# Jerusalem Mathematics Colloquium

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Thursday, 15th November 2001, 4:00 pm

Mathematics Building, Lecture Hall 2

##

Vladimir Lin

(Technion)

"Holomorphic maps of Riemann sphere's configuration spaces"

** Abstract: **

The *n*-th configuration space *C(n)* of the Riemann sphere
**CP**^1 is the irreducible non-singular affine algebraic
variety consisting of all *n*-point subsets
*Q={q_1,...,q_n}* of **CP**^1.
We study holomorphic (and, in particular, regular) mappings
*C(n)*-->*C(n)*.

Here is a simple example. Take any element *T* of
**PSL**(2,**C**)=Aut(**CP**^1)
and define *f: C(n) --> C(n)* by
*f(Q)=f({q_1,...,q_n})={Tq_1,...,Tq_n}*
for all *Q={q_1,...,q_n}* in *C(n)*.
This example may be generalized, so that from a holomorphic map
*T: C(n)* --> **PSL**(2,**C**)
one can define *f_T: C(n)* --> *C(n)* by
*f_T(Q)=f_T({q_1,...,q_n})={T(Q)q_1,...,T(Q)q_n}*
for all *Q={q_1,...,q_n}* in *C(n)*.
A holomorphic map of form *f_T* generated from some such holomorphic
*T* is said to be *tame*.

A continuous map *f: X --> Y*
of arcwise connected topological spaces is called
*non-cyclic* if the image *f*(pi_1(X))*
of the induced homomorphism *f*: pi_1(X) --> pi_1(Y)*
of the fundamental groups is not a cyclic group.

**Theorem** *For n<>4 every non-cyclic holomorphic map C(n) --> C(n)
is tame. In particular, for such n every automorphism of C(n) is tame.*

In view of the classical Cartan-Grauert theorem,
this implies that homotopy classes of non-cyclic holomorphic mappings
*C(n) --> C(n)* are in 1--1 correspondence with homotopy classes of all
continuous mappings *C(n)* --> **PSL**(2,**C**).
Here is another direct corollary:

**Corollary** * For n<>4 the orbits of the natural Aut(C(n))
action on C(n) coincide with the orbits of the diagonal
***PSL**(2,**C**) action on C(n). In particular,
C(n) / Aut(C(n)) == C(n) / **PSL**(2,**C**). The latter orbit space
may be viewed as the moduli space M(0,n) of the Riemann sphere
with n punctures.

Coffee, Cookies at the faculty lounge at 3:30.

You are invited to join the speaker for further discussion after the
talk at Beit Belgia.

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