Abstract : Given a topological space we can compute the cohomology groups(modules) of it. Call them , where is the ring of coefficients. We want to give a graded ring structure. In order to do so we have defined cup product,
We define this graded ring as , this ring can be a polynomial ring over . The natural question arose in our mind when can we write a graded commutative -algebra can as a cup-product algebra for some space . The answer is affirmative for any algebra over . In order to construct/ get such a space we will introduce Rational homotopy theory
(as it was done by mathematician D.Quillen).