On the Configuration Spaces of Grassmannian Manifolds
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 2, p. 353-359

Let h i (k,n) be the i-th ordered configuration space of all distinct points H 1 ,...,H h in the Grassmannian Gr(k,n) of k-dimensional subspaces of n , whose sum is a subspace of dimension i. We prove that h i (k,n) is (when non empty) a complex submanifold of Gr(k,n) h of dimension i(n-i)+hk(i-k) and its fundamental group is trivial if i=min(n,hk), hkn and n>2 and equal to the braid group of the sphere P 1 if n=2. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. k=n-1.

Soit h i (k,n) le i-ème espace de configuration ordonnée de tous les points distincts H 1 ,...,H h dans la Grassmannienne Gr(k,n) de sous-espaces de dimension k de n , dont la somme est un sous-espace de dimension i. Nous prouvons que h i (k,n) est (si non vide) une sous-variété complexe de Gr(k,n) h de dimension i(n-i)+hk(i-k) et que son groupe fondamental est trivial si i=min(n,hk), hkn et n>2 et égal au groupe de tresses de la sphère P 1 si n=2. Finalement, nous calculons le groupe fondamental dans le cas particulier des arrangements d’hyperplans, c’est-à-dire k=n-1.

@article{AFST_2014_6_23_2_353_0,
     author = {Manfredini, Sandro and Settepanella, Simona},
     title = {On the Configuration Spaces of Grassmannian Manifolds},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 23},
     number = {2},
     year = {2014},
     pages = {353-359},
     doi = {10.5802/afst.1410},
     mrnumber = {3205597},
     zbl = {06297896},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2014_6_23_2_353_0}
}
Manfredini, Sandro; Settepanella, Simona. On the Configuration Spaces of Grassmannian Manifolds. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 2, pp. 353-359. doi : 10.5802/afst.1410. http://www.numdam.org/item/AFST_2014_6_23_2_353_0/

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