I'm learning some algebraic topology books which introduce me to singular cubial homology and cohomology ( not simplicial as usual so sometimes it's hard for me . I think the difference between these theory is triangulation ) . I will go straight to main problem , if you have studied Poicare duality theorem you may know the way to define a orientable manifold through group homology . In higher dimension of manifold , it can't be imagined . My question is the intuition of homology and cohomology group in higher dimension . How these groups affect to orientation or somethings like this . For instance , rank of $H_{0}$ is the number of component , $H_{1}$ is abelianization of fundamental group . I try to find some ideas on Internet but the only I get is like " the number of $n$ dimension holes the space has " . I can't happily calculate when I don't really understand what they actually mean .

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  • Perhaps you can't happily calculate, but unhappy calculation is an excellent way to build your intuition. – Lee Mosher May 22 '17 at 18:56
  • What exactly are you looking for? If there were a simpler but meaningful definition of $H_*$, we'd just use that. For a more geometric setting of the construction, see de Rham cohomology (but not that it's only defined over $\mathbb{R}$ or something analogous, so you lose a lot of the information of simplicial homology in the bargain). – anomaly May 22 '17 at 18:58
  • The question might be phrased:"What is the meaning of singular cubical chains?". Massey's book gives the traditional answer. Our 2011 book partially titled "Nonabelian Algebraic Topology" (EMS) gives another answer, since there "chains" are defined as certain homotopy classes of maps, and the various operations are defined geometrically..It is not so easy to set up the basic properties which enable calculation, but the book does try to explain the history and intuition. – Ronnie Brown May 22 '17 at 19:35

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