Jerusalem Mathematics Colloquium

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Thursday, 15th November 2001, 4:00 pm
Mathematics Building, Lecture Hall 2

Vladimir Lin

"Holomorphic maps of Riemann sphere's configuration spaces"


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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