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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!
Sep
3
comment Entangled circle in a solid torus (follow up)
@Ryan: Yes thanks, I think that's a good suggestion that I'll stick to.
Sep
2
comment Surface of genus $g$ does not retract to circle (Hatcher exercise)
@group: Thank you! That is pretty neat! So the abelianization is only needed to prove the second part of the exercise?
Sep
2
revised Surface of genus $g$ does not retract to circle (Hatcher exercise)
Picture added.
Sep
2
asked Counter examples of cell complex
Sep
2
asked Surface of genus $g$ does not retract to circle (Hatcher exercise)
Sep
1
accepted Typo in Hatcher? $\mathbb{R}^n - \{x \} \cong S^{n-1} \times \mathbb{R}$?
Sep
1
asked Typo in Hatcher? $\mathbb{R}^n - \{x \} \cong S^{n-1} \times \mathbb{R}$?
Sep
1
accepted Entangled circle in a solid torus (follow up)
Sep
1
comment Entangled circle in a solid torus (follow up)
OK, that is only after embedding $A$ in $X$ via $i$!