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Evaluation Subgroups of Mapping Spaces over Grassmann Manifolds
Abdelhadi Zaim 경북대학교 자연과학대학 수학과 2023 Kyungpook mathematical journal Vol.63 No.1
Let Vk,n (C) denote the complex Steifel and Grk,n (C) the Grassmann mani folds for 1 ≤ k < n. In this paper, we compute, in terms of the Sullivan minimal models, the evaluation subgroups and, more generally, the relative evaluation subgroups of the fibration p : Vk,k+n (C) → Grk,k+n (C). In particular, we prove that G∗ (Grk,k+n (C), Vk,k+n (C) ; p) is isomorphic to G rel ∗ (Grk,k+n (C), Vk,k+n (C) ; p) ⊕ G∗ (Vk,k+n (C)).
On the rational cohomology of mapping spaces and their realization problem
Abdelhadi Zaim 대한수학회 2023 대한수학회논문집 Vol.38 No.4
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.