chosen orientation. If they disagreed this couldn’t happen as the fact that manifolds are normal and excision implies that $$H_n(V,V-p)rightarrow H_n(U,U-p)$$ is an isomorphism.

share|cite|improve this answer

answered Jul 18 at 13:16

Charlie Frohman

1,239611

add a comment | 

up vote
0
down vote

up vote
0
down vote

I believe the orientation class is characterized by the fact that it’s image under inclusion for each $pin U$,
$$H_n(U,U-K)rightarrow H_n(U,U-p)$$ is sent to the chosen orientation. If they disagreed this couldn’t happen as the fact that manifolds are normal and excision implies that $$H_n(V,V-p)rightarrow H_n(U,U-p)$$ is an isomorphism.

share|cite|improve this answer

answered Jul 18 at 13:16

Charlie Frohman

1,239611

I believe the orientation class is characterized by the fact that it’s image under inclusion for each $pin U$,
$$H_n(U,U-K)rightarrow H_n(U,U-p)$$ is sent to the chosen orientation. If they disagreed this couldn’t happen as the fact that manifolds are normal and excision implies that $$H_n(V,V-p)rightarrow H_n(U,U-p)$$ is an isomorphism.

share|cite|improve this answer

answered Jul 18 at 13:16

Charlie Frohman

1,239611

share|cite|improve this answer

share|cite|improve this answer

answered Jul 18 at 13:16

Charlie Frohman

1,239611

answered Jul 18 at 13:16

Charlie Frohman

1,239611

answered Jul 18 at 13:16

Charlie Frohman

1,239611

1,239611

add a comment | 

add a comment | 

 

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