“The 35th Symposium on Transformation Groups”
Abstracts of the Talks
November 11(Tue) – 13(Thu), 2008

November 11
13:30 – 14:20
Yoshinobu Kamishima, Two Rigidity Conjectures from Transformation Groups . . . . .2
14:35 – 15:25
Takeshi Ikeda, Degeneracy Loci Formulae from a Point of View of Equivariant Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
15:40 – 16:30
Kenji Tsuboi, Finite Transformation Groups and the Equivariant Determinant of Elliptic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
16:45 – 17:35
Ikumitsu Nagasaki, On the Isovariant Hopf Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

November 12
9:30 – 10:20
Norihiko Minami, Lurie’s Quasi Category Topos Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
10:35 – 11:25
Takao Satoh, On the Fourth Johnson Homomorphism of the Automorphism Group ofa Free Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
14:00 – 14:50
StanisÃlaw Spież, Borsuk-Ulam Type Theorems and Equilibria in a Class of Games . 9
15:05 – 15:55
Shigeyasu Kamiya, Shimizu’s Lemma for Complex Hyperbolic Space and its Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
16:10 – 17:00
Krzysztof PawaÃlowski, Current Trends in the Study of the Smith Equivalent Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

November 13
9:30 – 10:20
Tatsuhiko Yagasaki, Homeomorphism and Diﬀeomorphism Groups of Non-compact
Manifolds with the Whitney Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
10:35 – 11:25
Masaharu Morimoto, A New Theorem to Find Smith Equivalent Representations . .14

1

2

November 11 (Tue)Two Rigidity Conjectures from Transformation Groups
Yoshinobu Kamishima (Tokyo Metropolitan University)
Recall two types of Rigidity Conjecture/Result.
I. The Topological Borel conjecture (A. Borel 1960).
II. The Conformal rigidity of compact Riemannian manifold(Obata & Lelong-Ferrand 1970).
This is supported by the Vague conjecture by D’ambra and Gromov 1990. We shalldevelop this conjecture into the framework of Geometric topology from the viewpointof Transformation groups. In this talk, we study the following problems related to I,II.
• The Smooth Borel conjecture: Which compact aspherical smooth manifolds
with isomorphic fundamental groups must be diﬀeomorphic?
• The Obata & Lelong-Ferrand theorem to Lorentz manifolds - If a closed group
R acts conformally on a compact special Lorentz manifold, then is it conformal
to the conformally ﬂat Lorentz model S n−1,1 ≈ S n−1 × S 1 /Z?

Degeneracy Loci Formulae from a Point of View of Equivariant
Cohomology
Takeshi Ikeda (Okayama University of Science)
Let E, F be two vector bundles over a manifold X and ϕ : E → F a generic vector
bundle homomorphism. A question posed by Thom is, which cohomology class isdeﬁned by the set Σd (ϕ) ⊂ M consisting of points x where the linear map ϕx : Ex → Fx
has corank d? The answer, due to Porteous, is a determinant in terms of Chern classesof the bundles E, F . There are many generalization of this results. In this talk, I
would like to discuss these “degeneracy loci formulas” from a point of view of torus
equivariant cohomology.

Finite Transformation Groups and the Equivariant Determinant of Elliptic
Operators
Kenji Tsuboi (Tokyo University of Marine Science and Technology)
Assumption(1) A ﬁnite group G acts smoothly and eﬀectively on a closed smooth manifold
M.
(2) M has an almost complex structure and the G -action preserves the almost
complex structure.
(3) G contains an element g of order p whose ﬁxed point set consists of points.
Deﬁnition Let {q1 , · · · , qn } be the ﬁxed point set of g . Then g acts on the tangent
space Tqi M via multiplication by a diagonal matrix with diagonal entries (αpτi1 , · · · , αpτim )where αp is the primitive p-th root of unity.
We call the set τ = ((τ11 , · · · , τ1m ), · · · , (τn1 , · · · , τnm )) the rotation angle of g.

3

Equivalence of rotation angles is deﬁned bypermutation η1
permutation ηn
}|
{
z
}|
{
z
τ = ((τ11 , · · · , τ1m ), · · · , (τn1 , · · · , τnm )) ∼ τ = ( ( zτ11 , · · · , zτ1m ), · · · , ( zτn1 , · · · , zτnm ) )
|
{z
}
0

permutation ρ
where 1 ≤ z ≤ p − 1.
Equivariant determinant Let D be a G -equivariant elliptic operator. Then the equivariant determinant det(D, g) of D evaluated at g ∈ G is deﬁned bydet(D, g) := det(g| ker D)/ det(g|coker D) ∈ S 1 ⊂ C∗ .
Deﬁne a homomorphism ID ( • ) : G −→ R/Z by
1√
ID ( g) :=log det(D, g) .
2π −1`
Let L` be a complex G-line bundle deﬁned by L` = (∧m
C TM) ,

D` : Γ(S+ ⊗ L` ) −→ Γ(S− ⊗ L` ) (S± : half spinor bundles)
the Dirac operator deﬁned by the natural Spinc -structure of M , Td(T M ) the Toddclass of T M and [M ] the fundamental cycle of M .
Theorem Assume that p is an odd prime number. Then for 1 ≤ z ≤ p − 1 , wehavep − 1 `c1 (T M )ID` (g z ) ≡e
Td(T M )[M ]
2p
(Ã m
! p−1
)
p−1
n
Y j X
X
1 X
z
κij
fm,p (s) − p
fm,p (s)Φs (zτ )(mod Z)
+ m+2
p
s=0
s=0
j=1
i=1
where κij are integers determined by
κi1 = τi1 , 1 ≤ κij ≤ p − 1 , κij ≡ τ i j−1 τij (mod p) (i ≥ 2)
(1 ≤ τ i j−1 ≤ p − 1 , τi j−1 τ i,j−1 ≡ 1 (mod p)) ,fm,p (s) is an integral polynomial deﬁned by
µ
¶µ
¶
m+1
X m−κ
X
m−κ
−tp + s + m − ptfm,p (s) =(−1)
tm
κ=0 t=0
¶X
µ ¶µ
¶
m+1
κ
Xµ
s
κ
pv
v
(−1)
m+1−u
v
u
u=κ

v=0

and Φs (zτ ) is an integer deﬁned by¯¯ 0 ≤ j11 < zκi1 = zτi1




¯




¯
0
≤
j
,
·
·
·
,
j
<
κ


r1
rr
ir


¯




¯
µ
¶
(2
≤
r
≤
m)


¯
P
j11 , j21 , j22 , · · · ¯
m
zτ
1
+
(`
+
1)
Φs (zτ ) = #
.
ij
j=1
¯

 jm1 , · · · , jmm¯ +j11 + zτi1 (j21 + j22 ) + · · ·




¯




¯ +zτi m−1 (jm1 + · · · + jmm )




¯




¯ +szτim ≡ 0 (mod p)
Using the theorem above, we can obtain information on the rotation angle.

4

References[1] W. J. Harvey, Cyclic groups of automorphisms of a compact Riemann surface,
Quart. J. Math. 17 (1966), 86-97.
[2] S. P. Kerckhoﬀ, The Nielsen realization problem, Ann. of Math. 117 (1983),
235-265.
[3] K. Tsuboi, The ﬁnite group action and the equivariant determinant of elliptic
operators, J. Math. Soc. Japan. 57 (2005), 95-113.

On the Isovariant Hopf Theorem
Ikumitsu Nagasaki (Kyoto Prefectural University of Medicine)
Let M be an n-dimensional connected orientable closed manifold. It is a well-known
result, called the Hopf theorem or classiﬁcation theorem of Hopf, that homotopy classesof continuous maps from M to the n-dimensional sphere S n are classiﬁed by theirdegree. Namely, the degree of f : M → S n induces the map deg : [M, S n ] → Z, andthen the Hopf theorem states that
Theorem. The map deg : [M, S n ] → Z is a bijection.
Equivariant versions of the Hopf theorem and related topics have been studied by
Segal, Rubinsztein, Petrie, tom Dieck, Balanov, Ferrario, and others. We would like todiscuss an isovariant version of the Hopf theorem under certain conditions.
Let G be a ﬁnite group and X, Y G-spaces.
Deﬁnition. A G-isovariant map f : X → Y is called G-isovariant if f preserves theisotropy groups, i.e., Gf (x) = Gx for all x ∈ X.
A G-homotopy F : X × I → Y is called an isovariant homotopy if F is G-isovariant.
Let M be a connected, orientable, closed G-manifold with free and orientationpreserving action, and SW a representation sphere (i.e., the unit sphere of unitary
representation W of G). In this talk, using the multidegree introduced by [1], [2], we
determine the isovariant homotopy set [M, SW ]isov
G under the Borsuk-Ulam inequality:
dim M + 1 ≤ dim SW − dim SW >1 ,where SW >1 denotes the singular set of SW .
This work is joint with F. Ushitaki.
[1] I. Nagasaki and F. Ushitaki, Isovariant maps from free Cn -manifolds to representation spheres, Topol. and its Appl. 155 (2008), 1066-1076.
[2] I. Nagasaki and F. Ushitaki, Classiﬁcation of isovariant maps between free Gmanifolds and representation spheres, in preparation.

5

November 12 (Wed)Lurie’s Quasi Category Topos Theory
Norihiko Minami (Nagoya Institute of Technology)
In a series of papers:[BOOK]: Jacob Lurie, Higher Topos Theory, September 26, 2008, 731p.
[DAGI]: Jacob Lurie, DAGI - Stable Inﬁnity Categories, September 9, 2008, 72p.
[DAGII]: Jacob Lurie, DAGII - Noncommutative Algebra, September 9, 2008,
170p.
[DAGIII]: Jacob Lurie, DAGIII - Commutative Algebra, September 9, 2008,
80p.
[DAGIV]: Jacob Lurie, DAGIV - Deformation Theory, September 9, 2008, 73p.
[DAGV]: Jacob Lurie, DAGV - Structured Spaces, September 9, 2008, 137p;[DAGVI]: Lurie, J. Derived Algebraic Geometry VI: Spectral Schemes. In preparation.
Jacob Lurie embarked on the project of rewriting the Grothendieck theory of algebraic
geometry, using ∞-category (which was ﬁrst deﬁned by Boardman-Vogt, and called“quasi category” by Joyal). In spite of their obvious importance, their long page-length
made many reluctant to read these important series of works.
To remedy this situation, I wrote a survey paper on the ∞-category Yoneda’s lemmapresented in the thickest and most foundational [BOOK]:[米田]: 南範彦，”Lurie’s quasi category Yoneda’s lemma”, 京都大学数理解析研究
所講究録 1612 , 変換群の幾何とその周辺, 21-40 (2008).
Now, the present talk is a continuation of this survey paper, and the purpose is toexplain the following instructive analogies presented in the [BOOK]:¶³
Vague analogies between Higher Category Theory and Algebra
Remark (BOOK, Remark 6.1.1.3.). Let X be an ∞-category. The assumptionthat colimits in X are universal can be viewed as a kind of distributive law. We
have the following table of vague analogies:
Higher Category Theory
Algebra∞-Category
Set
Presentable ∞-Category
Abelian Group
Colimits
Sums
Limits
Products
lim→ (Xα ) ×S T ' lim→ (Xα ×S T ) (x + y)z = xz + yz∞-Topos
Commutative Ringµ´For this purpose, I shall give some ideas and background about the following deﬁnitions and theorems centering these analogies:¶

presentable ∞-category³

Deﬁnition 1 (BOOK, Deﬁnition 5.5.0.18.). An ∞-category C ispresentable if C is accessible and admits small colimits.
µ

´

6

¶

Simpson’s characterizqation of presentable ∞-category

Theorem 2 (BOOK, Theorem 5.5.1.1 (Simpson) ). Let C be an ∞-category. The
following conditions are equivalent:(1) The ∞-category C is presentable.
(2) The ∞-category C is accessible, and for every regular cardinal κ, the fullsubcategory C κ admits κ-small colimits.
(3) There exists a regular cardinal κ such C is κ-accessible and C κ admits κsmall colimits.
(4) There exists a regular cardinal κ, a small ∞-category D which admits κsmall colimits, and an equivalence Indκ D → C.
(5) There exists a small ∞-category D such that C is an accessible localizationof P(D).
(6) The ∞-category C is locally small, admits small colimits, and there exists a
regular cardinal κ and a (small) set S of κ-compact objects of C such thatevery object of C is a colimit of a small diagram taking values in the fullsubcategory of C spanned by S.

µ¶∞-topos

Deﬁnition 3 (BOOK, Deﬁnition 6.1.0.4.). Let X be an ∞-category. We will saythat X is an ∞-topos if there exists a small ∞-category C and an accessible left
exact localization functor P(C) → X .

µ¶Giraud’s theorem of ∞-topos

Theorem 4 (BOOK, Theorem 6.1.0.6.). Let X be an ∞-category. The following
conditions are equivalent:(1) The ∞-category X is an ∞-topos.
(2) The ∞-category X is presentable, and for every small simplicial set K andevery natural transformation α : p → q of diagrams p, q : K . → X , thefollowing condition is satisﬁed:- If q is a colimit diagram and α = α|K is a Cartesian transformation, then p is a colimit diagram if and only if α is a Cartesiantransformation.
(3) The ∞-category X satisﬁes the following ∞-categorical analogues of Giraud’s axioms:(i) The ∞-category X is presentable.
(ii) Colimits in X are universal.
(iii) Coproducts in X are disjoint.
(iv) Every groupoid object of X is eﬀective.

µFinally, I do not claim any originality in my talk. However, since the concept of ∞topos is, as the title of [BOOK] suggests, “the” most important concept in [BOOK], I
hope this survey talk would make audicnce feel familiar with [BOOK].

³

´
³

´
³

´

7

On the Fourth Johnson Homomorphism of the Automorphism Group of aFree Group
Takao Satoh (Kyoto University, takao@math.kyoto-u.ac.jp)
Keywords: Automrophism group of a free group, IA-automorphism group, Johnson homomorphism, Trace map
Abstract: In this talk we consider the Johnson homomorphism of the automorphism groupof a free group with respect to the lower central series of the IA-automorphism group of afree group. In particular, we determine the ratoinal cokernel of the forth Johnson homomorphism, and show that there appears a new obsturustion for the surjectivity of the Johnson
homomorphism. Furthermore we characterize this obstruction using trace maps.

Let Fn be a free group of rank n ≥ 2, and Aut Fn the automorphism group of Fn .
Let denote ρ : Aut Fn → Aut H the natural homomorphism induced from the abelianization H of Fn . The kernel of ρ is called the IA-automorphism group of Fn , denotedby IAn . The IA-automorphism group IAn reﬂects many richness and complexity ofthe structure of Aut Fn , and plays important roles on various studies of Aut Fn . The
purpose of our research is to clarlify the group structure of IAn .
Although the study of the IA-automorphism group has a long history since its ﬁnitely
many generators were obtained by Magnus [10] in 1935, the combinatorial group structure of IAn is still quite complicated. For instance, any presentation for IAn is notknown in general. Nielsen [12] showed that IA2 coincides with the inner automorphsim
group Inn F2 , hence, is a free group of rank 2. For n ≥ 3, however, IAn is much larger
than the inner automorphism group Inn Fn . Krstić and McCool [9] showed that IA3 is
not ﬁnitely presentable. For n ≥ 4, it is not known whether IAn is ﬁnitely presentable
or not.
In this talk, in order to study the group structure of IAn , we consider the Johnson
ﬁltration
IAn = An (1) ⊃ An (2) ⊃ · · ·consisting of certain normal subgroups of Aut Fn , and the Johnson homomorphisms
τk : grk (An ) → H ∗ ⊗Z Ln (k + 1)
deﬁned on each graded quotient of the Johnson ﬁltration. The study of the Johnson
homomorphisms was originally begun in 1980 by D. Johnson [6] who determined theabelianization of the Torelli subgroup of a mapping class group of a surface in [7]. Thenthe theory of the Johnson homomorphisms has been developed by many authors. Now,
there is a broad range of results for it. (For example, see [5] and [11].)
By virtue of the injectivity of the Johnson homomorphisms, considering their images,
we can divide IAn into inﬁnitely many free abelian groups of ﬁnite rank. These pieces
are regarded as one by one approximations of IAn . Hence, to determine the image of
the Johnson homomorphisms, or equivalently to determine their cokernel is one of themost basic problems. In this talk, in particular, we are interested in the irreducible
decomposition of the cokernel of τk,Q = τk ⊗ idQ as a GL(n, Z)-module. Now, for1 ≤ k ≤ 3, the cokernel of τk,Q is completely determined. (See [1], [13] and [14] fork = 1, 2 and 3 respectively.) In general, however it is quite hard problem to solve since
we can not obtain an explicit generating system of each grk (An ) easily.
To avoid this diﬃculty, we consider the lower central series A0n (1) = IAn , A0n (2),. . . of IAn . Since the Johnson ﬁltration is central, A0n (k) ⊂ An (k) for k ≥ 1. It isconjectured that A0n (k) = An (k) for each k ≥ 1 by Andreadakis who showed A02 (k) =8

A2 (k) for each k ≥ 1 and A03 (3) = A3 (3) in [1]. Now, we have A0n (2) = An (2) due toCohen-Pakianathan [2, 3], Farb [4] and Kawazumi [8]. Furthermore A0n (3) has at mostﬁnite index in An (3) due to Pettet [13].
For each k ≥ 1, set grk (A0n ) := A0n (k)/A0n (k + 1). Since IAn is ﬁnitely generated asabove, each grk (A0n ) is also ﬁntely generated as an abelian group. Then we can deﬁne
the Johnson homomorphisms
τk0 : grk (A0n ) → H ∗ ⊗Z Ln (k + 1)
with respect to the lower central series of IAn by a similar way of deﬁning τk . Since
grk (A0n ) is ﬁntely generated, it is easier to study the cokernel of τk0 than that of τk .
Furthermore, It is also important to determine Coker(τk0 ) from the view point of thestudy of the diﬀerence between the Johnson ﬁltration and the lower central series of
IAn . In this talk, as a consective result of our research [14], we determine the rational0
cokernel of the forth Johnson homomorphism τ4,Q:= τ40 ⊗ idQ .
Theorem 1. For any n ≥ 6,
[2,12 ]

0
Coker(τ4,Q) = S 4 HQ ⊕ HQ

[2,2]

⊕ HQ .

In the right hand side of the equation above, the ﬁrst term S 4 HQ is called the Morita
obstruction for the surjectivity of the Johnson homomorphism, which can be detectedby the Morita trace Tr[k] . The second term is an obstruction which can be detectedby the trace map Tr[2,12 ] constructed in our previous paper [14]. The ﬁnial term is anobstruction of new type. It seems that this obstruction appear due to the speciality ofthe degree 4. In this talk, we construct a GL(n, Z)-equivariant homomorphism Tr[2,2]
[2,2]
[2,2]
which detect HQ , and call it a trace map for HQ .
References[1] S. Andreadakis; On the automorphisms of free groups and free nilpotent groups, Proc. London
Math. Soc. (3) 15 (1965), 239-268.
[2] F. Cohen and J. Pakianathan; On Automorphism Groups of Free Groups, and Their Nilpotent
Quotients, preprint.
[3] F. Cohen and J. Pakianathan; On subgroups of the automorphism group of a free group andassociated graded Lie algebras, preprint.
[4] B. Farb; Automorphisms of Fn which act trivially on homology, in preparation.
[5] R. Hain; Inﬁnitesimal presentations of the Torelli group, Journal of the American MathematicalSociety 10 (1997), 597-651.
[6] D. Johnson; An abelian quotient of the mapping class group, Math. Ann. 249 (1980), 225-242.
[7] D. Johnson; The strucure of the Torelli group III: The abelianization of Ig , Topology 24 (1985),
127-144.
[8] N. Kawazumi; Cohomological aspects of Magnus expansions, preprint, arXiv:math.GT/0505497.
[9] S. Krstić, J. McCool; The non-ﬁnite presentability in IA(F3 ) and GL2 (Z[t, t−1 ]), Invent. Math.
129 (1997), 595-606.
[10] W. Magnus; Über n-dimensinale Gittertransformationen, Acta Math. 64 (1935), 353-367.
[11] S. Morita; Structure of the mapping class groups of surfaces: a survey and a prospect, Geometry
and Topology Monographs Vol. 2 (1999), 349-406.
[12] J. Nielsen; Die Isomorphismen der allgemeinen unendlichen Gruppe mit zwei Erzeugenden, Math.
Ann. 78 (1918), 385-397.
[13] A. Pettet; The Johnson homomorphism and the second cohomology of IAn , Algebraic and Geometric Topology 5 (2005) 725-740.
[14] T. Satoh; New obstructions for the surjectivity of the Johnson homomorphism of the automorphism group of a free group, Journal of the London Mathematical Society, (2) 74 (2006) 341-360.

9

Borsuk-Ulam Type Theorems and Equilibria in a Class of Games
StanisÃlaw Spież (Polish Academy of Sciences)
In 1968, R. Aumann, M. Mashler and R. Sterns posed a problem, whether any undiscounted inﬁnitely repeated two-person game of incomplete information on one side has
a Nash equilibrium. A brief description of these games is as follows. A game between
two players A and B proceeds in inﬁnitely many successive stages. In the 0-stage a k is
chosen from a ﬁnite set K of ”states of nature” according to a probability distribution
p0 ∈ int∆(K). In any subsequent stage each of the players selects a ”pure action” froma ﬁnite sets I (for A) and J (for B), gaining a stage-payoﬀ Ak (i, j) (for A) or Bk (i, j)(for B), which depends only on the pure actions i ∈ I and j ∈ J selected in this stage
and the ”true state of nature” k, chosen at stage 0. At any stage the players also know
the pure actions both of them took on proceedings stages and A (but not B) knows
the outcome k ∈ K of the 0-stage. We settle in the positive the problem stated above.
We also extend this result to more general games.
Several classical proofs in game theory depend on various ﬁxed point and related
theorems. Our proofs depend on new topological results. One of them, in its simplest
form, states that if x0 is a point of a compact subset C of Rn and f : C → Y is amapping such that dimension of f (intC) is less then n, then in the boundary of C
there exists a set C0 mapped by f into a singleton and containing x0 in its convex hull.
The resemblance with Borsuk-Ulam theorem is that if C is is an n-ball and Y = Rn−1 ,then the later says that C0 may be taken so as to consist of two points only. We also
prove a parametric version version of Borsuk-Ulam theorem.
This research is joint with T. Schick, R. S. Simon and H. Toruńczyk.

Shimizu’s Lemma for Complex Hyperbolic Space and its Application
Shigeyasu Kamiya (Okayama University of Science)
It is important to ﬁnd some conditions for a group to be discrete or non-discrete.
A well known result of Shimizu states the following:
Theorem. Let G be a discrete subgroup of PSL(2, R) containing the parabolic element
A(z) = z + t for some t > 0. If B(z) = (az + b)/(cz + d) is an element in G with c 6= 0,
then |c| ≥ 1/t.
This is called Shimizu’s Lemma.
Geometrically this result says that the radius rB of the isometric circle I(B) of B
satisﬁes rB ≤ t. Equivalently, the horoball Ut of height t is precisely invariant under
G∞ in G.
In this talk we show Shimizu’s lemma for complex hyperbolic space and we discuss
its application to complex hyperbolic triangle groups of type (n, n, ∞).

10Current Trends in the Study of the Smith Equivalent Representations
Krzysztof PawaÃlowski (Adam Mickiewicz University)
This is a joint work in progress of Krzysztof Pawalowski and Toshio
Sumi.
Tangent spaces as group modules
Let G be a ﬁnite group. By a real G-module we mean a ﬁnite dimensional real vector
space with a linear action of G given by a representation G → GL(n, R) of G. If G actssmoothly on a manifold M , then at any point a ∈ M ﬁxed under the action of G, thetangent space Ta (M ) can be considered as a real G-module. The action of G on Ta (M )is deﬁned by gv = Da θg (v) for all g ∈ G, v ∈ Ta (M ), where Da θg : Ta (M ) → Ta (M )is the derivative at the point a of the map θg : M → M given by θg (x) = gx for allx ∈ M.
According to the Slice Theorem, the action of G on the tangent space Ta (M ) is
equivalent to the action of G on some open neighborhood of a in M . Therefore, theG-module Ta (M ) describes the local behaviour of the action of G on M around theﬁxed point a.
The Smith isomorphism question
In 1960, P. A. Smith [16] posed the following question. If a ﬁnite group G actssmoothy on a sphere S with exactly two ﬁxed points a and b, is it true that as real
G-modules, the tangent spaces Ta (S) and Tb (S) are always isomorphic?
Let G be a ﬁnite group. Two real G-modules U and V are called Smith equivalent if
as real G-modules,
U∼= Ta (S) and V ∼= Tb (S)for a smooth action of G on a homotopy sphere S with exactly two ﬁxed points a andb. If in addition the ﬁxed point set S g is connected for any element g ∈ G of order 2k
for k ≥ 3, then U and V are called L-Smith equivalent.
In 1996, Laitinen has posed the conjecture that for a ﬁnite Oliver group G, there
exist pairs of noisomorphic, L-Smith equivalent real G-modules if and only if rG ≥ 2,
where rG denotes the number of real conjugacy classes in G of elements g ∈ G not ofprime power order.
The Smith question is answered in many cases of G. The articles [2], [3], [4], [12] and[13] contain long lists of related references, while [5], [6], [9], [10], [14] and [15] show
the current trends in the study of the Smith equivalence problem and the Laitinen
conjecture.
By [8], in the Laitinen conjecture, the condition that rG ≥ 2 is necessary, but Morimoto [9] proves that this condition is not suﬃcient for G = Aut(A6 ), the automorphismgroup of the alternating group A6 . In [15], we show that this condition is not suﬃcientalso for some ﬁnite solvable Oliver groups G, including the aﬃne group G = Aﬀ(2, 3).
Answers to Smith isomorphism question
Our goal is to impose a condition on G, stronger than rG ≥ 2, which implies that forany ﬁnite Oliver group G satisfying the condition, there exist pairs of nonisomorphic,
L-Smith equivalent real G-modules. We describe this condition in the following way.
For any ﬁnite group G, we denote by Gnil the smallest normal subgroup of G suchthat the quotient group G/Gnil is nilpotent. We say that two elements x and y of G
form a nil-pair if xGnil = yGnil and the following two conclusions hold.

11(1) The elements x and y are not real conjugate in G, and the orders |x| and |y| ofx and y, respectively, are not prime powers (and thus rG ≥ 2).
(2) The elements x and y are in some gap subgroup of G, or the orders |x| and |y|are even and the involutions of hxi and hyi are conjugate in G.
We say that a ﬁnite group G is of nil-type, or G is a nil-type group, if there exists a
nil-pair of elements of G. Recall that a ﬁnite group G is called an Oliver group if there
does not exist a series of normal subgroups P E H E G such that P is a p-group, G/H
is a q-group for some primes p and q, possibly p = q, and H/P is cyclic. According toOliver [11], this algebraic condition is necessary and suﬃcient for G to have a smooth
ﬁxed point free action on a disk, or as it is shown in [7], to have a smooth one ﬁxed
point action on a sphere.
Theorem A. For any ﬁnite Oliver nil-type group G, there exist pairs of nonisomorphic
real G-modules U and V such that the following three conclusions hold.
G∼(1) ResG
P (U ) = ResP (V ) for P ∈ P(G), the family of prime power order subgroupsof G.
(2) dim U L = dim V L = 0 for L ∈ L(G), the family of large subgroup of G.
(3) U and V both satisfy the weak gap condition and they are L-Smith equivalent.
Recall that rG = 2 in the case where G = Aut(A6 ) or P ΣL(2, 27), a semi-direct
product of P SL(2, 27) and the automorphism group of the ﬁeld of 27 elements. As
one can check directly, the two nonsolvable groups Aut(A6 ) and P ΣL(2, 27) are not of
nil-type.
Theorem B. Except for Aut(A6 ) and P ΣL(2, 27), a ﬁnite nonsolvable group G is of
nil-type if and only if rG ≥ 2.
For G = Aut(A6 ), Morimoto [9] proves that any two Smith equivalent real G-modules
are isomorphic. In contrast, for G = P ΣL(2, 27), Morimoto [10] proves that there exist
pairs of nonisomorphic, L-Smith equivalent real G-modules. As by the results of [8],the conditon that rG ≥ 2 is necessary in the Laitinen conjecture, Theorems A and B,
and the papers [9] and [10] yield the following theorem.
Theorem C. Except for the automorphism group of the alternating group on six letters,
the Laitinen conjecture is true for any ﬁnite nonsolvable group.
This extends the result of [8] showing that the Laitinen conjecture holds for any
ﬁnite perfect group. Now, we are able to answer the Smith isomorphism question as
follows.
Theorem D. In either of the cases (1)–(6) below, there exist pairs of nonisomorphic,
Smith equivalent real G-modules if and only if rG ≥ 2.
(1) G is a ﬁnite simple group.
(2) G = P SL(n, q) or SL(n, q) for any n ≥ 2 and any prime power q.
(3) G = P Sp(n, q) or Sp(n, q) for any even n ≥ 2 and any prime power q.
(4) G = An or Sn for any n ≥ 2.
(5) G = P GL(n, q) or GL(n, q) for any n ≥ 2 and any prime power q.
(6) G = Aﬀ(n, q) for any n ≥ 2 and any prime power q, and G 6= Aﬀ(2, 3).
By the work of Atiyah and Bott [1], for G = Zp where p is prime, any two Smithequivalent real G-modules are isomorphic and by [13], the same conclusion is true forﬁnite nonabelian simple groups G with rG = 0 or 1. Therefore, in the case (1) of
Theorem D, the condition that rG ≥ 2 is necessary. In the remaining ﬁve cases (2)–(6),the necessity of this condition follows from direct arguments for the values of n and q
in the cases where rG = 0 or 1.

12

In Theorem D, the suﬃciency of the condition that rG ≥ 2 follows from equivariant
surgery. The work [13] covers the cases (1)–(4) using surgery under the gap condition,and by applying surgery under the weak gap condition, we obtain the conclusion in
the cases (5) and (6).

References[1] Atiyah, M. F., and Bott, R., A Lefschetz ﬁxed point formula for elliptic complexes: II. Applications, Ann. of Math. 88 (1968), 451–491.
[2] Cappell, S. E., Shaneson, J. L., Representations at ﬁxed points, Group Actions
on Manifolds, Contemp. Math. 36 (1985), 151–158.
[3] Cho, E. C., Suh, D. Y., Induction in equivariant K-theory and s-Smith equivalence of representations, Group Actions on Manifolds, Contemp. Math. 36(1985), 311–315.
[4] Dovermann, K. H., and Petrie, T., Smith equivalence of representations for odd
order cyclic groups, Topology 24 (1985), 283–305.
[5] Ju, X.M., The Smith set of the group S5 ×C2 ×· · ·×C2 , submitted for publicationin Osaka Journal of Mathematics, on July 2, 2008.
[6] Koto, A., Morimoto, M., and Qi, Y., The Smith sets of ﬁnite groups withnormal Sylow 2-subgroups and small nilquotients, J. Math. Kyoto Univ. 48(2008), 219–227.
[7] Laitinen, E., and Morimoto, M., Finite groups with smooth one ﬁxed point
actions on spheres, Forum Math. 10 (1998), 479–520.
[8] Laitinen, E., and PawaÃlowski, K., Smith equivalence of representations for ﬁnite
perfect groups, Proc. Amer. Math. Soc. 127 (1999), 297–307.
[9] Morimoto, M., Smith equivalent Aut(A6 )-representations are isomorphic, Proc.
Amer. Math. Soc. 136 (2008), 3683–3688.
[10] Morimoto, M., Nontrivial P(G)-matched S-related pairs for ﬁnite gap Olivergroups, Preprint, 2008.
[11] Oliver, R., Fixed point sets of group actions on ﬁnite acyclic complexes, Comment. Math. Helv. 50 (1975), 155–177.
[12] PawaÃlowski, K., Smith equivalence of group modules and the Laitinen conjecture.
A survey, in Geometry and Topology (ed. K. Grove, I.H. Madsen, E.K. Pedersen), Contemp. Math. 258 (2000), 343–350.
[13] PawaÃlowski, K., and Solomon, R., Smith equivalence and ﬁnite Oliver groups
with Laitinen number 0 or 1, Algebr. Geom. Topol. 2 (2002), 843–895.
[14] PawaÃlowski, K., and Sumi, T., Finite groups with Smith equivalent, nonisomorphic representations, in Proc. of the 33rd Symposium on Transformation
Groups, Yokohama 2006 (ed. T. Kawakami), Wing Co. Ltd., Wakayama,
Japan, 2007, 68–76.
[15] PawaÃlowski, K., and Sumi, T., The Laitinen Conjecture for ﬁnite solvable Olivergroups, to appear in Proc. Amer. Math. Soc.
[16] Smith, P. A., New results and old problems in ﬁnite transformation groups, Bull.
Amer. Math. Soc. 66 (1960), 401–415.
[17] Sumi, T., Gap modules for direct product groups, J. Math. Soc. Japan 53(2001), 975–990.
[18] Sumi, T., Gap modules for semidirect product groups, Kyushu J. Math. 58(2004), 33–58.

13

November 13 (Thu)Homeomorphism and Diﬀeomorphism Groups of Non-compact Manifolds
with the Whitney Topology
Tatsuhiko Yagasaki (Kyoto Institute of Technology)(Joint Work with T. Banakh, K. Mine, K. Sakai)
In this talk we discuss conditions which ensure that topological groups are locally
homeomorphic to R∞ × `2 , and apply this characterization to determine the local/gloal
topological types of homeomorphism/diﬀeomorphism groups of noncompact manifolds
endowed with the Whitney topology.
§1. Topological groups locally homeomorphic to R∞ × `2 .
Suppose G is a topological group. Our main concern is the following fundamental
problem.
Problem. When is G locally homeomorphic to (≈loc ) a topological linear space L ?
The case L = Rn was solved by A. Gleason, D. Montgomery - L. Zippin and the caseL = `2 was studied by T. Dobrowolski - H. Toruńczyk. In this talk we consider thecase L = R∞ × `2 , where R∞ is the direct limit of the sequence R1 ⊂ R2 ⊂ R3 ⊂ · · ·A tower of closed subgroups of G is a sequence (Gn )n of closed subgroups of G suchthatS
G1 ⊂ G 2 ⊂ · · · ,G = n Gn .
This tower yields the small box product ¡n Gn and the multiplication map
p : ¡n Gn → G : p(x1 , x2 , · · · ) = x1 · x2 · · · .
We say that(i) G carries the box topology with respect to (Gn ) if the map p is an open map,and(ii) (Gn ) is (locally) topologically complemented (TC) in G if each quotient map
Gn → Gn /Gn−1 has a (local) section.
Theorem. G ≈(loc) R∞ × `2 if(∗1 ) G carries the box topology with respect to (Gn )(∗2 ) (Gn )n is (locally) TC in G(∗3 ) Gn /Gn−1 ≈(loc) `2 (n ≥ 1)
§2. Homeomorphism and diﬀeomorphism groups of noncompact manifolds
endowed with the Whitney topology.
§2.1. Homeomorphism groups.
Suppose M is a non-compact connected n-manifold without boundary. Let H(M )denote the group of homeomorphisms of M endowed with the Whitney topology andHc (M ) denote the subgroup of homeomorphisms of M with compact support. Since
the identity component H0 (M ) of H(M ) is included in Hc (M ), we have the mappingclass group Mc (M ) = Hc (M )/H0 (M ) (with the discrete topology).
Proposition. (1) Hc (M ) is locally contractible.

(2) Hc (M ) ≈ H0 (M ) × Mc (M ).

In the case n = 1, 2 we obtain the complete classiﬁcation of topological types ofHc (M ).
Theorem.
(1) Hc (R) = H0 (R) ≈ R∞ × `2

14((2) the case n = 2 :∞H0 (M ) ≈ R ×`2

Hc (M ) ≈R∞ × `2(#)

R∞ × `2 × Z

all other cases(#) M ≈ X \ K, where(i) X = an annulus, a disk or a Möbius band(ii) K is a non-empty compact subset of a boundary circle of X
§2.2. Diﬀeomorphism groups.
Suppose M is a non-compact connected smooth n-manifold without boundary. Let
D(M ) denote the group of diﬀeomorphisms of M endowed with the Whitney C ∞ topology and Dc (M ) denote the subgroup of diﬀeomorphisms of M with compact
support. The identity component D0 (M ) of D(M ) is included in Dc (M ) and induces
the mapping class group M∞
c (M ) = Dc (M )/D0 (M ) (with the discrete topology).
Theorem. (1) Dc (M ) ≈loc R∞ × `2 .

(2) Dc (M ) ≈ D0 (M ) × M∞c (M ).

Theorem.
(1) the case n = 1, 2 :D0 (M ) ≈ R∞ × `2 .
(2) the case n = 3 :If M is orientable and irreducible, then D0 (M ) ≈ R∞ ×`2 .
(3) If X is a compact connected smooth n-manifold with boundary andD0 (X, ∂X) has a homotopy type of a locally compact polyhedron L, thenD0 (Int X) ≈ D0 (X, ∂X) × R∞ × `2 ≈ L × R∞ × `2 .
Recently, we have obtained some results on the spaces of maps endowed with thegraph topology.
References[1] T. Banakh, K. Mine and K. Sakai, Classifying homeomorphism groups of inﬁnite
graphs, preprint.
[2] T. Banakh, K. Mine, K. Sakai and T. Yagasaki, Homeomorphism and diﬀeomorphism groups of non-compact manifolds with the Whitney topology, preprint,
arXiv math.GT/0802.0337
A New Theorem to Find Smith Equivalent Representations
Masaharu Morimoto (Okayama University)
Let G be a ﬁnite group. Two RG-modules V and W are Smith equivalent if there
exists a homotopy sphere Σ with a G-action such that ΣG = {a, b}, Ta (Σ) ∼=G V
and Tb (Σ) ∼=G W . Smith equivalent pairs (V, W ) obtained by various authors, e.g.
T. Petrie, S. Cappell-J. Shaneson, K. PawaÃlowski-R. Solomon and etc., satisfy
V N = 0 = W N for all N / G with |G/N | = prime.
Problem 1. Find Smith equivalent pairs (V, W ) satisfying
V N 6= 0 and W N = 0
for some N / G such that |G/N | is a prime.
We will invoke the next hypothesis on the group G.
Hypothesis 1. G is an Oliver group such that Gnil = G{p} 6= G for some odd primep.
We set N = Gnil and L(G) = {H ≤ G | H ⊇ G{p} }. We will invoke the next
hypothesis on the RG-modules V and W .

15

Hypothesis 2. V is an RG-module such that V G = R. W is an L(G)-free RG-module
P(G)-matched with V .
Let

R[G]L = R[G] − R[G/G{p} ].
Then we obtain the following results.
Lemma 1. For a suﬃciently large integer a, there exists a smooth G-action on a disk
D such that
(1) DG = {x},(2) DL x = P (V N )L (conn. component) for all L ∈ L(G),(3) (|π1 (DQ )|, q) = 1 for any Q ∈ P(G) and |Q| = q a > 1,
(4) Tx (D) = (V − V G ) ⊕ R[G]L ⊕a .
A P(G)-matched pair (V, W ) is called of type R[G/N ] if
VN ∼= R[G/N ] and W N = 0.
A P(G)-matched pair (V1 , V2 ) is called of type R[G/N ]G n if V2N = 0 and there existsa P(G)-matched pair (V, W ) of type R[G/N ] such that
V1N = (V N − V G )⊕n .
Lemma 2. Suppose N has a subquotient ∼= D2qr (q 6= r). For each i = 1, . . . m, let
(i)
(i)
(i)
(i)
(V1 , V2 ) be a P(G)-matched pair of type R[G/N ]G ni with respect to (U1 , U2 ) of(1)
(m)
type R[G/N ]. Suppose V1 , . . . , V1 are P(G)-matched with one another. Then fora suﬃciently large integer a there exits a smooth G-action on a disk D such that
(1) DG = {x1 , . . . , xm },(i)N(2) DN xi = P (U1 )×ni (conn. comp.), 1 ≤ i ≤ m,(i)⊕a(3) Txi (D) ∼= V1 ⊕ R[G]L .
Theorem 1. Let U be a gap G-module. Suppose N has a subquotient ∼= D2qr (q 6= r).
(i)(i)
For each 1 ≤ i ≤ m, let (V1 , V2 ) be a P(G)-matched pair of type R[G/N ]G ni with
(i)
(i)
(1)
(m)
respect to (U1 , U2 ) of type R[G/N ]. Suppose V1 , . . . , V1 are P(G)-matched with
one another. Then for suﬃciently large integers a and b, there exists a smooth G-action
on a sphere S such that
(1) S G = {x1 , . . . , xm },(i)N(2) S N xi = P (U1 )×ni (conn.comp.), 1 ≤ i ≤ m,(i)⊕a(3) Txi (S) ∼= V1 ⊕ R[G]L ⊕ U ⊕b .
Corollary 1. Let G be an Oliver group such that(1) |G/Gnil | = p (p odd prime),{G}(2) RO(G)P(G) 6= 0,(3) Gnil has a subquotient ∼= D2qr (q 6= r).
{G}
Then R[G]L(G) is a gap module and Sm(G) ⊃ Z. Particularly, if p = 3 and RO(G)P(G) ∼={G}

Z then Sm(G) = RO(G)P(G) .
We answer to problems posed by K. PawaÃlowski-R. Solomon and T. Sumi.
Corollary 2. If G = P ΣL(2, 27), SG(864, 2666), or SG(864, 4666) then Sm(G) ={G}
RO(G)P(G) ∼= Z.