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Complex Grassmann manifolds $G_{n,k}$ are a generalization of complex projective spaces and have many important features some of which are captured by the Pl\"{u}cker embedding $f:G_{n,k}\longrightarrow \mathbb C P^{N-1}$ where $N=\left (\begin{smallmatrix} n\\ k \end{smallmatrix} \right)$. The problem of existence of cross sections of fibrations can be studied using the Gottlieb group. In a more generalized context one can use the relative evaluation subgroup of a map to describe the cohomology of smooth fiber bundles with fiber the (complex) Grassmann manifold $G_{n,k}$. Our interest lies in making use of techniques of rational homotopy theory to address problems and questions involving applications of Gottlieb groups in general. In this paper, we construct the Sullivan minimal model of the (complex) Grassmann manifold $G_{n,k}$ for $2\leq k<n$, and we compute the rational evaluation subgroup of the embedding $f:G_{n,k}\longrightarrow \mathbb C P^{N-1}$. We show that, for the Sullivan model $\phi:A\longrightarrow B$, where $A$ and $B$ are the Sullivan minimal models of $\mathbb C P^{N-1}$ and $ {G_{n,k}}$ respectively, the evaluation subgroup $G_n(A,B;\phi)$ of $\phi$ is generated by a single element and the relative evaluation subgroup $G_n^{rel}(A,B;\phi)$ is zero. The triviality of the relative evaluation subgroup has its application in studying fibrations with fibre the (complex) Grassmann manifold.
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참고문헌 (14)
참고문헌 신청
E. H. Brown, Jr. / 1997 / On the rational homotopy type of function spaces / Trans. Amer. Math. Soc 349(12):4931~4951
U. Buijs / 2008 / The rational homotopy Lie algebra of function spaces / Comment. Math. Helv 83(4):723~739
Y. Feix / 1982 / Rational LS category and its applications / Trans. Amer. Math. Soc 273(1):1~38
D. H. Gottlieb / 1969 / Evaluation subgroups of homotopy groups / Amer. J. Math 91:729~756
W. Greub / 1973 / Connections, Curvature, and Cohomology. Vol. II / Academic Press
U. Buijs / 2008 / The rational homotopy Lie algebra of function spaces / Comment. Math. Helv 83(4):723~739
Y. Feix / 1982 / Rational LS category and its applications / Trans. Amer. Math. Soc 273(1):1~38
D. H. Gottlieb / 1969 / Evaluation subgroups of homotopy groups / Amer. J. Math 91:729~756
W. Greub / 1973 / Connections, Curvature, and Cohomology. Vol. II / Academic Press
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