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The Euler class of the tangent bundle of a compact, oriented manifold agrees with the evaluation of the top homology class on the fundamental class (which is represented by the manifold itself), and maybe also figure out how to do actual computations using Poincare duality (to figure out the top cohomology using 0-th homology).

I am trying to both find out the formal/actual evaluation here, say for the case of the spheres S^{2n} bring together different interpretations/perspectives of the Euler class of a vector bundle , i.e., so that , for the tangent bundle of a manifold M, its Euler class is the Poincare dual of the zero-set of a generic section. Also, if possible, we can consider that the odd-dimensional spheres have E(S^{2n+1})=0 , so that the Poincare dual of a generic section is the 0-class.

Let's fix the integers ## \mathbb Z ## as coefficient group.

So, first, say for M=S^n , how do we do the evaluation? By Poincare duality, ## H^n = H_0 ## ( with = meaning up to isomorphism ). So , since S^n is path-connected, ## H_0( S^n) = \mathbb Z = H^n## and we must evaluate this at the fundamental class of $S^n$ , which is generated by S^n itself. How is this done?

Sorry if this is too broad; any ideas welcome.