my previous question
was related to understanding of quotient group,i dont need to know too much detailed in group theore,just some part of algebraic topology,especially my interest is to define Homology and Cohomology ,also intuittive understanding about their,what is different between them,may be it is wrong but i was looking for video about homology group in this video
http://www.youtube.com/watch?v=2wn10l9qbJI
there was talking about cycles,kernels of cycles images, and was mentioned or represented Homology as a quotient of cycles and images,i have read also that Cohomology acts as a dual of homology,but what does mean dual term in group meaning?are Homology/Cohomology same besides only with minor difference?thanks in advance
If I asked you to tell me how to speak Russian -- I don't want to learn the vocabulary or the grammar, just the general structure of the language -- you'd laugh at me. The question doesn't even make sense. In a way, what you're asking is the same. The classic ideas of topology are things like "in some ways, a donut and a coffee-mug are 'the same', because each has one hole," but formalizing these isn't so simple. By the time I've explained the formalism, you've learned the vocabulary and grammar. :) I'll do my best, though, with some broad strokes:
The big ideas are that (1) "sameness" for objects $A$ and $B$ means that there is a continuous function from $A$ to $B$ whose inverse is also continuous, and (2) you can find functions that take objects to numbers in such a way that two objects that are the "same", in the sense just defined, get the same numbers assigned to them. Such a number is called a "topological invariant." (And the notion of sameness is called "homeomorphism" -- we say "$A$ is homeomorphic to $B$.")
As an example, if you have two triangulated surfaces (the definition requires some care), you could count the number of vertices, edges, and faces in each, and compute V - E + F for each object. If these two numbers are different for the two surfaces, then the surfaces cannot be homeomorphic. (That's a big theorem, by the way!) If the numbers are equal, you don't, a priori, know whether the surfaces are the same or different. (If you allow the surfaces to have boundary, like a single triangle, then there are different (nonhomeomorphic) surfaces that have the same V-E+F number.)
What if you computed V+E+F instead? Well, you'd get a number, but it's not an invariant, because for a cube, you'd get $8 + 12 + 6$, while for an tetrahedron, you'd get $4+6+4$, but these two shapes are in fact homeomporphic. So finding invariants requires some skill...not just anything will work!
More generally, topologists try to assign not just numbers to shapes, but algebraic objects like groups. If YOUR homology groups and MY homology groups are different, then you and I are not homeomorphic.
Why do we try to shift from shapes to numbers or groups? Because numbers and groups are easy to compare.
Now the bad news: proving that the number you compute, or the group you compute, will be the same for any two shapes that are homeomorphic --- that's often difficult. It's easy to get the definitions wrong, and hard to get the proofs right. That's why there are big fat topology books like Hatcher's.