Rational homotopy type of mapping spaces via cohomology algebras

2020/10/09 by Xie, Sang, Liu, Jian, Liu, Xiugui
#54C35 (Secondary) #55P62 (Primary) #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2010.04579

Abstract

In this paper, we show that for finite CW-complexes X and two-stage space Y (for example n-spheres Sⁿ, homogeneous spaces and F₀-spaces), the rational homotopy type of \map(X, Y) is determined by the cohomology algebra H^*(X; \Q) and the rational homotopy type of Y. From this, we deduce the existence of H-structures on a component of the mapping space \map(X, Y), assuming the cohomology algebras of X and Y are isomorphism. Finally, we will show that \map(X, Y; f)≃\map(X, Y; f') if the corresponding Maurer-Cartan elements are connected by an algebra automorphism of H^∗(X, \Q).

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