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Jul
25
accepted With field coefficients homology and cohomology coincide
Jul
25
comment With field coefficients homology and cohomology coincide
Ok i see what you mean now. you are saying that a finite dimensional vector space is isomorphic to its dual space. Here our vector space is $H_n(X,F)$ which when finite dimensional becomes isomorphic to its dual space $hom_F(H_n(X,F),F)$. Thank you so much!!
Jul
25
comment With field coefficients homology and cohomology coincide
Also i don't understand your sentence "and dual of a (finite dimensional ) vector space is isomorphic to itself" could you please explain what do you mean ?
Jul
25
comment With field coefficients homology and cohomology coincide
Also $H_n(X;F)$ is an $F-$ module so when you talk about finite rank, you mean finite dimension as an $F-$vector space ?
Jul
25
comment With field coefficients homology and cohomology coincide
I'm writing so to say that it is the set of $F-$module homomorphisms from the $F-$module $H_n(X;F)$ to the $F-$module $F$. This is to differentiate it from group homomorphism or any other less structure so we need homomorphisms that respect $F-$module structure. If this is what you mean buy your simpler notation by putting $F$ instead of $F-$modules ?
Jul
25
asked With field coefficients homology and cohomology coincide
Jul
25
comment Homology and Reduced homology coincide on non trivial pair.
I think you mean $C_0(A)\stackrel {\epsilon_A }{\rightarrow}\mathbb Z$ is surjective from the complex associated to $A$ and $C_0(X)\stackrel {\epsilon_X }{\rightarrow}\mathbb Z$ is surjective from the complex associated to $X$.. but $\epsilon_A$ is just the restriction homomorphism of $\epsilon_X$ hence the induced surjective homomorphism $\epsilon :C_0(X)/C_0(A)\stackrel{\epsilon}{\rightarrow} \mathbb Z$ must be zero.
Jul
25
comment Homology and Reduced homology coincide on non trivial pair.
why is the mapping $C_0(A)\stackrel {\epsilon }{\rightarrow}\mathbb Z$ surjective ? and what is $\epsilon $ here ?
Jul
25
comment Homology and Reduced homology coincide on non trivial pair.
@WillardZhan Hatcher is saying they are the same, I quoted the paragraph in my question. Could you please be more specific.. thanks!
Jul
25
asked Homology and Reduced homology coincide on non trivial pair.
Jul
21
accepted Deleting a contractible subspace is the same as deleting a point
Jul
21
comment Deleting a contractible subspace is the same as deleting a point
Yes I see thank you archipelago !!
Jul
21
comment Deleting a contractible subspace is the same as deleting a point
So you are saying there is no known result giving standard conditions on $X$ and $A$ under which removing $A$ from $X$ is the same as removing a point from $X$.
Jul
21
asked Deleting a contractible subspace is the same as deleting a point
Jul
12
awarded  Nice Question
May
31
awarded  Popular Question
May
30
awarded  Notable Question
May
27
accepted A morphism of principal bundles is an isomorphism.
May
26
asked A morphism of principal bundles is an isomorphism.
May
23
accepted Confusion between an element and its preimage