Alladi Ramakrishnan Hall

Configuration spaces, products, and fibre

Daciberg L. Goncaves

University of Sao Paulo

We study the inclusion i of the n-th configuration space of a
surface M in the $n$-fold Cartesian product $M^n$ for the cases where M is
the 2-sphere $S^2$ or the projective plane $RP^2$. If $M=RP^2$, we show that
the kernel of the homomorphism induced by i on the level of fundamental
groups
coincide wih the commutator subgroups of the domain. Further this groups can
be written as a direct sum of $Z_2$ and another groups which is an iterated
semi-direct product of free groups. We then go on to analyse the homotopy
fibre of i, and we determine its homotopy type. It turns out they are
product of two types of space: loop spaces of the 2-spheres and spaces
which have universal covering contractible. We review the case where the
closed surface is distint from $S^2$ and $RP^2$.