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Sep
6
comment $H_1(\mathbb{R}, \mathbb{Q})$ is free abelian
thank you! But what about my computations? Is $H_0(\mathbb{Q}) = \mathbb{Z}^\mathbb{N}$?
Sep
6
revised $H_1(X,A) = 0 \iff H_1(A) \rightarrow H_1(X)$ surjective and $X_i$ contains no more than one path-component of $A$
Typo corrected.
Sep
5
asked $H_1(\mathbb{R}, \mathbb{Q})$ is free abelian
Sep
5
asked $H_1(X,A) = 0 \iff H_1(A) \rightarrow H_1(X)$ surjective and $X_i$ contains no more than one path-component of $A$
Sep
5
accepted $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$
Sep
5
comment $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$
Ah no it's needed to show that the long relative homological sequence is exact!
Sep
5
comment $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$
It's needed to define $\partial_\ast$ in the chain complex $C(X)/C(A)$?
Sep
5
comment $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$
What do you mean by "it defines the complex"? If I compute $H(X,A)$ from $\dots C_n(X)/C_n(A) \xrightarrow{\partial_n} C_{n-1}(X) / C_{n-1}(A) \dots$ then I'm not sure what I need $0 \rightarrow C_0(A) \rightarrow C_0(X) \rightarrow C_0(X)/C_0(A) \dots$ for.
Sep
5
comment $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$
Yeah, I just looked at it. But then why do they mention said sequence on Wikipedia in that context?
Sep
5
comment $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$
But that's what I proved in question 15 on page 114: if $H_n(X,A) = 0$ for all $n$ then $i_\ast$ is an isomorphism for all $n$!
Sep
5
comment $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$
If $i_\ast$ is an isomorphism $A$ has to meet all path components of $X$.
Sep
5
comment $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$
Thank you! I think I might have found something shorter: I could use the previous question in Hatcher which states that $H_n(X,A) = 0 \iff i:A\hookrightarrow X$ induces an isomorphism....
Sep
5
comment $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$
But they mention it on Wikipedia: en.wikipedia.org/wiki/Relative_homology. They don't explain why so I assumed they use it to compute $H_n(X,A)$. Why do they mention it there?
Sep
5
revised $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$
added 202 characters in body
Sep
5
accepted Determining whether there is a short exact sequence
Sep
5
asked $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$
Sep
4
comment Determining whether there is a short exact sequence
Thanks! I have to look at that contradiction again.
Sep
4
comment Determining whether there is a short exact sequence
Yes, I can list the groups of order $p^{n+m}$. So you're saying what I wrote is all correct?
Sep
4
asked Determining whether there is a short exact sequence
Sep
3
comment Counter examples of cell complex
OK, so in case one, there should indeed be another black dot. And in case two, the entirety of $\partial e^2$ should be mapped, not just one dot on it! Thanks for your help!