Realizing Algebra over rational as cohomology of some topological space
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Abstract : Given a topological space \(X\) we can compute the cohomology groups(modules) of it. Call them \(H^k(X;R)\), where $R$ is the ring of coefficients. We want to give $\bigoplus_k H^{k}(X,R)$ a graded ring structure. In order to do so we have defined cup product, $$\smile : H^{k}(X;R) \times H^{n}(X;R)\to H^{n+k}(X;R)$$
We define this graded ring as $H^*(X;R)$, this ring can be a polynomial ring over $R$. The natural question arose in our mind when can we write a graded commutative $R$-algebra can as a cup-product algebra $H^\ast(X;R)$ for some space $X$. The answer is affirmative for any algebra over $\mathbb{Q}$. In order to construct/ get such a space we will introduce Rational homotopy theory (as it was done by mathematician D.Quillen).