Communications of the
Korean Mathematical Society

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024



Commun. Korean Math. Soc. 2023; 38(4): 1309-1320

Online first article October 11, 2023      Printed October 31, 2023

Copyright © The Korean Mathematical Society.

On the rational cohomology of mapping spaces and their realization problem

Abdelhadi Zaim

University Hassan II


Let $f:X\rightarrow Y$ 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$ \lbrace i, \pi_{i }(Y)\otimes \mathbb{Q}\neq 0 \rbrace$ is an even number. Moreover, we are interested in determining the rational homotopy type of map$\left( \mathbb{S}^{n}, \mathbb{C} P^{m};f\right) $ and we deduce its rational cohomology as a consequence. The paper ends with a brief discussion about the realization problem of mapping spaces.

Keywords: Rational homotopy theory, mapping space, rational cohomology, Sullivan minimal model, derivation

MSC numbers: Primary 55P62; Secondary 54C35