Cfrgtkky

Let $X$ be a smooth manifold and $Osubset X$ be an closed set containing a non-trivial neighbourhood of $xin X$. The reason to ask the question is to clarify the relationship between limit and relative homology. Recall that in algebraic geometry, an affine scheme $Spec(R)$ has stalk $R_p$ with $pin Spec(R)$ where $R_p$ can be realized as direct limit over open sets $pin D(f)subset Spec(R)$. I wish to see whether there is connection between limit and relative homology.

Suppose $O$ is small enough say a small ball around $x$. Then $H_star(X,X-O)cong H_star(X,X-{x})$ via deformation retraction. Take any homology class $gammain H_star(X,X-O)$. I can always lift to a $gamma'$ cycle of $Z_star(X,X-O)$ first and then perform subdivision. Note that I am basically invoking subdivision procedure and identifying the chain complex level quasi isomorphism and this does not change the relevant homology class information. Now $H_star(X,X-O)$ will throw away the cycles sitting inside $X-O$. So I can keep dividing $gamma'$ and in hope to get a cycle "totally" lying in $O$ as all cycles in $X-O$ are removed by quotient.

$textbf{Q1:}$ Does this infinite subdivision procedure makes sense? It looks like I am considering open covering of $X$ and looking at the cycles supported on those coverings. In other words, eventually, I have deformed original $gamma'$ cycle to a cycle lying on $O$ and this $gamma'$ may very likely touch boundary of $O$ as $O$ is closed.

Now $H_star(X,X-O)to H_star(X,X-{x})$ is by excision first and then following deformation retraction.

$textbf{Q2:}$ Can I say "$H_star(X,X-{x})$ is directed limit of $H_star(X,U)$ with $xnotin U$? It is clear that if $U$ is small enough, one can do excision again and deformation retraction argument. Here I wish to draw analogy with algebraic geometry setting.

$textbf{Q3:}$ What is the weakest assumption on $X,O$ to keep $Q1,Q2$ holding?

$H_star(X,X setminus O)cong H_star(X,X setminus {x})$ is a serious requirement on $O$. For example, it is wrong if $O = X$. An adequate choice for $O$ is this: If $X$ is $n$-dimensional, choose a chart $phi : U to mathbb{R}^n$ such that $phi(x) = 0$, where $U$ is an open neighborhood of $x$. Let $D(0,r) = { x in mathbb{R}^n mid lVert x rVert le r } subset mathbb{R}^n$ be a closed ball and $mathring{D}(0,r)$ its interior which is an open ball. For $O = O(phi,r) = phi^{-1}(D(0,r))$ both $X setminus O$ and $X setminus { x }$ have $X setminus phi^{-1}(mathring{D}(0,2r))$ as a strong deformation retract, therefore the inclusion $i : X setminus O to X setminus { x }$ is a homotopy equivalence. Hence $i_* : H_*(X setminus O) to H_*(X setminus { x })$ is an isomorphism. The long exact sequences of the pairs $(X,X setminus O)$ and $(X,X setminus { x })$ and the five lemma gives us then an isomorphism $H_star(X,X setminus O) to H_star(X,X setminus {x})$.

Now it seems that you consider the set $mathcal{U}$ of all open sets $U subset X$ not containing $x$. This is set is partially ordered by inclusion (i.e. $U le U'$ if $U subset U'$). It is clearly a directed set and it has $X setminus {x}$ as maximum. Hence the directed system $H_*(X, U)$ indexed by $mathcal{U}$ trivially has $ H_star(X,X setminus {x})$ as its direct limit.

But perhaps you want to admit only open $U subset X$ such that $O_U = X setminus U$ is a closed neighborhood of $x$. The set $mathcal{U}'$ of all these $U$ is a directed subset of $mathcal{U}$ which does not have a maximum. We know that the $O(phi,r)$ constitute a neigborhood basis for $x$. Hence the $U(phi,r) = X setminus O(phi,r)$ form a cofinal subset of $mathcal{U}'$, hence the direct limits $text{dirlim}_{U in mathcal{U}'} H_*(X, U)$ and $text{dirlim}_{r >0} H_*(X,U(phi,r))$ are isomorphic. Since all $i_* : H_*(X,U(phi,r)) to H_star(X,X setminus {x})$ are isomorphisms, we end again with $text{dirlim}_{U in mathcal{U}'} H_*(X, U) cong H_star(X,X setminus {x})$.