Oct. 24, 2015
The 42th Symposium on Transformation Groups
Abstrac ( PDF )
 Tomohiro Kawakami (Wakayama University)
Title : Definable C^r submanifolds in a definable C^r manifold
Abstract:
Let $X$ be a definable $C^r$ manifold, $Y_1, Y_2$ definably compact definable $C^r$ submanifolds of $X$
such that $\dim Y_1+\dim Y_2<\dim X$ and $Y_1$ has a trivial normal bundle.
We prove that there exists a definable isotopy $\{h_p\}_{p \in J}$ such that
$h_0=id_X$ and $h_1(Y_1) \cap Y_2=\emptyset$.

Koh Ohashi (The University of Tokyo)
Title : Equivariant maps between representation spheres of cyclic pgroups
Abstract:
We will consider necessary conditions for
the existence of equivariant maps between the unit spheres of unitary
representations of a cyclic $p$group $G$.
Bartsch gave a necessary condition
for some unitary representations of $G$ by using equivariant $K$theory.
In this talk, we will give two necessary conditions following Bartsch's
approach.
One is a generalization of Bartsch's result for all unitary
representations of $G$ with fixed point free actions.
The other gives a stronger necessary condition under stronger
assumptions.

Takahiro Matsushita （The University of Tokyo）
Title : On the lower bounds obtained from Hom complexes
Abstract:
A coloring of a simple graph $G$ is to assign a color to each vertex of
$G$ so that adjacent vertices have different colors. The smallest number
of colores we need to color $G$ is called the chromatic number of $G$,
and is denoted by $\chi(G)$.
Hom complex ${\rm Hom}(T,G)$ is a CWcomplex associated to a pair of
graphs $T$ and $G$. The graph $T$ is called a homotopy test graph if the
inequality
$$\chi(G) > {\rm conn}({\rm Hom}(T,G)) + \chi(T) $$
holds for every graph $G$, where ${\rm conn}(X)$ is the largest number $
n$ such that $X$ is $n$connected. (However, we put ${\rm conn}(X)= \
infty$ if $X = \emptyset$.) It is known that the complete graph $K_n$
with $n \geq 2$ is a homotopy test graph (Lov\'asz, BabsonKozlov), and
the odd cycle $C_{2n+1}$ with $n \geq 1$ (BabsonKozlov).
However, there is a graph $G$ such that the difference between the above
inequality is quite large, when $T=K_2$. In particular, Walker (1983)
showed that for each positive integer $n$, there is a graph $G$ such
that
$$\chi(G)  {\rm conn}({\rm Hom}(K_2,G)) > n,$$
and he also showed that there are graphs $G_1,G_2$ such that ${\rm Hom}(
K_2,G_1) \simeq {\rm Hom}(K_2,G_2)$ but $\chi(G_1) \neq \chi(G_2)$.
In this talk we showed the following result strengthening Walker's
result. Let $T$ be a finite graph, $G$ a graph whose chromatic number is
greater than 2, and $n$ a positive integer. Then there is a graph $H$
which contains $G$ as a subgraph and satisfies the following properties:
the inclusion ${\rm Hom}(T,G) \rightarrow {\rm Hom}(T,H)$ is a homotopy
equivalence and $\chi(H) \geq n$.
In particular, any homotopy invariant of ${\rm Hom}(T,G)$ does not give
an upper bound for the chromatic number of $G$.

Takashi Sakai (Tokyo Metropolitan University)
Title : The intersection of two real flag manifolds in a complex flag manifold
Abstract:
Tanaka and Tasaki studied the antipodal structure of the intersection of two real
forms in Hermitian symmetric spaces of compact type. An orbit of the adjoint
representation of a compact connected Lie group $G$ admits a $G$invariant K\"ahler
structure, and called a complex flag manifold. Furthermore, any simplyconnected
compact homogeneous K\"ahler manifold is a complex flag manifold. Using a torus
action, we can define (generalized) antipodal sets of a complex flag manifold.
An orbit of the linear isotropy representation of the compact symmetric space $G/K$
is called a real flag manifold, and is embedded in a complex flag manifold as
a real form. In this talk, we will give a necessary and sufficient condition for
two real flag manifolds, which are not necessarily congruent with each other,
in a complex flag manifold to intersect transversally in terms of symmetric triads.
Moreover we will show that the intersection is an orbit of a certain Weyl group
and an antipodal set, if the intersection is discrete.
This talk is based on a join work with Osamu Ikawa, Hiroshi Iriyeh, Takayuki Okuda
and Hiroyuki Tasaki.

Takayuki Okuda (Hioshima University)
Title : Totally geodesic surfaces in Riemannian symmetric spaces
and nilpotent orbits
Abstract:
Let X = G/K be a Riemannian symmetric space of noncompact type with connected G.
We are interested in classifications of totally geodesic (complete) submanifolds in X = G/K.
It is known that any totally geodesic submanifold of X is homogeneous,
more precisely, any such submanifold can be realized as an orbit of a reductive subgroup L of G acting on X.
In this talk, we give a onetoone correspondence between
the set of Gconjugate classes of nonflat totally geodesic oriented surfaces in the Riemannian symmetric space X = G/K
and the set of nilpotent orbits of the real semisimple Lie algebra Lie G, which were already classified in Representation theory.
This is joint work with Akira Kubo (Hiroshima Shudo Univ.), Katsuya Mashimo (Hosei Univ.) and Hiroshi Tamaru (Hiroshima Univ.).

Takayuki Masuda (Osaka University)
Title : Combination of Lorentzian transformation groups
Abstract:
We will consider classification of affine Lorentzian transformation groups
acting on $(2+1)$Minkowski spacetime properly discontinuously. All
noncocompact affine Lorentzian transformations acting properly
discontinuously are obtained by affine deformations of noncompact
hyperbolic surfaces. Let a hyperbolic surface fixed. We introduce a new
parameter, the affine twist parameter. Then we show that the affine
deformation space can be parametrized by Margulis invariants and affine
twist parameters. We will also talk about some topics associated with this
theory.

Ryoma Kobayashi (Ishikawa National College of Technology)
Title : A normal generating set for the Torelli group of a compact nonorientable surface
Abstract:
The mapping class group M(N) of a compact nonorientable surface N is defined as the group consisting of isotopy classes of diffeomorphisms over N fixing the boundary pointwise.
The Torelli group I(N) of a compact nonorientable surface N is defined as the subgroup of M(N) consisting of mapping classes acting trivially on the integral first homology group of N.
Hirose and the speaker have obtained a normal generating set for I(N), where N is a genus g(>3) closed nonorientable surface. In this work, we obtain a normal generating set for I(N),
where N is a genus g(>3) compact nonorientable surface with b(>0) boundary components.

Haozhi Zeng (Osaka City University)
Title : Cohomology of nonorientable toric origami manifolds
Abstract:
Toric origami manifolds, introduced by A. Canas da Silva, V. Guillemin and A. R. Pires, are generalization of symplectic toric manifolds.
In this talk we will discuss cohomology groups of some kinds of toric origami manifolds.
This talk is based on the joint work with Anton Ayzenberg, Mikiya Masuda and Seonjeong Park.

Norihiko Minami (Nagoya Institute of Technology)
Title : Homotopy theoretical methods for rational points
Abstract:
In recent years, homotopy theory is applied to study
rational points. In this talik, I shall survey this
situation from the homotopy theoretical point of view.
If possible, I would like to mention a possibility
of some homotopy theoretical method for applications
to rational points.

Masafumi Sugimura (Okayama University)
Title : Remark on the Burnside ring of A_5
Abstract:
Let $G$ be a finite nontrivial group and
$\cF$ a set of subgroups of $G$
which is closed under conjugations
by elements in $G$ and under taking subgroups.
Let $\mathfrak{F}$ denote the category whose objects are elements in $\cF$
and whose morphisms are triples $(H,g,K)$
such that $H$, $K$ $\in \cF$ and $g \in G$
with $gHg^{1} \subset K$.
We denote by $A(G)$ the Burnside ring of $G$.
For each morphism $(H,g,K)$, we have the associated
homomorphism $(H, g, K)^{*} : A(K) \to A(H)$.
In particular, if $H \leq K$ then
$(H, e, K)^{*}$ agrees with ${\rm{res}}^K_H : A(K) \to A(H)$.
We denote by $A(\mathfrak{F})$ the inverse limit
$$
{\text{invlim}}_{\mathfrak{F}}
A(\thinspace \bullet \thinspace ) \ \ \left(
\subset \prod_{H \in \cF} A(H) \right)
$$
associated with the category $\mathfrak{F}$.
We denote by ${\rm{res}}_{\cF}$
the restriction homomorphism $A(G) \to A(\mathfrak{F})$ and
by $A(G)_{\cF}$ the image of the map ${\rm{res}}_{\cF}$.
It is interesting to ask whether $A(G)_\cF$ coinsides with
$A(\mathfrak{F})$.
Y.Hara and M.Morimoto showed that in the case of $G = A_4$,
alternating group on four letters,
the answer is affirmative.
We consider that issue in the case of $G = A_5$.
( PDF )

Masaharu Morimoto （Okayama University）
Title : Realization of closed manifolds as A_5fixed point sets
Abstract:
Let $G$ be the alternating group on $5$ letters and
let $\fM$ denote the family of closed smooth manifolds
which can be obtained as the $G$fixed point sets
of smooth $G$actions on disks.
Let $S^n$ denote the sphere of dimension $n$, let
$P_{\bC}^n$ and $P_{\bR}^n$ denote the complex and
real projective space of dimension $n$, respectively,
and let $L^{2n1}_m$ denote the lens space
$S(\bC^n)/C_m$, where $m$ is an integer $\geq 3$ and
$$C_m = \{ z \in\bC \  \ z^m = 1 \}.$$
Let $M \in \fM$.
We will discuss whether $M$
can be realized as the $G$fixed point sets of
smooth $G$actions on $S^n$, $P_{\bC}^n$,
$P_{\bR}^n$ and $L^{2n1}_m$.
( PDF )

Takahito Naito (The University of Tokyo)
Title : String topology of the Borel constructions
Abstract:
The theory of string topology is a study of algebraic structures
on the homology of the free loop space (called the loop homology).
It is known that the loop homology of a manifold has many algebraic
structures, for example graded commutative algebra, BatalinVilkovisky
algebra and 2dimensional TQFT.
In this talk, we will study string topology of the Borel constructions,
especially TQFT structure on the loop homology. Moreover, we will
introduce some properties and give some computational examples of
the loop homology.

Masaki Nakagawa (Okayama University)
Title : Gysin formulas for the universal HallLittlwood functions
Abstract:
For certain kinds of maps (e.g., smooth maps between compact, oriented manifolds, or projections of fiber bundles),
the Gysin maps (sometimes called push forwards, Umkehr maps, integration over the fiber etc.) can be defined in ordinary cohomology.
In Schubert calculus, there are many formulas for Gysin maps for Grassmann and flag bundles which relate Schubert classes with Schur S and Pfunctions (Damon, Fulton, HarrisTu, Pragacz). Recently Pragacz generalized the above formulas to the Hall Littlewood functions which interpolate Schur Sfunctions and Pfunctions.
Our main goal is to generalize the above formulas in ordinary cohomology to the complex cobordism theory which is universal among complexoriented generalized cohomology theories.
More precisely, we introduce the universal analogue of the HallLittlewood functions,
which we call the universal HallLittlwood functions,
and give analogous Gysin formulas in complex cobordism theory. This is joint work with H. Naruse.
( PDF )