Given a smooth manifold $M$, we have what are called deRham cohomology groups $H^i(M,\mathbb{R})$.

deRham cohomology ring $H^*(M,\mathbb{R})$ is as a set $\bigoplus_{i=0}^{\text{dim(M)}} H^i(M,\mathbb{R})$. This is made a ring by giving multiplication. Given $1$-form $\omega$ and a $2$-form $\tau$, the product $\omega.\tau$ is the wedge product $\omega\wedge\tau$.

Is there any reference where this is mentioned?

  • $\begingroup$ See $\S$ 3.2 of Hatcher's Algebraic Topology. pi.math.cornell.edu/~hatcher/AT/ATpage.html $\endgroup$ – Travis Dec 23 '18 at 3:17
  • $\begingroup$ @Travis It does not say anything about deRham cohomology but it does say something about cup product... Thank you :) Any reference where deRham cohomology is specifically mentioned? $\endgroup$ – Praphulla Koushik Dec 23 '18 at 3:25
  • $\begingroup$ The notion of cup product is a general feature of cohomology theory. I don't know offhand a reference that exclusively treats the cup product of de Rham cohomology. $\endgroup$ – Travis Dec 23 '18 at 3:42
  • $\begingroup$ @Travis Yes, I know that cup product is in other cohomology theories.. Some one asked me for reference only for deRham cohomology (may be because it is easier than others).. I could not think of any.. So, asked as question here.. $\endgroup$ – Praphulla Koushik Dec 23 '18 at 3:45
  • $\begingroup$ Look at Warner's Foundations of Manifolds book or Spivak's Differential Geometry, volume 1. $\endgroup$ – Ted Shifrin Dec 23 '18 at 6:12

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