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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