Tuesday, 13 November 2007

at.algebraic topology - What is the difference between homology and cohomology?

On a closed, oriented manifold, homology and cohomology are represented by similar objects, but their variance is different and there is an important change in degrees. For simplicity, consider homology or cohomology classes represented by submanifolds. Then if f:MtoNf:MtoN is a smooth map between manifolds of dimension mm and nn respectively and WW is a submanifold of MM representing a homology class then f(M)f(M) represented (really, f([M])f([M]) is) a homology class of NN, in the same dimension. On the other hand, if VV is a submanifold of NN, then we can consider f1(V)f1(V), which is a manifold if ff is transverse to FF. Its codimension is the same as that of VV (that is, ndim(V)=mdim(f1(V))ndim(V)=mdim(f1(V))).



Note that this preimage is generically at least as "nice" as VV (smooth is VV is, with reasonable singularities if VV has, etc.) whereas little can be said about the geometry of f(W)f(W). That's one reason I think that cohomology is of more use in algebraic geometry.



If you want to relate this view of cohomology to the standard one, a codimension dd submanifold of MM (that is, one of dimension mdmd) generically intersects a dimension dd one in a finite number of points which can be counted with signs. The former defines a class in Hd(M)Hd(M) while the latter a class in Hd(M)Hd(M), and this intersection count is evaluation of cohomology on homology.



This point of view is more applicable than it might seem since in a manifold with boundary cohomology classes are similarly defined by submanifolds whose boundary lies on the boundary of the ambient manifold. Since any finite CW complex is homotopy equivalent to a manifold with boundary, one can view cohomology in this way for finite CW complexes and often infinite ones as well.

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