Using Short Exact Sequence to Long Exact Sequence to determine properties of a space of solutions

Question

Coming from physics (postdoc) with some knowledge of mathematics, I'm trying to use exact sequences and Homology/Cohomology to try to understand the space of solutions for a research problem I'm working with.

I begin by constructing a short exact sequence of topological spaces: $$0 \rightarrow A \xrightarrow{\varphi} B \xrightarrow{\pi} C \rightarrow 0$$

Here $A$ is my unknown space of solutions, I know this is a subset of a vector space (which I will call $X$), and will define it as a topological space using the topology of the vector space. The goal is to understand if this space can only be point-like or if there can be a continuum of solutions.

$\varphi$ is a map from solutions to some vector space $B$ and has a sensible definition over all of $X$, it is also continuous (though non-linear). The result in $B$ is considered valid if the affine function $\pi: B \rightarrow C$ maps to 0 in a second vector space $C$ (which has a dimension strictly less than $B$). This means if I enforce exactness at $B$, then $A$ must be a subset of all possible solutions.

Since I'm thinking of all of these spaces as topological spaces then they can also be viewed as a chain complex, allowing me to convert the short exact sequence above into the homological or cohomological long exact sequence below:

$$\cdots \rightarrow H^{n-1}(C) \rightarrow H^{n}(A) \rightarrow H^{n}(B) \rightarrow H^{n}(C) \rightarrow H^{n+1}(C)\rightarrow \cdots$$

$$\cdots \rightarrow H_{n-1}(C) \rightarrow H_{n}(A) \rightarrow H_{n}(B) \rightarrow H_{n}(C) \rightarrow H_{n+1}(C)\rightarrow \cdots$$

Now I should be able to determine all the maps in these long exact sequences from $\varphi$ and $\pi$ (though I haven't gone through and computed them yet). But my idea is that this long exact sequence should give me some information regarding the homology/cohomology of my space of solutions, such as if there is a continuum of solutions or not (i.e. is the set of solutions uncountably infinite or countable/finite). Specifically since I believe the homology/cohomology of vector spaces tend to be quite simple. So in summary I have 3 core questions:

1. Does the construction I've provided make sense?
2. Assuming this argument makes sense in the broad strokes, I feel that this construction is a little vague, what do I need to clarify (and is there something sensible that I seem to have assumed)?
3. How should I go about extracting the information regarding the space $A$ after I have constructed the long exact sequence(s) above? My intuition is that in the case that there are only a countable number of solutions I should find that the only sensible $H^{n>0}(A)$ should be trivial, but I don't know enough about this field to trust my intuition yet.

I'm also unfamiliar with the tags here and so am wondering if there are any other tags I should include? Or if this question fits more with mathoverflow than math stack exchange?

Answer 1

I am sure somebody will end up giving a better answer, but here it goes:

1. To define the term short exact sequence you need to know how to define the kernel and the image of a map. There are no kernels for maps of topological spaces. In topology, people usually use sequences of the form

   $$F\to E\to B,$$

   where $E$ is the total space of fiber bundle over $B$ with fiber $F$. This is not really a short exact sequence, even though it is ideologically similar and does lead to a long exact sequence.

2. If your topological spaces were actually vector spaces with linear maps between them, then you could define what short exact sequences are. But this wouldn't give you any exact sequences in topological (i.e. singular) homology/cohomology. You write that the homology/cohomology of vector spaces tend to be quite simple. Indeed, they are so simple actually, that for a vector space $\mathbb{R^n}$ all $H^i(\mathbb{R^n})$ and $H_i(\mathbb{R^n})$ are simply zero for $i>0$ and $H^0(\mathbb{R^n})=\mathbb{Z}=H_0(\mathbb{R^n}).$ As you can see, these groups don't form a long exact sequence.

In short, I think you seem to be confusing the algebraic incarnation of homology, which just associates a series of subquotients to a chain complex, with topological homology/cohomology theories, which associate groups to topological spaces.