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I am trying to get some basic idea about characteristic numbers. from the wikipedia https://en.wikipedia.org/wiki/Characteristic_class The definition contians an pairing between cohomology and homology "one can pair a product of characteristic classes of total degree n with the fundamental class"

But how is this pairing defined? please help.

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    If you think about de Rham cohomology (as indicated in the comment below) then the relevant pairing is simply given by integrating top-degree forms over the manifold. – 2017-02-13

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Let $\sigma$ be a singular chain of degree $k$ and $\eta$ a singular co-chain of degree $k$. You have an "obvious" bilinear pairing $\eta(\sigma)\in R$ of such elements.

Now if $\eta\in\ker(\delta)$ and $\sigma=\partial\tau$, then $\eta(\sigma)=\eta(\partial\tau)=(\delta\eta) (\tau) = 0$. If $\eta=\delta\omega$ and $\sigma\in\ker(\partial)$ then you have $\eta(\sigma)=(\delta\omega)(\sigma)=\omega(\partial\sigma)=0$.

The conclusion is that if you restrict the paring to the kernels of the differentials, then $\eta(\sigma)$ depends only the cohomology class of $\eta$ and the homology class of $\sigma$. This induces a pairing: $H_k\times H^k\to R$.

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    Thanks very much. But, I am still confused about the "obvious" definition of paring. From what is described here, I feel that if we are talking about de Rham cohomology and the paring is defined by integrate some differential forms over some cirlcles, the paring is well defined. But your definition does not need de Pham cohomology, right? Can you more about how is the paring defined here exactly? – 2017-02-12
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    is it defined this way? If the cochain group is defined as $C_n^ * = Hom({C_n},R)$ and let $\alpha \in Hom({C_n},R)$and $\beta \in {C_n}$, then I may define a pair as $< \alpha ,\beta > = \alpha (\beta ) \in R$. Is this right? – 2017-02-13
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    @ChenLi: You can use any (co)homology theory over the real numbers, where cochains are defined as elements of the dual of the space of chains. – 2017-02-13