# Reference for cup product in deRham cohomology

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?

• See $\S$ 3.2 of Hatcher's Algebraic Topology. pi.math.cornell.edu/~hatcher/AT/ATpage.html – Travis Dec 23 '18 at 3:17
• @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? – Praphulla Koushik Dec 23 '18 at 3:25
• 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. – Travis Dec 23 '18 at 3:42
• @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.. – Praphulla Koushik Dec 23 '18 at 3:45
• Look at Warner's Foundations of Manifolds book or Spivak's Differential Geometry, volume 1. – Ted Shifrin Dec 23 '18 at 6:12