On the Configuration Spaces of Grassmannian Manifolds

Sandro Manfredini; Simona Settepanella

Annales de la faculté des sciences de Toulouse Mathématiques (2014)

  • Volume: 23, Issue: 2, page 353-359
  • ISSN: 0240-2963

Abstract

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

How to cite

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Manfredini, Sandro, and Settepanella, Simona. "On the Configuration Spaces of Grassmannian Manifolds." Annales de la faculté des sciences de Toulouse Mathématiques 23.2 (2014): 353-359. <http://eudml.org/doc/275362>.

@article{Manfredini2014,
abstract = {Let $\{\mathcal\{F\}\}_h^i(k,n)$ be the $i$-th ordered configuration space of all distinct points $H_1,\ldots ,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of $\scriptstyle \{\mathbb\{C\}\}^n$, whose sum is a subspace of dimension $i$. We prove that $\{\mathcal\{F\}\}_h^i(k,n)$ 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 $i=min(n,hk)$, $hk \ne n$ and $n&gt;2$ and equal to the braid group of the sphere $\scriptstyle \{\mathbb\{C\}\}$$P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.},
author = {Manfredini, Sandro, Settepanella, Simona},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
month = {3},
number = {2},
pages = {353-359},
publisher = {Université Paul Sabatier, Toulouse},
title = {On the Configuration Spaces of Grassmannian Manifolds},
url = {http://eudml.org/doc/275362},
volume = {23},
year = {2014},
}

TY - JOUR
AU - Manfredini, Sandro
AU - Settepanella, Simona
TI - On the Configuration Spaces of Grassmannian Manifolds
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2014/3//
PB - Université Paul Sabatier, Toulouse
VL - 23
IS - 2
SP - 353
EP - 359
AB - Let ${\mathcal{F}}_h^i(k,n)$ be the $i$-th ordered configuration space of all distinct points $H_1,\ldots ,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of $\scriptstyle {\mathbb{C}}^n$, whose sum is a subspace of dimension $i$. We prove that ${\mathcal{F}}_h^i(k,n)$ 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 $i=min(n,hk)$, $hk \ne n$ and $n&gt;2$ and equal to the braid group of the sphere $\scriptstyle {\mathbb{C}}$$P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.
LA - eng
UR - http://eudml.org/doc/275362
ER -

References

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  1. Berceanu (B.), Parveen (S.).— Braid groups in complex projective spaces, Adv. Geom. 12, p. 269-286 (2012). Zbl1250.55007
  2. Birman (J. S.).— Braids, Links, and Mapping Class Groups, Annals of Mathematics 82, Princeton University Press (1974). Zbl0305.57013MR375281
  3. Fadell (E.R.).— Homotopy groups of configuration spaces and the string problem of Dirac, Duke Math. J. 29, p. 231-242 (1962). Zbl0122.17803MR141127
  4. Fadell (E.R), Husseini (S.Y.).— Geometry and Topology of Configuration Spaces, Springer Monographs in Mathematics (2001), Springer-Verlarg Berlin. Zbl0962.55001MR1802644
  5. Fadell (E.R.), Neuwirth (L.).— Configuration spaces, Math. Scand. 10, p. 111-118 (1962). Zbl0136.44104MR141126
  6. Manfredini (S.), Parveen (S.), Settepanella (S.).— Braid groups in complex spaces, arXiv: 1209.2839 (2012). 

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