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Realizing Algebra over rational as cohomology of some topological space

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Abstract : Given a topological space XX we can compute the cohomology groups(modules) of it. Call them Hk(X;R)H^k(X;R), where RR is the ring of coefficients. We want to give kHk(X,R)\bigoplus_k H^{k}(X,R) a graded ring structure. In order to do so we have defined cup product, :Hk(X;R)×Hn(X;R)Hn+k(X;R)\smile : H^{k}(X;R) \times H^{n}(X;R)\to H^{n+k}(X;R)

We define this graded ring as H(X;R)H^*(X;R), this ring can be a polynomial ring over RR. The natural question arose in our mind when can we write a graded commutative RR-algebra can as a cup-product algebra H(X;R)H^\ast(X;R) for some space XX. The answer is affirmative for any algebra over Q\mathbb{Q}. In order to construct/ get such a space we will introduce Rational homotopy theory (as it was done by mathematician D.Quillen).

Introduction
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