Bratislava Topology Symposium“Group Actions and Homogeneous Spaces”
Comenius University, 7-11 September 2009

ABSTRACTS OF LECTURES

Comenius University
Faculty of Mathematics, Physics, and Informatics
Bratislava, SlovakiaOn the first homology of automorphism groups ofG-manifolds with codimenison one orbit
Kōjun Abe (Shinshu Univ.)
We have been calculated the first homology group of the equivarinat diffeomorphism, the equivariant Lipschitz homeomorphism group and the equivariant homeomorphism group of G-manifolds. Those first homology groups
are quite different by the category of the automorphism group. We shalldescribe the results and the outline of the proof.
References[1] K. Abe and K. Fukui, On the structure of the group of equivariant diffeomorphisms of G-manifolds with codimension one orbit, Topology, 40(2001), 1325-1337.
[2] K. Abe and K. Fukui, On the structure of the group of Lipschitz homeomorphisms and its subgroups II, J. Math. Soc. Japan, 55 (2003), 947-956.
[3] K. Abe, K. Fukui and T. Miura, On the first homology of the group ofequivariant Lipschitz homeomorphisms, J. Math. Soc. Japan, 58 (2006),
1-15.
[4] K. Abe and K. Fukui, The first homology of the group of equivariant
diffeomorphisms and its applications, Journal of Topology, 1 (2008), 461476.
[5] K. Abe and K. Fukui, On the first homology of the group of equivariant
Lipschitz homeomorphisms of G-manifolds with codimension one orbit,
preprint.
[6] K. Abe, K. Fukui and T. Rybicki, On the first homology of the group ofequivariant homeomorphisms of G-manifolds with codimension one orbit,
in preperation.
Department of Mathematical Sciences, Shinshu University,
Matsumoto 390-8621, JAPAN.
e-mail : kojnabe@shinshu-u.ac.jp

REAL BOTT MANIFOLDS AND ACYCLIC DIGRAPHSSUYOUNG CHOI

Abstract
A Small cover, defined by Davis and Januszkiewicz [2], is an n-dimensional
closed smooth manifold M with a smooth action of standard real torus
Zn2 action such that the action is locally isomorphic to a standard actionof Zn2 on Rn and the orbit space M/Zn2 can be identified with a simple(combinatorial) polytope. In this talk, we assume that the orbit polytope
is a hypercube. Small covers over cubes, called the real Bott manifolds, areobtained as iterated RP 1 bundles starting with a point, where each stage isthe projectivization of a Whitney sum of two real line bundles. In this talk,we discuss about the topological classification of real Bott manifolds. In fact,by Kamishima and Masuda [3], two real Bott manifolds are diffeomorphic
if their cohomology rings with Z2 coefficients are isomorphic. In addition,
Masuda [4] has explained the diffeomorphism classes of them by 3 operationsof the set of real Bott manifolds. In other words, two Bott manifolds arediffeomorphic if one is transformed to the other through a sequence of thethree operations.
On the other hand, Choi [1] has found a 1-1 correspondence between
the set of real Bott manifolds and the set of combinatorial objects, acyclicdigraphs. So we may regard operations of real Bott manifolds as operationsof digraphs. One of operations is well-known for the local complementation.
We investigate many invariants under the these operations. This work is
jointly with Sangil Oum.
References
1. Suyoung Choi, The number of small covers over cubes, Algebr. Geom. Topol. 8 (2008),
no. 4, 2391–2399. MR MR2465745
2. Michael W. Davis and Tadeusz Januszkiewicz, Convex polytopes, Coxeter orbifolds andtorus actions, Duke Math. J. 62 (1991), no. 2, 417–451.
3. Yoshinobu Kamishima and Mikiya Masuda, Cohomological rigidity of real Bott manifolds, arXiv:0807.4263 (2008).
4. Mikiya Masuda, Classification of real Bott manifolds, arXiv:0809.2178 (2008).
Department of Mathematical Sciences, KAIST, 335 Gwahangno, Yuseonggu, Daejeon 305-701, Republic of Korea
E-mail address: choisy@kaist.ac.kr
URL: http://topology.kaist.ac.kr/schoi

Date: June 30, 2009.
1

Symplectomorphisms, diffeomorphisms
and circle actions on symplectic manifolds
BogusÃlaw Hajduk

University of WrocÃlaw and University of Warmia and Mazury
In this talk I will discuss relations between smooth and symplectic objects on closed manifolds. Basic problems in this topic include: existenceof symplectic forms, relations between symplectomorphisms and diffeomorphisms, existence of symplectic circle actions. A lot is known on such questions in dimension four, but in dimensions ≥ 6 they are wide open.
I will explain relations between the following questions:• If a symplectic manifold is homeomorphic to the torus T 2n , is it diffeomorphic to T 2n ?
• When a diffeomorphism of T 2n is isotopic to a symplectomorphism?
? Does M × S 1 admit a symplectic structure only if M fibers over S 1 ?
? Does any closed symplectic manifold with a (smooth) circle action
admit a symplectic circle action?
I will give examples to show that answers to these questions might be
not simple in higher dimensions. In particular there are diffeomorphisms of
T 2n supported in a disc which are not isotopic to symplectomorphisms andcircle actions on symplectic manifolds which are not equivalent to symplectic
actions.
Large part of my talk is a report on a joint project with Aleksy Tralle
and RafaÃl Walczak.

1

On the first Pontrjagin class ofhomotopy complex projective spaces
Yasuhiko KITADA(Yokohama National University)
Let M 2n be a closed smooth manifold homotopy equivalent to the complex projective space CP(n). Pontrjagin classes have been studied to classify homotopy
complex projective spaces. It is known that the first Pontrjagin class p1 (M ) has theform
p1 (M ) = ((n + 1) + 24γ(M ))x2
for some integer γ(M ) ∈ Z, where x ∈ H 2 (M ; Z) is the generator. When n isodd, many facts are known; if there exists a framed (n + 1)-manifold with Kervaireinvariant one, then γ(M ) can take any integer. In other dimensions, γ(M ) can takeany even number. Conversely, γ(M ) is even at least when n ≡ 1 mod 4. However
it is hard to get meaningful results when n is even.
Let ν2 (n) denote the 2-order of n, i.e. the exponent of the prime factor 2 in theprime decomposition of n. We have been doing calculations of the surgery index
obstruction of normal maps of CP(n) when n is even. The calculation gets harderas ν2 (n) increases. We have now succeeded in proving the following theorem:
Theorem. If n is even and ν2 (n) < 6, then γ(M ) is even.
The restriction on the 2-order of n is not best possible. By direct calculation,
we know that the theorem is also true for n = 64, but we do not know the general
proof for other dimensions with larger 2-orders. We hope that we can remove therestriction on the 2-order of n in the future.
Let n = 2k and f : M 4k → CP(2k) be a homotopy equivalence and φ : CP(2k) →F/O be its normal map. Since γ(M ) is even, we can conclude that the pullbackof the two dimensional Kervaire class φ∗ (K2 ) vanishes. Then we know that otherpullbacks φ∗ (K2j+1 −2 )’s simultaneously vanish in dimensions less than 2k. Let us
consider the case where the normal map φ has an extension CP(n + 1) → F/O.
Unless k + 1 is not a power of 2, by the simultaneous vanishing of the Kervaire
classes, using Sullivan’s formula, we conclude that the (4k + 2) dimensional Kervaire
surgery obstruction is also zero. This implies :Corollary. If k + 1 is not a power of 2 and ν2 (k) < 5, then the (4k + 1) dimensional
Kervaire sphere does not admit any free S 1 action.
E-mail : ykitada@ynu.ac.jp

A new bound for the cup-length ofzero-cobordant manifolds
Július Korbaš,Faculty of Mathematics, Physics, and Informatics,
Comenius University, Bratislava
Abstract
In this talk a new upper bound for the Z2 -cup-length of unorientedly zero-cobordant smooth closed connected manifolds will be presented. This bound is similar in its spirit to another upper boundfor the Z2 -cup-length, published in J. Korbaš, Topology Appl. 153(2006), 2976-2986, which works (in particular) for all smooth closed
connected manifolds. As an illustration, it will be shown that the
new bound leads to interesting estimates, and in infinitely many caseseven to exact values, of the Z2 -cup-length of the Grassmann manifolds
SO(n)/SO(k) × SO(n − k) of oriented k-dimensional vector subspacesin Euclidean n-space with n odd. [Since these homogeneous spaces are
zero-cobordant thanks to an obvious stationary point free action of
the group Z2 on them, one can see here one of the many interplays
between the two main topics of the present conference.]
To use the upper bound for the Z2 -cup-length of zero-cobordant
manifolds optimally, one should know the value of a numerical homotopy invariant of smooth closed connected manifolds, apparently
not studied up to now, called the characteristic rank (J. Korbaš, Bull.
Belgian Math. Soc., to appear): the characteristic rank of a smoothclosed connected d-dimensional manifold M is defined to be the largest
integer k (0 ≤ k ≤ d) such that each element of the cohomology groupH j (M ; Z2 ) with j ≤ k can be expressed as a polynomial in the StiefelWhitney classes of M . Some results on the characteristic rank of theGrassmann manifolds SO(n)/SO(k) × SO(n − k) will also be mentioned.

1

Simply connected asymmetric manifolds
Matthias Kreck,
Director of HIM
http://www.hausdorff-research-institute.uni-bonn.de/
http://www.hausdorff-research-institute.uni-bonn.de/kreck
Abstract
A closed smooth manifold M is called asymmetric if no compact
Lie group can act effectively or equivalently if for all Riemannian metrics the group of self-isometries is trivial. About 40 years ago Borel
gave a criterion for aspherical manifolds which guarantees that the
manifold is asymmetric. Since many people believe that a manifold”picked at random” is asymmetric there should also be simply connected examples. But this is an open problem for about 30 years.
Volker Puppe has shown that there are many 6-dimensional simply
connected manifolds which are almost asymmetric in the sense that
there are no orientation preserving actions. I showed that all thesemanifolds are actually asymmetric. The key is some sort of index
formula which I will explain.

1

ON PROJECTIVE BUNDLES OVER SMALL COVERSSHINTARÔ KUROKI

A small cover M is an n-dimensional manifold with locally standard
Zn2 -action whose orbit space is a simple polytope (e.g. the real projective space RP (n) is a small cover). Small cover can be regarded as areal version of quasitoric manifolds.
Let ξ be an equivariant vector bundle over a small cover M . If ξ decomposes into a Whitney sum of line bundles, then its projectivization
P (ξ) becomes a small cover again. We call P (ξ) a projective bundle
over small cover.
In the paper [1], this bundle was used to find counter examples ofcohomological rigidity of small covers, i.e., the cohomology ring over Z2 coefficient can not distinguish homeomorphism types of small covers.
Such counter examples are constructed from projective bundles overRP (n) by classifying their homeomorphism types. So we are naturally
led to study homeomorphism types of projective bundles over every
small cover.
In this talk, we characterize such projective bundles over 2-dimensional
small covers by using combinatorial argument and introducing a new
k−1
combinatorial operation ]∆ . The main theorem is as follows:
Theorem 1. Let P (ξ) be a projective bundle over 2-dimensional small
cover M 2 . Then P (ξ) can be constructed from projective bundles P (κ)
over the real projective space RP 2 and P (ζ) over the torus T 2 by using
k−1
]∆ .
k−1

Here, a combinatorial operation ]∆ , where k is a fibre dimensionof ξ, corresponds with the fibre sum along the fibre as the geometric
operation. Moreover, homeomorphism types of P (κ) and P (ζ) are
known by easy computations.
This is a joint work with Zhi Lü.
References[1] M. Masuda, Cohomological non-rigidity of generalized real Bott manifolds ofheight 2, arXiv:0809.2215.
Dept. of Mathematical Science, Korea Advanced Institute of Science and Technology (KAIST), 335 Gwahangno (373-1 Guseong-dong),
Yuseong-gu, Daejeon 305-701, Republic of Korea
E-mail address: kuroki@kaist.ac.kr
1

SURGERY CLASSIFICATION OF LENS SPACESBRATISLAVA TOPOLOGY SYMPOSIUMSEPTEMBER 2009
TIBOR MACKO

Let G be a finite cyclic group and let V be a finite-dimensional G-representation
which is free away from 0 ∈ V . Such a representation gives a free action of G on thesphere SV . The orbit space is a manifold called a lens space. The classification oflens spaces up to homotopy equivalence is a standard topic of courses in algebraic
topology. The classification up to simple homotopy equivalence and up to homeomorphism goes via the Reidemeister torsion and is a standard topic of courses inalgebraic K-theory.
A fake lens space is the orbit space of any free action of a finite cyclic group ona finite dimensional sphere, that means not necessarily one coming from a representation. It turns out that the homotopy and simple homotopy classification ofthese topological manifolds can be obtained using the same methods as for the standard lens spaces. However, the homeomorphism classification is more complicated,in addition methods of surgery theory are needed. Wall and others in the 1960’s
treated the case when the order of the group G is N = 2 and N is odd, the results
are presented in chapter 14 of Wall’s book Surgery on compact manifolds. However, the case N even remained open, due to lack of knowledge of certain surgery
L-groups (which were, however, calculated in the meantime) and due to computational difficulties. In a joint work with Christian Wegner we recently addressed thecalculations for the remaining cases (arXiv:0810.1196 and arXiv:0805.0965).
In the talk I will survey the above methods and calculations. Besides surgery theory an important tool is the so-called rho-invariant, which is an invariant of closed
odd-dimensional manifolds closely related to the G-signature of even-dimensional
manifolds.
The surgery classification is stated in terms of the so-called surgery structure set.
It turns out that this set is just π0 of a certain automorphism space of lens spaces.
The above methods can be used to study also the higher homotopy groups of thatspace. In cases N = 2 and N odd this has been done by Madsen and Rothenberg
in the 1980’s. I will discuss this aspect as well.
It turns out that when the automorphism spaces are studied, yet another tool
can be employed, the orthogonal calculus of functors of Weiss. If time permits I
will sketch some ideas in this direction.

Mathematisches Institut, Universität Münster, Einsteinstraße 62, Münster, D-48149,
Germany, and Matematický Ústav SAV, Štefánikova 49, Bratislava, SK-81473, Slovakia
E-mail address: macko@uni-muenster.de
URL: http://www.math.uni-muenster.de/u/macko

1Homotopical theory of periodic points ofperiodic homeomorphisms on closed surfaces
Waclaw Marzantowicz & Xuexhi Zhao
Abstract
In this talk we show that the Wecken theorem for periodic points
holds for periodic homeomorphisms on closed surfaces, which therefore
completes the periodic point theory in such a special case. Using it wederive the set of homotopy minimal periods for such homeomorphisms.
Moreover we show that the results hold for homotopically periodic
self-maps of closed surfaces. This lets us to re-formulate our results
as a statement on properties of elements of finite order in the groupof outer automorphisms of the fundamental group of a surface with
non-positive Euler characteristic.

1

SYMMETRY OF TORUS MANIFOLDSMIKIYA MASUDA

A fundamental result in toric geometry says that there is a bijective correspondence between toric varieties and fans, so all algebrogeometrial information on a toric variety is encoded in the associatedfan. Among toric varieties, compact smooth toric varieties, which wecall toric manifolds, are well studied. If X is a toric manifold, then thegroup Aut(X) of automorphisms of X is known to be a (ﬁnite dimensional) algebraic group and Demazure introduced a root system for the
fan associated with X and proved that it agrees with the root systemof the identity component Aut0 (X) of Aut(X). He also described themapping class group Aut(X)/ Aut0 (X) in terms of the fan associatedwith X.
A symplectic toric manifold, which is a compact symplectic manifold (M, ω) with a Hamiltonian action of a compact torus T where
2 dim T = dim M , is a symplectic counterpart to a toric manifold, but
the group Symp(M, ω) of symplectomorphisms of (M, ω) is inﬁnite dimensional unlike in the toric case. According to Delzant, symplectic
toric manifolds are classiﬁed by their moment polytopes. In this talkI introduce a root system R(P ) for the moment polytope P and showthat any irreducible subsystem of R(P ) is of type A and that if G isa compact Lie subgroup of Symp(M, ω) containing the torus T , thenthe root system of G is a subsystem of R(P ), so that G is of type A.
We can also estimate the mapping class group G/G0 in terms of an
automorphism group of P .
A similar story goes in the topological category, to be more precise,for a torus manifold M and the group Diﬀ(M ) of diﬀeomorphisms of
M , where a torus manifold M is a compact smooth manifold with asmooth action of a torus T where 2 dim T = dim M and M T 6= ∅.
The story also goes for a torus manifold M with a stably complex
structure J invariant under the action of T and the group Diﬀ(M, J)of diﬀeomorphisms of M preserving the J.
References[1] M. Masuda, Symmetry of a symplectic toric manifold, arXiv:0906.4479.
Department of Mathematics, Graduate School of Science, Osaka
City University, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
E-mail address: masuda@sci.osaka-cu.ac.jp

1

On The Smith Equivalent Representations of Oliver Groups
Masaharu Morimoto and Yan Qi(Okayama University)

Let G be a finite group. Two real G-modules V and W are called S ht -related (orSmith equivalent) and written V ∼Sht W if there exists a homotopy sphere Σ withsmooth G-action such that ΣG = {a, b}, and Ta (Σ) ∼= V and Tb (Σ) ∼= W . This Σ iscalled an Sht -realization of V and W . Since late 1970’s, we have obtained various
results which provide Sht -realizations of certain nontrivial pairs (V, W ). Let RO(G, Sht )denote the Smith set{[V ] − [W ] ∈ RO(G) | V ∼Sht W }and let RO(G, Sht )P denote the primary Smith set∼ G{[V ] − [W ] ∈ RO(G, S) | resG
P V = resP W ∀P ∈ P(G)}where P(G) is the set of all subgroups of G of prime power order.
Nontrivial Sht -equivalent pairs (V, W ) were found by Petrie, Cappell-Shaneson, PetrieRandall, Petrie-Dovermann, Suh, . . . , but their pairs satisfy the condition(C)

dim V N = dim W N for any N / G with prime index.

We would like to find Sht -equivalent pairs (V, W ) not satisfying Condition (C). Let
N = Gnil be the smallest normal subgroup of G such that G/N is nilpotent.
Theorem 1. Let G be an Oliver group with |G/N | = 3 and N has a subquotient groupisomorphic to a dihedral group of order 2q with an odd prime q. If (V, W ) is a P-matchedpair of real G-modules such that V G = 0, V N 6= 0, and W N = 0, then V ⊕ U and W ⊕ U
are Sht -equivalent for sufficiently large real G-modules U with U N = 0. Moreover in thecase aG = 2, RO(G, Sht )P is isomorphic to Z.
This enables us to determine the Smith sets of certain groups G, e.g. G = P ΣL(2, 27),
SG(864, 2666), and SG(864, 4666). Furthermore, we have a generalization of the theorem
above and can compute the Smith sets of certain groups G with |G/N | = 6, e.g. G =(D6 × D6 × D6 ) o C3 (a gap Oliver group).
———————————————————————
E-mail address: morimoto@ems.okayama-u.ac.jp

Isovariant maps from free G-manifolds torepresentation spheres
Ikumitsu NAGASAKI
Kyoto Prefectural University of Medicine (KPUM)

Abstract
This research is joint with F. Ushitaki.
Let M be an n-dimensional connected orientable closed manifold. It is a wellknown result, called the Hopf classiﬁcation theorem, that homotopy classes of continuous maps from M to the n-dimensional sphere S n are classiﬁed by their degree.
We treat an isovariant version of the Hopf classiﬁcation theorem under the following situation: Let M be a connected, orientable, closed manifold with free G-action,and SW a G-representation sphere (i.e., the unit sphere of unitary representation
W of G), and assume
dim M + 1 = dim SW − dim SW >1 ,where SW >1 denotes the singular set of SW .
A G-equivariant map f : X → Y is called G-isovariant if f preserves the isotropy
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.
We determine the isovariant homotopy set [M, SW ]isovby calculating the equiGvariant obstruction group. As a result, isovariant maps are classiﬁed by the multidegree of isovariant maps ([1], [2]) if G-action on M is orientation-preserving, but
not are necessary classiﬁed by the multidegree if G-action on M is not orientationpreserving.
We also discuss the case of non-orientable manifolds if time allows.
References[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 G-manifolds andrepresentation spheres, preprint.

ON THE SIGNATURE OF GRASSMANNIANS VIA SCHUBERTCALCULUS
MASAKI NAKAGAWA
Abstract. Let M = M 4k be a compact oriented manifold of dimension 4k. Thenthe symmetric bilinear form deﬁned by
QM : H 2k (M ; R) × H 2k (M ; R) −→ R,(x, y) 7−→ (x ∪ y) [M ]is non-degenerate by the Poincaré duality theorem. The signature σ(M ) of M isby deﬁnition the signature of QM . The signature of a manifold whose dimension
is not divisible by 4 is deﬁned to be zero. The celebrated Hirzebruch signature
theorem asserts that the signature σ(M ) of a compact oriented manifold M = M 4k
can be represented as a certain linear combination of Pontrjagin numbers of M .
However, the explicit computation of the signature of a manifold M is usually adiﬃcult task, even if its cohomology ring H ∗ (M ; R) and characteristic classes arewell known.
In [4], P. Shanahan computed the signature of a complex Grassmannian. He
used the Atiyah-Bott Lefschetz formula [1], and the result is given as follows:
Theorem 0.1. Let G(k, n) be the complex Grassmannian of k-dimensional linear
subspaces of Cn . Then the signature of G(k, n) is
 (n)
[2]

if k(n − k) even,[k2]
σ(G(k, n)) = 0
if k(n − k) odd.
Using the Schubert calculus technique (see [3], [2]), we will give a quite simple
proof of the above result. More explicitly, the Schubert classes behave nicely under
the Poincaré duality, and the product of two Schubert classes can be describedcombinatorially. From this, the computation of σ(G(k, n)) reduces to a simple
enumerative problem, and we obtain the above result.

References[1] M. F. Atiyah and R. Bott, A Lefschetz ﬁxed point formula for elliptic complexes: II. Applications, Ann. of Math., 88 (1968), 451-491.
[2] W. Fulton, Young tableaux, London Math. Soc. Student Texts 35, Cambridge Univ. Press,
Cambridge, 1997.
[3] P. Griﬃths and J. Harris, Principles of algebraic geometry, Pure and Applied Mathematics,
Wiley-Intescience, New York, 1978.
[4] P. Shanahan, On the signature of Grassmannians, Paciﬁc J. Math. 84 (1979), 483-490.
Department of General Education
Takamatsu National College of Technology
Takamatsu 761-8058, Japan
E-mail address: nakagawa@takamatsu-nct.ac.jp

0

September, 7-11, 2009, Bratislava Topology Symposium
1Torus actions and comblex cobordism
Taras Panov
Faculty of Mathematics and Mechanics, Moscow State University, Leninskie
Gory, Moscow 119991 RUSSIA
e-mail: tpanov@mech.math.msu.su
This is a joint work with Victor M. Buchstaber and Nigel Ray.
The ideas and methodology of the emerging field of Toric Topology canbe applied to one of the classical subjects of algebraic topology: finding nicerepresentatives in complex cobordism classes. Quasitoric manifolds are thekey players; they constitute a sufficiently wide class of stably complex manifolds to represent the whole complex cobordism ring. In other words, every
stably complex manifold is cobordant to a manifold with a nicely behaving
torus action. More precise statement is as follows.
Theorem 1. In dimensions > 2, every complex cobordism class contains
a quasitoric manifold, necessarily connected, whose stably complex structure
is compatible with the action of the torus.
The first ingredient of the proof is an explicit construction of a seriesof algebraic smooth toric varieties additively generating the complex cobordism ring; this was completed in an earlier work of Buchstaber-Ray. We
proceed by defining a generalisation of the equivariant connected sum operation, which is used to replace the disjoint union of manifolds (representing
the sum operation in cobordism) by a single connected quasitoric manifold.
By way of application, we give an explicit construction of a quasitoric
representative for every complex cobordism class as the quotient of a free
torus action on a real quadratic complete intersection. The latter is a yet
another disguise of the moment-angle manifold, another familiar object of
toric topology. We therefore obtain a systematic description for quasitoric
manifolds and their cobordism classes in terms of combinatorial data.
Originally, these considerations take place in non-equivariant complex
cobordism; but the combinatorial description allows for calculations withquasitoric representatives in equivariant cobordism also. These theme iscurrently under development [2].

References[1] V. Buchstaber, T. Panov and N. Ray, “Spaces of polytopes and cobordismof quasitoric manifolds,” Moscow Math. J., 7, No. 219242 (2007).
[2] V. Buchstaber, T. Panov and N. Ray, “Toric genera,” Preprint (2009).

CLASSIFICATION OF QUASITORIC MANIFOLDS OF b2 = 2
SEONJEONG PARK

Abstract
A quasitoric manifold, introduced by Davis and Januszkiewicz in [1], of
dimension 2n is a smooth manifold with locally standard half-dimensional
torus (S 1 )n action whose orbit space is diffeomorphic to a simple polytopeof dimension n as manifolds with corners. A quasitoric manifold can bethought of topological generalization of a compact smooth toric variety,which we call a toric manifold. Among varieties, toric manifolds are wellstudied. In particular, toric manifolds of the second Betti number 2 arealready classified as varieties by Kleinschmidt [2]. In this talk, we classify
topologically quasitoric manifolds of b2 = 2; they are distinguished by theircohomology rings up to homeomorphism. Especially, we focus on quasitoric
manifolds of b2 = 2 which are not homeomorphic to toric manifolds. This
work is jointly with Suyoung Choi and Dong Youp Suh.
References
1. Michael W. Davis and Tadeusz Januszkiewicz, Convex polytopes, Coxeter orbifolds andtorus actions, Duke Math. J. 62 (1991), no. 2, 417–451.
2. P. Kleinschmidt, A classification of toric varieties with few generators, Aequationes
Mathematicae 35 (1988), 254–266.
Department of Mathematical Sciences, KAIST, 335 Gwahangno, Yu-sung
Gu, Daejeon 305-701, Korea
E-mail address: sjeongp@kaist.ac.kr

Date: June 30, 2009.
1

MAPS BETWEEN GRASSMANN MANIFOLDSPARAMESWARAN SANKARAN

Abstract
Let F Gn,k denote the Grassmann manifold of k-dimensional F -vector
subspaces of F n where F denotes the field of real numbers or complex
numbers. It is an interesting problem to classify maps between two
Grassmann manifolds of the same dimension.
Specifically we shall address the question: For what values of k, l, n, m
with k(n − k) = l(m − l) is there a non-zero degree map CGn,k −→CGm,l ? We shall give an exposition of recent results on this problem.
Our results apply to the case of quaternionic Grassmann manifolds as
well.
In the case when F = R, the manifold Gn,k is not orientable unless
n is even. However, the same question can be addressed for maps
en,k which is the (universal)
between oriented Grassmann manifolds G
double cover of Gn,k (when n > 2). Although in this case the results
were obtained over twelve years ago, we shall recall them (if there istime) for the sake of completeness.
References
1. V. Ramani and P. Sankaran, On degrees of maps between Grassmannians, Proc. Indian Acad. Sci. (Math. Sci), 107 (1997), 13–19.
2. P. Sankaran and S. Sarkar, Degrees of maps between complex Grassmann manifolds, Osaka Jour. Math., to appear.
The Institute of Mathematical Sciences, Chennai 600113.
E-mail address: sankaran@imsc.res.in1

Abstract of paper to be presented in
BRATISLAVA TOPOLOGY SYMPOSIUM”GROUP ACTIONS AND HOMOGENEOUS SPACES”
to be held at
COMENIUS UNIVERSITY,
FACULTY OF MATHEMATICS, PHYSICS AND INFORMATICS,
BRATISLAVA, SLOVAKIASEPTEMBER 07-11, 2009

PARAMETRIZED BORSUK-ULAM PROBLEM FOR
PROJECTIVE SPACE BUNDLES
Abstract. Let π : E → B be a fiber bundle with fibers having themod 2 cohomology algebra of a real or a complex projective space′′and let π : E → B be a vector bundle. Let Z2 act fiber preserving′
and freely on both E and E −0, where 0 stands for the zero section′′of the bundle π : E → B. For a fiber preserving Z2 -equivariant′
map f : E → E , we estimate the cohomological dimension of thezero set Zf = {x ∈ E | f (x) = 0}. As an application, we also
estimate the size of the Z2 -coincidence set Af = {x ∈ E | f (x) =′f (T (x))} of a fiber preserving map f : E → E , where T : E → E′
is the generator of the Z2 -action and no action on E is specified.

Mahender Singh, School of Mathematics, Harish-Chandra ResearchInstitute, Chhatnag Road, Jhunsi, Allahabad-211019, INDIA.
E-mail address: msingh@mri.ernet.in

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SURGERIES ON KNOTS BOUNDING DEFINITE 4-MANIFOLDSSAŠO STRLE

Abstract
In the study of various properties or structures on a closed 3-manifold Y (eg. fillable
contact structures) it is often important to know whether Y bounds a compact definite
4-manifold (of given definiteness). We study this question for surgeries on knots. It
turns out that in this case the information can be encoded in a numerical invariant
that is a concordance invariant of the surgery knot and has a subadditivity property.
We compute this invariant for torus knots.
There are two basic approaches to problems of this sort, both essentially relaying ongauge theory. Though the more numerical approach using correction term invariants
in Heegaard-Floer homology of Ozsváth and Szabó (or related invariants in otherversions of gauge theoretic invariants) seems more promissing at first it turns out thatthe computation for torus knots can be carried out using Donaldson’s diagonalization
theorem.
This is joint work with Brendan Owens.
Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
E-mail address: saso.strle@fmf.uni-lj.siPROPERTIES OF BOTT MANIFOLD AND
COHOMOLGICAL RIGIDITYDONG YOUP SUH

A Bott tower is a sequence Bn → Bn−1 → · · · → B1 → B0 where B0
is a point and Bi is a CP 1 -bundle over Bi−1 for i = 1, . . . n. Each Bi isa called the i-th stage Bott manifold. One can extend this definition todefine a generalized Bott tower and manifold to be a sequence of complex space bundles. One of the interesting question in toric topology
asks whether two (generalized) Bott manifolds Bn and Bn0 are homeomorphic (or diffeomorphic) provided their cohomologies are isomorphic
as graded rings. This is called the cohomological rigidity question for(generalized) Bott manifolds. In this talk we give some topological andcohomological properties of Bott manifolds, and using these properties
we give some positive answers to cohomological rigidity question for
Bott manifolds.
Department of Mathematical Sciences, KAIST, 335 Gwahangno, Yuseong Gu, Daejeon 305-701, Korea
E-mail address: dysuh@math.kaist.ac.kr

Date: June 30, 2009.
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SMITH EQUIVALENT MODULES AND THE WEAK GAP
CONDITIONTOSHIO SUMIIn the real representation ring RO(G) of G, we consider the subset CSm(G) consistingof the diﬀerences U − V of real G-modules U and V which are strong Smith equivalent;i.e., there exists a smooth action of G on a (homotopy) sphere Σ with exactly two ﬁxed
points x and y such that
Tx (Σ) ∼= U ⊕ W and Ty (Σ) ∼=V ⊕W
as real G-modules for some real G-module W and Σ P is connected for a subgroup P of
G of prime power order.
In this talk we discuss what CSm(G) is for an Oliver group G. Here G is an Olivergroup if there exists no sequence of subgroups P  H  G such that P and G/H are groups
of prime power order and H/P is cyclic.
Let P(G) be the set of all subgroups of G of prime power order and let Op (G) for aprime p denote the smallest normal subgroup G with index a p-power. We denoteOp (G)pker(ResG
:
RO(G)
→
RO(P
))
∩
ker
Fix
:
RO(G)
→
RO(G/O(G))
PP ∈P(G)pby LO(G). A group G is called a gap group if there exists a G-module W such thatp
FixO (G) (W ) = 0 for any prime p and dim W P > 2 dim W H for all pairs (P, H) of subgroups of G with P ∈ P(G) and P < H.
Theorem. Let G be an Oliver group. Suppose that for any subgroup K with [K : O2 (G)] =2 it holds that K is a gap group or that all elements x of K  O2 (G) of order 2 such that
CK (x) is not a 2-group are conjugate in G. Then it holds that LO(G) is a subset ofCSm(G). Furthermore, if Op (G) = G for any odd prime p, then it holds that LO(G) =CSm(G).
We remark that all groups G with |G| ≤ 2000 satisfy the assumption.
Theorem. Let G be an Oliver group satisfying that |G| ≤ 2000 and that G/(∩p Op (G))
is an elementary abelian 2-group, and let K be afinite elementary abelian 2-group. Then
CSm(G × K) = LO(G × K).
Theorem. Sm(P GL(2, q)) = CSm(P GL(2, q)) = LO(P GL(2, q)).

TOPOLOGY OF THE GROUP OF HAMILTONIANSYMPLECTOMORPHISMS OF COMPACT HOMOGENEOUSSPACESALEKSY TRALLE

By a, now classical, result of Gromov, the symplectomorphism group
Symp0 (S 2 × S 2 , ω)for the standard product symplectic form, has the homotopy type of SO(3)×SO(3).
If one changes the product symplectic form ωS 2 ⊕ ωS 2 by rescaling one ωS 2 , one
can get the symplectomorphism group of a different homotopy type. However, thegenerators of π∗ Symp0 are obtained from the map π∗ SO(3) → π∗ Symp0 inducedby the inclusion SO(3) → Symp0 in both cases. The following general question ismotivated by these results: is the symplectic action of a compact Lie group G on aclosed symplectic manifold (M, ω) visible on the level of π∗ Symp(M, ω)? We prove
the following results in this direction.
Theorem 1. Let K/H be a hermitian symmetric space with a standard K-invariant
symplectic form. Then the homomorphism K → Ham(M, ω) induces a surjection
on rational cohomology.
Theorem 2. Let G/K be a Riemannian symmetric space of non-compact type,
and M/K be the Riemannian symmetric space dual to G/K in the sense of Matsushima. Let Γ denote a lattice in G. Consider the Matsushima homorphimsµ∗ : H ∗ (M/K) → H ∗ (BΓ) = H ∗ (Γ \ G/K). Let K/H be any compact hermitian symmetric space. Let c : BΓ → B Ham(K/H) by a map classifying the bundle
K/H → Γ \ G/H → Γ \ G/K.
Then the image of the induced homomorphismc∗ : H ∗ (B Ham(K/H), R) → H ∗ (Γ, R)
is equal to the image of the Matsushima homomorphism. In other words, all Matsushima classes are Hamiltonian characteristic classes.
These results detect nontrivial elements in π∗ Ham(M, ω), if M = K/H is ahermitian symmetric space. In this way, in particular, we generalize several previous
results by Reznikov and Kȩdra-McDuff. An essential feature of this work is aconstruction of a new hamiltonian fiber bundle, whose fiber is a compact symplectic
homogeneous space. This enables us to use a general machinery of classifying
bundles for the group Ham(M, ω), and to establish a link between the cohomologyof B Ham(M, ω) and the cohomology of lattices in Lie groups.
This is a joint work with Jarek Kȩdra.

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Quaternionic Toric Varieties
Dariusz M. Wilczyński
Utah State University

Abstract
We are going to discuss a comparison theorem for the algebraic and topological classifications of quaternionic toric varieties. It turns out that every topological type of an 8-dimensional quaternionic toric variety contains infinitely
many distinct algebraic types of such manifolds (joint work with Piotr Runge).

Span of Projective Stiefel Manifolds
Peter Zvengrowski
Abstract
In this talk some of the known results on the span of the real projective Stiefel manifolds Xn,r will be summarized, and some recent
improvements due to work of P. Sankaran, J. Korbaš, and the author
will be presented. A wide variety of techniques have been applied tothis problem, including vector bundles, characteristic classes, primaryand secondary cohomology operations, K-theory, Cayley-Dickson algebras, normal bordism, etc., leading to sharp and even exact resultsin many cases. The case Xn,2 with n odd seems to be the most difficult, and will receive special attention in the talk.

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