Braid groups in complex projective spaces

Abstract

Abstract. We describe the fundamental groups of ordered and unordered k−point sets in CP n generating a projective subspace of dimension i. We apply these to study connectivity of more complicated configurations of points.

Braid groups in complex spaces

“…From the above Lemma and the exact sequence in (2) we get that the image in π 1 (F k (C)) of the generator of π 2 (Graff 1 (C 2 )) is D k and the following theorem is proved.…”

“…, p k > gives a point in the affine grassmannian manifold Graff i (C n ) parametrizing the i-dimensional affine subspaces of C n . Analogously to the projective case in [2], we have the following:…”

“…whose restriction to S 1 gives the map [2]) The homotopy class of the mapγ corresponds to the following pure braid in PB k = π 1 (F k (C)):…”

“…In order to prove this theorem we use, as in [2], the homotopy long exact sequence of fibrations and the Van Kampen theorem applied to an open cover. But, unlike the projective case in which the fiber of the key fibration and the open sets of the cover are simply connected, in the affine case the fundamental groups are non trivial and more work is needed.…”

Cohomology of 3-points configuration spaces of complex projective spaces

“…In Section 6 we compute the Betti numbers of the spaces of configurations of three collinear points and configurations of three non-collinear points (see [4] for the fundamental groups of these spaces):…”