#### 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 $RP^2$.

Done