TITLE: What is the geometric meaning of homology group and cohomology group?
QUESTION [7 upvotes]: I am reading some algebraic topological book, where they said that the p-th homology group tells us how many p-dimensional holes inside the set. How should I understand this? I know that in the planar case, the 1-th order homology group tells us exactly how many 1-holes the set has, but I do not have a good feeling about the p-holes, could anybody explain this by some very simple examples? 
Also they said the cohomology groups are dual to homology group, is there also a good geometric meaning of cohomology group? I heard some duality result, like the Alexander duality between homology group and cohomology group, but I do not know the real meanings.

REPLY [9 votes]: I recommend you to have a look at the first chapter of Roger Fenn's book "Techniques of geometric topology". 

Going back to Poincaré's first approach to homology (where homology is defined as a geometric calculus), you can think to $n$-cycles of a space $X$ as maps from $n$-manifolds (with singularities) to $X$ up to a bordism relation. A nice reference is Mathias Kreck's book: "Differential algebraic topology", in that text manifolds with singularities are treated as stratifolds. 
There are several ways to think to cohomology in a geometric way. One goes back to Dan Quillen's treatment of complex cobordism and is described in the case of a closed manifold in Kreck's book (in this setting the duality between homology and cohomology is transparent). 
I also want to emphasize on Dennis Sullivan's principle that says that cohomology is represented by geometric cocycles. Geometric cocyles are cycles with "normal geometry". Roughly speaking, if your space is stratified they are cycles transverse to the stratification. In that geometric setting, the cup product is given by the geometric intersection, going back to the roots of algebraic topology. This principle is described in Roger Fenn's book, in Clint MacCrory's PhD thesis (available on his homepage), in Mark Goresky's paper "Whitney stratified chains and cochains" and in Buoncristiano-Rourke-Sanderson's monography "A geometric approach to homology theory".   

Thus you can think of the homology of a space $X$ as equivalences classes of maps $V\rightarrow X$ where $V$ is a "manifold with singularities". And you must play with singular spaces in order to recover singular homology, this is a deep and beautiful result due to René Thom. For "reasonable" spaces $X$ (including polyedra) together with and additional geometric structure: a stratification, these geometric cycles can be taken as sums of subspaces of $X$ (substratified spaces). The second lesson is that cohomology can be build from particular cycles those who are in general position with respect to this stratification. This last fact is deeply rooted into a geometric interpretation of Poincaré duality in terms of the polyedral dual decomposition $C$ to a triangulation $T$ of a manifold $M$.

Thanks to $T$ you get cycles for the homology of $M$,
Thanks to $C$ you get geometric cocycles for the cohomology of $M$, by definition they are transverse to $T$.