Rudy the Reindeer
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 Sep5 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 $n$! Sep5 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 $X$. Sep5 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.... Sep5 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? Sep5 revised $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall$ path-components $X_i$ added 202 characters in body Sep5 accepted Determining whether there is a short exact sequence Sep5 asked $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall$ path-components $X_i$ Sep4 comment Determining whether there is a short exact sequence Thanks! I have to look at that contradiction again. Sep4 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? Sep4 asked Determining whether there is a short exact sequence Sep3 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! Sep3 comment Entangled circle in a solid torus (follow up) @Ryan: Yes thanks, I think that's a good suggestion that I'll stick to. Sep2 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? Sep2 revised Surface of genus $g$ does not retract to circle (Hatcher exercise) Picture added. Sep2 asked Counter examples of cell complex Sep2 asked Surface of genus $g$ does not retract to circle (Hatcher exercise) Sep1 accepted Typo in Hatcher? $\mathbb{R}^n - \{x \} \cong S^{n-1} \times \mathbb{R}$? Sep1 asked Typo in Hatcher? $\mathbb{R}^n - \{x \} \cong S^{n-1} \times \mathbb{R}$? Sep1 accepted Entangled circle in a solid torus (follow up) Sep1 comment Entangled circle in a solid torus (follow up) OK, that is only after embedding $A$ in $X$ via $i$!