## Tuesday Seminar on Topology

Seminar information archive ～06/09｜Next seminar｜Future seminars 06/10～

Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |

### 2007/01/23

16:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

The twistor correspondence for self-dual Zollfrei metrics

----their singularities and reduction

On the homology group of $Out(F_n)$

**中田 文憲**(東京大学大学院数理科学研究科) 16:30-17:30The twistor correspondence for self-dual Zollfrei metrics

----their singularities and reduction

[ Abstract ]

C. LeBrun and L. J. Mason investigated a twistor-type correspondence

between indefinite conformal structures of signature (2,2) with some properties

and totally real embeddings from RP^3 to CP^3.

In this talk, two themes following LeBrun and Mason are explained.

First we consider an additional structure:

the conformal structure is equipped with a null surface foliation

which has some singularity.

We establish a global twistor correspondence for such structures,

and we show that a low dimensional correspondence

between some quotient spaces is induced from this twistor correspondence.

Next we formulate a certain singularity for the conformal structures.

We generalize the formulation of LeBrun and Mason's theorem

admitting the singularity, and we show explicit examples.

C. LeBrun and L. J. Mason investigated a twistor-type correspondence

between indefinite conformal structures of signature (2,2) with some properties

and totally real embeddings from RP^3 to CP^3.

In this talk, two themes following LeBrun and Mason are explained.

First we consider an additional structure:

the conformal structure is equipped with a null surface foliation

which has some singularity.

We establish a global twistor correspondence for such structures,

and we show that a low dimensional correspondence

between some quotient spaces is induced from this twistor correspondence.

Next we formulate a certain singularity for the conformal structures.

We generalize the formulation of LeBrun and Mason's theorem

admitting the singularity, and we show explicit examples.

**大橋 了**(東京大学大学院数理科学研究科) 17:30-18:30On the homology group of $Out(F_n)$

[ Abstract ]

Suppose $F_n$ is the free group of rank $n$,

$Out(F_n) = Aut(F_n)/Inn(F_n)$ the outer automorphism group of $F_n$.

We compute $H_*(Out(F_n);\\mathbb{Q})$ for $n\\leq 6$ and conclude

that non-trivial classes in this range are generated

by Morita classes $\\mu_i \\in H_{4i}(Out(F_{2i+2});\\mathbb{Q})$.

Also we compute odd primary part of $H^*(Out(F_4);\\mathbb{Z})$.

Suppose $F_n$ is the free group of rank $n$,

$Out(F_n) = Aut(F_n)/Inn(F_n)$ the outer automorphism group of $F_n$.

We compute $H_*(Out(F_n);\\mathbb{Q})$ for $n\\leq 6$ and conclude

that non-trivial classes in this range are generated

by Morita classes $\\mu_i \\in H_{4i}(Out(F_{2i+2});\\mathbb{Q})$.

Also we compute odd primary part of $H^*(Out(F_4);\\mathbb{Z})$.