It's said here https://mathworld.wolfram.com/Homology.html that homology studies relations between two maps (from two manifolds to a manifold). Does it suggest that it's to study if two manifolds can be 'projected' into the same manifold, and a homology group is such that all elements (manifolds, or maps) in the group can be projected into the same manifold. Cohomology is said to complement homology, in what way? Does it study two manifolds that can be 'less precisely' projected into a manifold, or does it study manifolds that can't be projected into a manifold.
The question: Can anyone give the simplest examples in homology (group) and cohomology (group)?
The following is a note for myself and not the answer.
An explanation seems to be all these groups seem to describe boundary or 'segmented boundary' (with these phrase I imply sth like polygon with directed edges) of a manifold (or other spaces); an 'edge' here is a one-segment curve with direction. For example, we can have $C^k(M)$ for k-chain (boundary of k-cube, e.g. 1-chain is an edge, 2-chain is like edges of a square, 3-chain is like edges of a cube, etc.), a boundary of any number of edges; $Z^k(M)$ for k-circle which is like a 'boundary' of one edge or zero edge (in differential forms, it's group of 'closed' k-forms); $H^k(M)$ for the usual boundary of a manifold (in differential forms, it's group of 'exact' k-forms). (A difficulty here is that it seems hard to imagine a k-circle, one edge, as 'boundary' of a manifold of n-manifold ($n \geq 3$) since, for example, 3-cube looks like definitely need more than two edges to make a boundary.)
So we see $H^k(M)$ (0-segment boundary) is subgroup of $Z^k(M)$ (0,1-segment boundary), which is a subgroup of $C^k(M)$ (any number segment boundary), and we can make quotient groups, and by studying these quotient groups (about 'boundary' of a manifold) (one among the so called 'invariants') we can know something about the manifold.
This seems to make sense, but personally I still quite suspect that the name bears an implication that it has something to do about two manifolds' (or other kinds of geometric shape') projection in to one manifold (geometric shape).
With further thought, it seems easy that the two kinds of understanding may be reconciled by noticing that if manifolds $M_1, M_2,\dots$ can be projected into the same manifold $M_0$, then the former all shares some property of the latter, which implies there are 'invariants' common in all these manifolds $M_0, M_1, M_2,\dots$. (Notice that $M_0$ can be surely projected into itself--and even more, all other manifolds can be projected into any $M_i$--so all these manifolds form a set satisfying certain 'invariants', instead of regarding $M_1, M_2,\dots$ as separated from $M_0$.)
Therefore, the concept of projection can fade away a bit from the definition of homology, while the concept of set (or group) emerges in or enters the definition.) Basically, I guess homology and cohomology is to study all these manifolds (sharing some common property) together, but for it's sometimes difficult to study the manifolds themselves, homology and cohomology become subjects studying mainly 'invariants' (of 'boundaries') of these manifolds (e.g. the quotient groups mentioned above), which perhaps lead to confusion about these names which ought to suggest related manifolds (geometric shapes), not just the relation itself. But such a switch of perspectives also makes (to me) the content of modern homology 'obvious'/easily acceptable.
Besides, because of the pervasive and unifying 'abstract algebra' methods in the last century, it's natural that algebraic concept like groups are used significantly to describe such 'invariants' among ('boundaries' of) manifolds (geometric shapes), which leads to a heavily 'algebraic' impression/flavor of homology and related subjects. But even if so (and if my understanding is correct), I would like to say it seems to me, still, 'invariants' and manifolds (geometric shapes) are core of these subjects, and algebraic content is more like a tool to study them, or a representation of, a way of expressing these invariants (i.e. there may be other ways as well).
Basically these seem to explain particularly homology, then what is cohomology? It seems it is just another (somehow better) version of homology, like an optimized version of an app.
