Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON THE RATIONAL COHOMOLOGY OF MAPPING SPACES AND THEIR REALIZATION PROBLEM

  • Abdelhadi Zaim (Department of Mathematics and Computer Sciences Faculty of Sciences Ain Chock University Hassan II)
  • Received : 2023.01.28
  • Accepted : 2023.05.09
  • Published : 2023.10.31
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Abstract

Let f : X → Y be a map between simply connected CW-complexes of finite type with X finite. In this paper, we prove that the rational cohomology of mapping spaces map(X, Y ; f) contains a polynomial algebra over a generator of degree N, where N = max{i, πi(Y)⊗ℚ ≠ 0} is an even number. Moreover, we are interested in determining the rational homotopy type of map(𝕊n, ℂPm; f) and we deduce its rational cohomology as a consequence. The paper ends with a brief discussion about the realization problem of mapping spaces.

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Acknowledgement

The author is very grateful to Professor Gregory Lupton for fruitful discussions and helpful suggestions. The author would also like to thank the referee for his valuable comment and suggestions which helped to improve the manuscript.

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