# Can cohomologies always be defined as the quotient of the kernel and the image of a certain operation?

+ 2 like - 0 dislike
178 views

For example, concerning de Rham cohomology as explained on the lower part of page 5 here, one considers the vector space of all forms on a manifold $M$, $A_{DR}(M)$ in the notation of the paper (?).

The $p$-th de Rham cohomology is then defined as the quotient of the kernel and the image of the exterior derivative $d$ acting on the space of $p$ and $p-1$ forms respectively as

$H_{DR}^p(M) = \frac{Ker(d: A_{DR}^{p}(M)\rightarrow A_{DR}^{p+1}(M))} {Im(d: A_{DR}^{p-1}(M)\rightarrow A_{DR}^{p}(M))}$

Can cohomology always be defined as the kernel divided by the image of a certain operation?

What does cohomology mean or "measure" in as simple as possible intuitive terms in the de Rham case but also more generally?

PS: I tried to read wikipedia of course, but it was (not yet?) very enlightening for me ...

The $n$th cohomology group is always the quotient of a vector space of $n$-cocycles by the corresponding vector space of $n$-coboundaries (which are special $n$-cocycles constructed from arbitrary $(n-1)$-cocycles. They are a way to measure how nontrivial the topology is from which they are constructed. (They are topological invariants.)

Some of the low order cohomology groups measure in some applications the possibilities for extensions of an object, or the obstructions for possible deformations. in the latter case the nonvanishing is called an anomaly.

+ 2 like - 0 dislike

As in the math.stackexchange answer linked by an anonymous user in a comment to your question, certain (topological) cohomology theories like K theory do not come from cohomology of a chain complex functor.

These generalized cohomology theories always come from a spectrum, however, which is a sequence of spaces analogous to the Eilenberg-Maclane spaces K(G,n) which represent ordinary cohomology in the sense that for any CW complex X, the homotopy classes of maps X -> K(G,n) form a group (!) isomorphic to $H^n(X,G)$.

The spectrum of K complex theory is 2-periodic, alternating between $BU \times \mathbb{Z}$ and its delooping.

answered Jan 26, 2017 by (1,895 points)
edited Jan 26, 2017

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:$\varnothing\hbar$ysicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.