Involuç˜oes fixando Fn ∪ F3 Resumo

Involuções fixando F n ∪ F 3
Évelin Meneguesso Barbaresco ∗
[email protected]
Pedro Luiz Queiroz Pergher
[email protected]
†
Resumo
Dada uma involução T atuando sobre uma variedade suave, fechada,
n
[
m
m-dimensional M , cujo conjunto de pontos fixos F =
F i não borda, é
i=0
um fato bastante conhecido que, nestas condições, m não pode ser muito
grande em relação à n. Este fato tornou-se público através do famoso
Teorema 5/2 de Boardman (On manifolds with involutions, Bull Am. Math.
Soc., (1967)), o qual determinou, nas condições acima, que m ≤ 5/2n;
além disso, Boardman mostrou, através de exemplos, que esse valor não
pode ser melhorado nas condições gerais segundo as quais ele é formulado.
A generalidade do resultado de Boardman independe de n e abrange a
possibilidade de F possuir componentes com todas as dimensões possı́veis,
de 0 a n; neste contexto surge a seguinte questão: este limitante m pode
ser melhorado quando alguma restrição sobre n é imposta, ou ainda quando
algumas dimensões de F são omitidas inicialmente? Nesta direção, R. C.
Koniowski e R. E. Stong, em 1978, estabeleceram que, quando F = F n ,
então m ≤ 2n. Ainda nesta direção , temos alguns outros resultados que
melhoraram o limitante m quando F = F n ∪ F j , (por exemplo, quando F j é
igual ao ponto, ou igual à F 1 , ou à F 2 ). O objetivo neste trabalho é comentar
à respeito do caso F = F n ∪ F 3 , mostrando que nesse caso tambem é possı́vel
obter limitantes melhores; mais ainda, através de exemplos, pode-se mostrar
que em várias situações os limitantes obtidos são os melhores possı́veis.
∗
†
Aluna de Doutorado - Bolsista CAPES
Orientador
1
Referências
[1] J. Boardman, On manifolds with involutions, Bull Am. Math. Soc. 73
no. 2, 136-138, (1967).
[2] R. C. Koniowski and R. E. Stong, Involutions and characteristic
numbers, Topology 17, no.4, 309-330 (1978).
[3] P.L.Q. Pergher and R. E. Stong, Involutions fixing {ponto} ∪ F n ,
Transform. Groups 6, 78-85, (2001).
[4] Suzanne M. Kelton, Involutions fixing RP j ∪ F n , Topology Appl. 142,
197-203, (2004).
[5] P.L.Q. Pergher and Fabio G. Figueira, Involutions fixing F n ∪ F 2 ,
Topology Appl. 153 no. 14, 2499-2507, (2006).
DEFORMATIONS OF HYPERBOLIC CONE-STRUCTURES COLLAPSING CASE
ALEXANDRE PAIVA BARRETO
This talk is devoted to the study of deformations of hyperbolic cone structures under the assumption that the lengths of the singularity remain uniformly
bounded over the deformation. Given a sequence (Mi , pi ) of pointed hyperbolic
cone-manifolds with topological type (M, Σ), where M is a closed, orientable and
irreducible 3-manifold and Σ an embedded link in M . Assuming that the lengths
of the singularity remain uniformly bounded and that the sequence Mi collapses,
we prove that M is Seifert fibered or a Sol manifold. We apply this result to study
sequences of representations whose volume shrinks down to zero.
1
A CLASSIFICATION OF BRANCHED COVERINGS OF RP2
OVER RP2
NATALIA A VIANA BEDOYA, ELENA KUDRYAVTSEVA AND DACIBERG LIMA
GONÇALVES
In the present work we show some partial results about the following classification
problem: Given a branched covering of RP2 over RP2 of degree d, classify the one’s
which are always decomposable and the one’s that can be indecomposable. Here
we discuss the possibility of classification for an arbitrary degree.
References
[1] N. V. Bedoya. Revestimentos ramificados e o problema de decomponibilidade, Tesis Universidade de São Paulo, IME-USP 2008.
[2] N. V. Bedoya and D. L. Gonçalves. Decomposability problem on branched coverings. 2009.
Submitted.
[3] S. A. Bogatyi, D. L. Gonçalves, E. A. Kudryavtseva and H. Zieschang, Realization of primitive
branched coverings over closed surfaces, Advances in topological quantum field theory, 179,
297-316 2004.
[4] S. Bogatyi, D. L. Gonçalves, E. Kudryavtseva and H. Zieschang. Realization of primitive branched coverings over closed surfaces following the Hurwitz approach. Cent. Eur. J.
Math.,1, 2003.
[5] K. Borsuk and R. Molski, On a class of continuous mappings, Fund. Math., 45, 84-98 1957.
[6] A. L. Edmonds, R. S. Kulkarni and R. E. Stong, Realizability of branched coverings of surfaces, Trans. Amer. Math. Soc., 282, 1984.
[7] A. Hurwitz, Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann.,
39, 1-60 1891.
Departamento de Matemática, CCET UFSCar, Universidade Federal de São Carlos
E-mail address: [email protected]
Department of Mathematics and Mechanics, Moscow State University Moscow 119992,
Russia
E-mail address: [email protected]
Instituto de Matemática e Estatı́stica, IME USP, Universidade de São Paulo
E-mail address: [email protected]
1
PERSISTENCE, AN INVITATION TO “TOPOLOGY FOR DATA
ANALYSIS”
DAN BURGHELEA
Persistence is an important new topic in Computational Topology. In this talk I
will explain what “Persistence” is, what is this good for and how can be calculated.
(This is a summary of work of Edelsbrunner, Letcher, Zomorodian, Carlsson.) To
the extent the time permits more “topology for data analysis” related to persistence
will be presented. From a larger perspective the lectures is an invitation to use
Combinatorial topology, Homology and Morse-Novikov theory to Data Analysis.
1
ON THE NO EXISTENCE OF G-EQUIVARIANT MAPS
DENISE DE MATTOS, EDIVALDO L. DOS SANTOS AND FRANCIELLE R. DE C.
COELHO
Results on the no existence of G-equivariant maps often lead to important consequences. For example, the classical Borsuk-Ulam theorem, which states that each
continuous map of an n-dimensional sphere to n-dimensional Euclidean space takes
the same values at pair of antipodal points, is equivalent to the assertion that there
exists no Z2 -map S n → S n−1 , where Z2 acts on spheres by antipodal involution.
On the other hand, results of this kind can be interesting in themselves. We can
prove that one space cannot be embedded in another it is sufficient to show that
there exists no equivariant map between their deleted products considered with the
natural action of a free involution; or one can consider in a more general manner other configuration spaces connected with the spaces in question and endowed
with actions of suitable groups. One example of such a result is the well-known
van Kampen-Flores theorem [1], which states that the k-dimensional skeleton of a
(2k + 2)-dimensional simplex cannot be embedded in R2k .
Let R be a PID and G a compact Lie group. We denote by βi (X; R) the i-th
Betti number of X and i(X) the numerical index defined in [2].
In [4, Theorem 1.1], it was proved that if X, Y are free G-spaces, Hausdorff, pathwise connected and paracompact such that for some natural m ≥ 1, H q (X; R) = 0
for 0 < q < m, H m+1 (Y /G; R) = 0, where Y /G is the orbit space of Y by G and
βm (X; R) < βm+1 (BG; R) then there is no G-equivariant map f : X → Y .
In this work, our objective is to generalize this result. Specifically, we prove
Theorem 1. Let G be a compact Lie group and X, Y free G-spaces, Hausdorff,
pathwise connected and paracompact. Suppose that for some natural m ≥ 1, i(X) ≥
m + 1 and H m+1 (Y /G; R) = 0. Then, if βm (X; R) < βm+1 (BG; R), there is no
G-equivariant map f : X → Y .
In [2], Volovikov showed that :“If H q (X) = 0 for 0 < q < m, then i(X) ≥ m+1”.
Then, Theorem 1 generalizes [4, Theorem 1.1].
References
[1] A. Yu. Volovikov, On the van Kampen-Flores theorem, Mat. Zametki 59 (1996), 663 − 670;
English transl., Math. Notes 59 (1996), 477 − 481.
[2] A. Yu. Volovikov, On the index of G-spaces, Sb. Math. 191 (9 − 10) (2000) 1259 − 1277.
[3] Bredon, G., Introduction to Compact Transformation Groups, Academic Press, INC., New
York and London (1972).
[4] C. Biasi, D. de Mattos, A Borsuk-Ulam theorem for compact Lie group actions, Bull Braz
Math Soc, New Series 37(1) (2006) 127 − 137.
[5] C. T. Yang, On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujob and Dyson. I, Ann. of
Math. 60 (1954), 262 − 282.
(Denise de Mattos) ICMC - USP - São Carlos
E-mail address: [email protected]
(Edivaldo L. dos Santos) DM - UFSCar
E-mail address: [email protected]
(Francielle R. de C. Coelho) DM - UFSCar
E-mail address: [email protected]
1
Involutions whose fixed set has three or four
components: a small codimension phenomenon
Patricia E. Desideri ∗and Pedro L. Q. Pergher
†
9 de maio de 2010
Let F =
n
[
F j be a disjoint union of smooth and closed manifolds and M be
j=0
a smooth and closed manifold equipped with a smooth involution T : M → M
whose fixed point set is F . We say that an involution (M, T ) is essential when,
for every union of components of F , P ⊂ F , the normal bundle over P does
not bound as a bundle. Roughly speaking, no part of the fixed data of an
essential involution can be equivariantly removed; essential involutions are the
appropriate objects to study small codimension phenomenons. In this setting,
the following results were obtained: let (M, T ) be an essential involution having
fixed point set of the form F = F n ∪ P , where n ≥ 4 is even and P has
the possible forms: 1) P = F 3 ∪ F 2 ∪ {point}; or 2) P = F 3 ∪ F 2 ; or 3)
P = F 3 ∪ {point}; or 4) P = F 3 . Then, if k is the codimension of F n , k ≤ 4.
Further, there are essential involutions showing that this bound is best possible
in the cases 2) and 4), and in the cases 1) and 3) with n of the form n = 4t,
t ≥ 1.
∗
†
Aluna de Doutorado - Bolsista FAPESP
Orientador
1
MORSE-BOTT THEORETICAL SETTING FOR THE SEIBERG-WITTEN
4-DIM THEORY
CELSO MELCHIADES DORIA
UFSC - DEPTO. DE MATEMÁTICA
Let (X, g) be a closed, smooth riemannian 4-manifold. For any fixed spinc structures α on
X, the Seiberg-Witten functional
Z
(0.1)
SWα (A, φ) =
1
1
1
{ | FA+ |2 + | OA φ |2 + kg | φ |2 + | φ |4 }dvg
2
4
8
X
satisfies the Palais-Smale condition. There are two classes of critical points for the SWα functional (i) irreducibles: (A, φ), φ 6= 0, (ii) reducibles: (A, 0). For the purpose of studying
smooth invariants on X, it only matters the existence of irreducible stable critical points of SWα
(SWα -monopoles) which exist only for a finite set of spinc classes named basic classes. If the
scalar curvature satisfies kg ≥ 0, then there is no irreducible critical points. The motivation to
set up the SWα -functional in a Morse-Bott theoretical framework is to understand the existence
of SW-monopoles from a analytical point of view, since in the presence of a SWα -monopole the
Morse-Bott index of the reducibles is greater than 0 . In order to achieve transversality conditions, the following perturbation of the Seiberg-Witten functional is considered: let η be a closed,
smooth self-dual 2-form;
(0.2)
SWαη (A, φ)
Z
1
1
1
{ | FA+ |2 + | OA φ |2 + kg | φ |2 + | φ |4 }dvg +
2
4
8
ZX 1
−
< FA+ − σ(φ), η > + | η |2 dvg .
2
X
=
It is shown that for a large set of self-dual closed 2-forms η, the SWαη functional fits into a MorseBott framework. The reducibles critical points define a critical set diffeomorphic to the jacobian
torus JX = H 1 (X, R)/H 1 (X, Z). The 2nd variation formula (hessian) of SWαη is obtained and
the Morse-Bott index of reducible solutions (A, 0) is shown to be the dimension of the largest
k
negative eigenspace of the elliptic linear operator LA,η = 4A + 4g + η, hence is finite. Moreover,
for a large set of self-dual closed 2-forms η, it is shown that the hessian’s null space is exactly the
tangent space to JX . In [1], they prove the gradient flow lines always converge to a critical point
allowing to define a sort of Floer Complex. By using the blow-up ideas of Kronheimer-Mrowka
d (X; α), HF
ˇ (X; α) and HF (X; α).
in [2], it is possible to define Floer Homology Groups HF
References
[1] HONG, MIN-CHUN and SCHABRUN, LORENZ - Global Existence for the Seiberg-Witten Flow arXiv:0909.1855v3.
[2] KRONHEIMER,P. and MROWKA, T. - Monopoles and Three Manifolds, New Mathematical Monographs
10, 2007.
[3] DORIA, CELSO M - Variational Principle for the Seiberg-Witten Equations, Progress in Nonlinear Diffrential
Equations and Their Applications, 66, pp 247-261, 2005, Birkhäuser Verlag.
[4] JOST, J., PENG, X. and WANG, G. - Variational Aspects of the Seiberg-Witten Functional, Calculus of
Variation 4 (1996) , 205-218.
E-mail address: [email protected]
1
PSEUDO-ANOSOV FLOWS IN TOROIDAL 3-MANIFOLDS
SERGIO FENLEY
We first discuss the following rigidity results:
1) A pseudo- Anosov flow in a Seifert fibered manifold is up to finite covers
topologically conjugate to a geodesic flow;
2) A pseudo-Anosov flow in a solv manifold is topologically conjugate to a suspension Anosov flow.
We describe an interaction of a pseudo-Anosov flow with possible Seifert fibered
pieces in the torus decomposition: if the fiber is associated to a periodic orbit of the
flow, we produce a standard form for the flow in the piece using Birkhoff annuli.
We also produce a very large class of new examples in graph manifolds. This is
joint work with Thierry Barbot.
1
COINCIDENCE OF PAIR OF MAPS BETWEEN TWO SPHERES
DACIBERG LIMA GONALVES AND D. RANDALL
The purpose of this talk is describe mainly the recent results about the classification for which pair of maps between 2 spheres one can deform the pair to be
coincidence free. The special case where the two maps are are equal then the deformation can be considered of two types, which leads to a quite different approach
and results. To illustrate some of the results obtained, for the latter problem we
recall the connection of this problems with the Kervaire invariant one problem and
we show that the remain questions are mod 2 problem.
1
Proposed talk-EBT
Ring cohomology of Seifert manifolds
A. Bauval, C. Hayat
We define a simplicial decomposition of a Seifert manifold with a description
of the words defining the boundaries and a quasi-isomorphism from the simplicial
cochain complex to the cellular cochain complex. This quasi-isomorphism helps
shorting the computations of the cup-products. We get a complete description of
the ring cohomology with coefficients in Zp in terms of the Seifert invariants of
the manifold. Only the ring cohomology in Z2 will be presented here. The new
results concern the non-orientable Seifert manifolds. The cases of orientable Seifert
manifolds were studied by many authors as for example K. Aaslep, J. Bryden, C.
Hayat, S. Tomoda, H. Zieschang, P. Zvengrowski.
1
Planos rápidos de energia finita e aplicações à
topologia simplética
Umberto Hryniewicz (UFRJ)
Curvas pseudo-holomorfas, originalmente introduzidas em geometria simplética por Gromov em 1985, ocupam papel central como ferramenta na solução
de diversos problemas na área. Nesta palestra introduzimos novas ferramentas
da teoria de curvas pseudo-holomorfas e discutimos aplicações ao estudo da
topologia de 4-variedades simpléticas com bordo de contato. Em um primeiro
momento descrevemos como estes novos métodos fornecem novas demonstrações
de alguns célebres teoremas provados por Dusa McDuff na década de 1990 sobre
a influência da estrutura de contato do bordo no seu tipo de difeomorfismo.
Em um segundo momento usamos tais métodos para estudar domı́nios convexos em R4 munido de sua estrutura canônica. O problema de classificação
desta simples e rica famı́lia de 4-variedades simpléticas continua em aberto e
é tido como difı́cil pelos especialistas da área. No intuito de estudá-los definimos um certo invariante em termos das ações de uma famı́lia especial de
órbitas periódicas da dinâmica Hamiltoniana no bordo e exploramos algumas
consequências.
1
THE PROPERTIES AND COMPUTATIONS OF THE
GEOMETRIC INVARIANT Ωn
NICHOLAS KOBAN
The main subject of this talk will be the relatively new Ω-invariants of a finitely
generated group G. We will motivate these invariants by first introducing the
invariant Σ1 (developed by Robert Bieri and Ross Geoghegan) of an action ρ of G
by isometries on a CAT (0) space M . An example of such spaces would be simply
connected manifolds of non-positive sectional curvature. We then will show how
this is a generalization of the Bieri-Neumann-Strebel Σ-invariants of a group which
will lead us to the analogous Ω-invariants. Once defined, we will examine certain
properties of these Ω-invariants such as the connection to the property R∞ of a
group and the connection to the finite generation of the commutator subgroup. The
Σ-invariants have proven rather difficult to compute. We will present some recent
computations of the Ω-invariants that have been elusive for the Σ-invariants. Many
of these computations and properties are joint work with Peter Wong.
1
FIXED POINTS AND COINCIDENCES OF FIBERWISE MAPS
ULRICH KOSCHORKE
Minimum numbers of fixed points or of coincidence components (realized by
maps in given homotopy classes) are the principal objects of study in topological
fixed point and coincidence theory. In this lecture we investigate fiberwise analogies
and determine both types of minimum numbers for all maps between torus bundles
of arbitrary (possibly different) dimensions over spheres. Our results are based on a
careful analysis of the geometry of generic coincidence manifolds and involve mainly
the orbit behavior (e.g. the number of odd order orbits) of a certain selfmap of an
abelian group. In particular, this allows us to decide when maps can be deformed
to become fixed point or coincidence free.
The same geometric approach yields also an isomorphism which describes (in
terms of normal bordism theory) the ring of all stabilized fiberwise maps between
sphere bundles over a closed manifold B; this ring is graded by the K-theoretic
group KO(B).
Research in these areas has a long and active tradition in Brasil and, in particular,
in the state of São Paulo.
1
Various residues in K-theory
Daniel Lehmann
In this talk, I would like to present a synthesis between various classical results (and a few new
ones).
In general, residues of characteristic classes appear when some ”vanishing theorem”, due to the
existence of some ”geometrical structure” on a space V , does not apply because of the presence of a
non-empty ”singular set” Σ in V : then, the concerned characteristic class may not vanish, but it will
however ”localize” near Σ (the prototype result in this spirit is the Poincaré-Hopf theorem, writing
the Euler-Poincaré invariant of a compact manifold as the sum of the indices of a vector field around
its singular points).
There are essentially two kinds of situations giving rise to residues in K-theory :
- the first one happens when the vanishing theorem concerns characteristic classes of a bundle which
is defined only off Σ, but whose stable class in K-theory extends naturally to all of V ; in this category
are for instance the Baum-Bott residues for singular holomorphic foliations, the Diop-Fulton residues
for holomorphic maps which are submersions almost everywhere (including the Milnor numbers for
hypersurfaces),...
- the second one occurs when a sequence of vector bundles, all defined above the whole space V , is
exact only off Σ ; here we find for instance the genus formula for singular algebraic curves, the mexican index (GomezMont-Seade-Verjovski) for foliations preserving some singular subvariety (including
applications to Poincaré inequality for algebraic solutions of an algebraic differential system), the
index of Camacho and L. for foliations which are transversal almost everywhere to some submanifold
(including in particular the index for dicritical singularities of vector fields), ....
2
Coincidences of fiberwise maps
between sphere bundles over the circle
D. Gonçalves
U. Koschorke
A. Libardi
O. Neto
Abstract
Let AM : FM −→ FM and AN : FN −→ FN be selfdiffeomorphisms of smooth, closed,
connected manifolds FM and FN (of dimensions m−1 and n−1, resp.). Consider the fiberwise
maps
f1 , f2 : M m := (I × FM /(1, x) ∼ (0, AM (x)) −→ N n := (I × FN )/(1, y) ∼ (0, AN (y))
which commute with the obvious fiber projections pM and pN onto the unit circle S 1 .
When can the maps f1 and f2 be deformed in a fiberwise fashion until they are coincidence
free? If this can be done we say that the pair (f1 , f2 ) is loose over S 1 . The looseness obstruction
ωB (f1 , f2 ) ∈ Ωm−n+1 (M ; ϕ) was introduced by Koschorke and Gonçalves and it depends only
on the fiberwise homotopy classes of f1 and f2 and vanishes for loose pairs.
We assume, in this work, that the fibers FM and FN are spheres of strictly positive dimensions and the gluing maps AM and AN are orthogonal. Let us consider also
F := {f : M −→ N fiberwise map} /{fiberwise homotopy}
and the map degree degB : F −→ Ωm−n+1 (M ; ϕ) which sends [f ] to ωB (f, a ◦ f0 ), where a
denotes the fiberwise antipodal map in a sphere bundle and f0 = soN ◦ pM , soN the zero
section in N .
In this talk we present the following result involving these two invariants ωB and degB :
Theorem 0.1 Given sphere bundles M and N over S 1 , the following two conditions are
equivalent.
(i) for all [f1 ], [f2 ] ∈ F we have ωB (f1 , f2 ) = 0 if and only if the pair(f1 , f2 ) is loose overS 1 .
(ii) the map degB : F −→ Ωm−n+1 (M ; ϕ) is injective.
QUANTUM FIELD THEORY, FEYNMAN DIAGRAMS, AND
KNOT THEORY
PEDRO LOPES
In this talk, we revisit the progression from the path integral to Feynman diagrams in perturbative Quantumm Field Theory. We then recall the correspondence
between Feynman diagrams and Knot Theory, ‘a la Kreimer, and report on our work
on the subject.
1
Ações Anosov de Rk integráveis e de codimension
um
Carlos Maquera
ICMC - USP
Resumo/Abstract:
Neste trabalho, consideramos ações Anosov de Rk , k ≥ 2, de codimensão um sobre variedades fechadas de dimensão n+k with n ≥ 3.
Mostramos que, sob condições sobre a diferenciabilidade dos subfibrados E ss and E uu (subfibrados forte estável e forte instável), a
ação é topologicamente equivalente à suspensão de uma ação linear
hiperbólica de Z k sobre T n . Isto mostra parcialmente a Conjetura
de Verjovsky para ações de Rk . Este é um trabalho conjunto com
T. Barbot.
Hopf invariants in W -topology
Howard J. Marcum
joint work with Keith A. Hardie and Nobuyuki Oda
Let T op∗ denote the 2-category of based topological spaces, base point
preserving continuous maps, and based track classes of based homotopies.
Fix a space W and consider the 2-functor − ∧ W : T op∗ → T op∗ defined by
taking the smash product with W . The categorical full image of this functor,
denoted W T op∗ , is called the W -topology category. The study of W -topology
was initiated by Hardie, Marcum and Oda [1]. Of course W -topology and
stable homotopy theory, while related, are distinct.
In the associated W -homotopy category the morphism sets are the sets
π W (X, Y ) = π(X ∧ W, Y ∧ W ) of homotopy classes between the respective
smash products. The r-th W -homotopy group πrW (X) is defined to be π(S r ∧
W, X ∧ W ). These groups have been recognized as rather significant. For
example, Barratt (1955) studied πnW (S n ) for W = S 1 ∪p e2 . Toda (1963)
considered the suspension order of a complex Yk having the same homology
as the (n − 1)-fold suspension Σn−1 P 2k of the real projective 2k-space P 2k ,
namely the order of the identity class of π1W (S 1 ) when W = Yk .
In [1] some non-trivial elements in W -homotopy groups were detected by
making use of W -Hopf invariants. This talk focuses on a general proceedure
for introducing Hopf invariants into W -topology. When W is a mod p Moore
space, namely W = S 1 ∪p e2 , we show that it is possible to detect elements
W
in πr+1
(ΩS m+1 ) which have connection with known stable periodic families
of the homotopy groups of spheres. In particular we show that some of
the families discovered by Brayton Gray (1984) are also non-zero in the W W
homotopy groups πr+1
(ΩS m+1 ).
References
[1] K. Hardie, H. Marcum and N. Oda, The Whitehead products and powers in
W -topology, Proc. Amer. Math. Soc. 131 (2003), 941–951.
The Ohio State University at Newark,
1179 University Drive,
Newark, Ohio 43055
USA
[email protected]
1
On the geometry of surfaces in 3-space
L.F. Martins and J.J. Nuño Ballesteros
(Work partially supported by a FAPESP grant 08/02622 and a Capes grant 0399/08-6)
Abstract
Surfaces in 3-space are often defined explicitly as the image of a smooth mapping f : U → R3 (possibly with singularities), where U is an open subset of R2 . We
are interested in the study of the differential geometry of germs of surfaces which
are image of smooth simple map-germs (R2 , 0) → (R3 , 0) under A-equivalence.
Clearly these surfaces have diffeomorphic image but we are interested in the study
of the geometry of the image, and not merely in its diffeomorphism type (map
germs at the same A-orbit may have differing local differential geometry).
Mond present in [4] some results on the classification of smooth map-germs
2
(R , 0) → (R3 , 0) under A-equivalence. More precisely, he showed that each of the
germs in the following list is A-simple, and every A-simple germ of a map from a
2-manifold to a 3-manifold is equivalent to one of the germs on the list.
Germ
A-codimension
Name
0
2
k+2
k+2
k+2
6
k+2
Immersion
Cross-cap (S0 )
Sk±
Bk±
Ck±
F4
Hk
(x, y, 0)
(x, y 2 , xy)
(x, y 2 , y 3 ± xk+1 y), k ≥ 1
(x, y 2 , x2 y ± y 2k+1 ), k ≥ 2
(x, y 2 , xy 3 ± xk y), k ≥ 3
(x, y 2 , x3 y + y 5 )
(x, xy + y 3k−1 , y 3 ), k ≥ 2
The geometry of cross-cap was carried out in [1, 2, 3, 5, 6]. We shall look
at the local differential geometry of surfaces parametrised by simple map germs
R2 , 0 → R3 , 0 that are in the A-orbit of the map germ with normal form being S1±
in the above list.
References
[1] J.W. Bruce and J.M. West, Functions on cross-caps, Math. Proc. Cambridge
Philos. Soc. 123 (1998), 19-39.
[2] T. Fukui and J.J. Nuño-Ballesteros, Isolated roundings and flattenings of
submanifolds in euclidean spaces, Tohoku Math. J. (2) 57 (2005), 469-503.
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[3] R. Garcia, J. Sotomayor and C. Gutierrez, Lines of principal curvature around
umbilics and Whitney umbrellas, Tohoku Math. J. (2) 52 (2000), 163-172.
[4] D. Mond, On the classification of germs of maps from R2 to R3 , Proc. London
Math. Soc., (3), 50 (1985), 333-369.
[5] J.J. Nuño-Ballesteros and F. Tari. Surfaces in R4 and their projections to
3-spaces, Proc. Royal Soc. Ed., 137A (2007), 1313-1328.
[6] J. West, The differential geometry of the cross-cap. Ph.D. Thesis, Liverpool
University, 1995.
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Minimal C 1 -diffeomorphisms of the circle which admit
measurable fundamental domains
Hiroki Kodama and Shigenori Matsumoto
The concept of ergodicity is important not only for measure preserving dynamical systems but also for systems which admits a natural quasi-invariant
measure. Given a probability space (X, µ) and a transformation T of X, µ
is said to be quasi-invariant if the push forward T∗ µ is equivalent to µ. In
this case T is called ergodic with respect to µ, if a T -invariant Borel subset
in X is either null or conull.
A diffeomorphism of a differentiable manifold always leaves the Riemannian volume (also called the Lebesgue measure) quasi-invariant, and one can
ask if a given diffeomorphism is ergodic with respect to the Lebesgue measure or not. It is shown by Denjoy [1] that a C 1 -diffeomorphism of the circle
with derivative of bounded variation is ergodic provided its rotation number
is irrational. Contrarily Oliveira and da Rocha [2] constructed a minimal C 1
diffeomorphism of the circle which is not ergodic.
At the opposite extreme of the ergodicity lies the concept of measurable
fundamental domains. Given a transformation T of a standard probability
space (X, µ) leaving µ quasi-invariant, a Borel subset C of X is called a
measurable fundamental domain if T n C (n ∈ Z is mutually disjoint and
the union ∪n∈Z T n C is conull. In this case any Borel function on C can
be extended to a T -invariant measurable function on X, and an ergodic
component of T is just a single orbit. The purpose of my talk is to show the
following theorem.
Theorem For any irrational number α, there is a minimal C 1 -diffeomorphism
of the circle with rotation number α which admits a measurable fundamental
domain with respect to the Lebesgue measure.
References
[1] A. Denjoy, Sur les courbes défini par les équations différentielle à la
surfase du tore. J. Math. Pures Appl. 9(11) (1932), 333-375.
[2] F. Oliveira and L. F. C. da Rocha, Minimal non-ergodic C 1 diffeomorphisms of the circle. Erg. Th. Dyn. Sys. 21(2001), 1843-1854.
APLICAÇÕES ESTÁVEIS DE SUPERFÍCIE NO PLANO
PROJETIVO
CATARINA MENDES DE JESUS
Seja f uma aplicação estável entre as duas superfı́cies M e N. Se M é uma
superfı́cie compacta, então o conjunto singular de f , Σf , são formados por curvas
fechadas, mergulhadas e disjuntas sobre M . Sempre podemos associar um grafo G,
com pesos nos vértices, à aplicação f , onde as arestas de G correspondem as curvas
de Σf , os vértices às componentes de M − Σf e os pesos no vértice corresponde
ao gênero da região que ele está associado. O objetivo aqui é apresentar condições
necessárias e suficientes para que um grafo dado esteja associado a alguma aplicação
estável de alguma superfı́cie compacta M no plano projetivo, em especial para o
caso de aplicações estáveis sem cúspides.
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Abstract
Modifications of 2-dimensional foliations
on 4-manifolds and tautness
Yoshihiko Mitsumatsu (Chuo University, Tokyo)
This is a report on joint work with Elmar Vogt, which is in progress.
We have studied the existence of a 2-dimensional foliation for a given 4manifold and a given embedded closed surface, as a prescribed compact leaf
or closed transversal. Thanks to Thurston’s h-principle, we know that the
cohomological constraints give rise to a necessary and sufficient condition for
the existence of such foliations.
Especially for a trivial S 2 -bundle over a surface of genus ≥ 2 and an embedded surface which is a section, the cohomology equation is unexpectedly
beautifully solved and only in the case where the self intersection of the embedded surface divides the euler characteristic of the base surface (=2 − 2g),
there exist three anormalous solutions.
This work comes from the effort to realize these solutions geometrically
without relying on the h-principle. The constructions are strongly related to
the geodesic flow which is of algebraic Anosov, of the hyperbolic structure
of the embedded surface. The cohomological constraint is understood from
this point of view.
Moreover, it turns out that such geometrically constructed foliations describe some modification procedures for foliations around a compact leaf or
a closed transversal. One of such modifications is regarded as an analogue of
the ‘turbulization’ in the codimension 1 case.
If the time allows, we will further discuss the (geometric or homological)
tautness of such modifications.
Yoshihiko MITSUMATSU
Department of Mathematics, Chuo University
1-13-27 Kasuga Bunkyo-ku, Tokyo, 112-8551, Japan
e-mail : [email protected]
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HOMOTOPIC GENERATORS, GAUSS-BONNET-CHERN TYPE
FORMULAE AND THEIR VARIATIONAL PROPERTIES ON
FLAG MANIFOLDS
CAIO NEGREIROS
In this talk we will discuss variational properties of some generators of homotopy
groups of flag manifolds that are well known to have infinite classes of geometries, in
contrast of the Grassmannians that have only one invariant geometry: the FubinyStudy one. These results rely on several Gauss-Bonnet-Chern type formulae and
also allow us to treat certain classes of geodesics as well as harmonic maps on
certain flags.
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A COINCIDENCE THEOREM FOR COMMUTING
INVOLUTIONS
PEDRO L. Q. PERGHER
Coincidence theory concerns itself with the following basic question: given
two maps f, g : X → Y of topological spaces, when is there a coincidence point,
that is, a point x ∈ X such that f (x) = g(x)? If so, another question concerns
with the size of the set of coincidence points, Coinc(f, g). In this generality, one
can say almost nothing of interest; good questions arise in this context when we
focus attention on specific situations. In this setting, consider X = Y = M m
an m-dimensional, connected, closed and smooth manifold, and f and g two
commuting diffeomorphisms on M m . In this case, f and g generate a smooth
action of Z × Z, Z × Zr or Zs × Zr on M m , and Coinc(f, g) is a disjoint
n
m
union of closed submanifolds of M , Coinc(f, g) =
F j (n < m), where
j=0
F j denotes the union of those components of Coinc(f, g) having dimension j.
A natural question is the determination of the largest dimension that occurs
in Coinc(f, g). In this talk we discuss a particular case, obtained by taking
r = s = 2, that is, with f and g being involutions. This case is motivated
by the following classic example: consider f, g : S m → S m , where S m is the
m-dimensional sphere, f(x0 , x1 , ..., xm ) = (−x0 , −x1 , ..., −xm ) (the antipodal
map) and g(x0 , x1 , ..., xm ) = (−x0 , −x1 , ..., −xm−1 , xm ). The fixed point set of
f is empty, and the fixed point set of g consists ot two points. Note that the
largest dimension that occurs in Coinc(f, g) is m − 1. We will see that in a
much more general situation the behavior is very similar to the classic example;
specifically, we have the following generalization of this classic example: let M m
be an m-dimensional, connected, closed and smooth manifold, and f and g two
commuting involutions on M m . Suppose the fixed point set of f is empty and
the number of points of the fixed point set of g is even and of the form 2p, with
1
p odd. Then the largest dimension that occurs in Coinc(f, g) is m − 1. The
proof is based on facts related to the cobordism of free Z2 × Z2 actions, that is,
to the Conner and Floyd cobordism group N∗ (RP ∞ × RP ∞).
References
[1]
[2]
P. E. Conner and E. E. Floyd, Differentiable periodic maps, Springer-Verlag, Berlin,
(1964).
R. E. Stong, Equivariant bordism and (Z2 )k -actions , Duke Math. J. 37, (1970), 779-785.
2
THE KERVAIRE INVARIANT ONE PROBLEM
IN DIMENSION 126
DUANE RANDALL
The celebrated Kervaire Invariant One Problem has been recently solved in all
dimensions except 126. We first explain briefly the significance of this problem
in both differential topology and in the stable homotopy groups of spheres. We
then communicate equivalent formulations of this open problem in dimension 126.
Work with Kee Lam equates the existence of a Kervaire invariant one element in
dimension 126 with the vanishing of a specific Whitehead product involving the
ImJ generator in dimension 11. An unpublished result with Ulrich Koschorke
affirms that the existence of KI one elements in dimension 126 is equivalent to a
property in coincidence theory for mappings between spheres of dimensions 241 and
116.
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THE HOMOTOPY TYPE OF SPACES
OF LOCALLY CONVEX CURVES IN THE SPHERE
NICOLAU C. SALDANHA
A smooth curve γ : [0, 1] → S2 is locally convex if its geodesic curvature is positive
at every point. J. A. Little showed that the space of all locally positive curves γ
with γ(0) = γ(1) = e1 and γ 0 (0) = γ 0 (1) = e2 has three connected components
L−1,c , L+1 , L−1,n .
A motivation for considering this space comes from differential equations. Consider the linear ODE of order 3:
u000 (t) + h1 (t)u0 (t) + h0 (t)u(t) = 0,
t ∈ [0, 1];
the set of pairs of potentials (h0 , h1 ) for which the equation admits 3 linearly independent periodic solutions is homotopically equivalent to LI . These spaces and variants have been discussed, among others, by B. Shapiro, M. Shapiro and B. Khesin.
These spaces are also the orbits of the second Gel’fand-Dikki brackets and therefore
have a natural symplectic structure. Furthermore, these spaces are related to the
orbit classification of the Zamolodchikov Algebra.
Our aim is to describe the homotopy type of these spaces. The connected component L−1,c is known to be contractible. We construct maps from L+1 and L−1,n
to ΩS3 ∨ S2 ∨ S6 ∨ S10 ∨ · · · and ΩS3 ∨ S4 ∨ S8 ∨ S12 ∨ · · · , respectively, and show
that they are (weak) homotopy equivalences.
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PARAMETER RIGID ACTIONS OF THE HEISENBERG
GROUPS
NATHAN M. DOS SANTOS
A locally free action of a Lie group is parameter rigid if for each other action
with the same orbit foliation there exists a smooth orbit-preserving diffeomorphism
which conjugates the action to a reparametrization of other by an automorphism
of the Lie group. We show that for actions of the Heisenberg groups, if the first
leafwise cohomology of the orbit foliation is isomorphic to the first cohomology of
the Lie algebra of the group, then the action is parameter rigid. Using this, we give
examples of parameter rigid actions for all the Heisenberg groups.
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THE ISOMORPHISM PROBLEM FOR PLANAR POLYGON
SPACES
DIRK SCHUETZ
Given a length vector consisting of n positive numbers, we consider configurations
of a robot arm in the plane with n bars, whose lengths are given by the length
vector. Such a configuration is called a closed linkage, if the endpoint of the arm
coincides with the starting point. The space of such configurations up to rotations
and translations is called the planar polygon space of the length vector.
Generically, this space is a manifold, and the set of length vectors falls naturally
into finitely many regions such that two length vectors in the same region have
diffeomorphic planar polygon spaces.
We give a proof of a Conjecture of Walker which states that one can recover
the region of a length vector (up to permutation) from the cohomology ring of the
planar polygon space. For a large class of length vectors, this has been shown by
Farber, Hausmann and Schuetz. In the remaining cases, we use Morse theory and
the fundamental group to describe a subring of the cohomology invariant under
graded ring isomorphism. From this subring the conjecture can be derived by
applying a result of Gubeladze on the isomorphism problem of monoidal rings.
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Normal subgroups of diffeomorphism groups of
Rn and other open manifolds
Paul A. Schweitzer, S.J.
Pontifı́cia Universidade Católica do Rio de Janeiro
[email protected]
July 30, 2010
Abstract
Mather showed that the group of C r diffeomorphisms of an nmanifold M , Diff r (M ), is simple if M is closed and r 6= n + 1,
1 ≤ r ≤ ∞. When M = Rn , we determine all the normal subgroups of
the group Diff r (Rn ) of C r diffeomorphisms of Rn except when r = n+1
or n = 4, and also of the group of homeomorphisms, Homeo(Rn ). The
normal subgroups of Diff r (Rn ) are related to the group of homotopy
(n + 1)-spheres θn+1 (except possibly in dimension 4). We also study
the group A0 = Diff r0 (M ) of those diffeomorphisms of an open manifold M that are isotopic to the identity. If M is the interior of a
compact manifold with nonempty boundary, then we show that the
quotient of A0 by the normal subgroup of diffeomorphisms that coincide with the identity near to a given end e of M is simple; the proof
of this result of the speaker’s from the 1970’s, which was was used by
Dusa McDuff in a study of the lattice of normal subgroups of such
manifolds, has never been published, but all these results will appear
in a forthcoming article in Ergodic Theory and Dynamical Systems
(available as arXiv 0911.4835).
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RELATIONSHIPS BETWEEN 3-MANIFOLDS AND
HOPF-ALGEBRA OBJECTS
FERNANDO SOUZA
Quantum topology is the interdisciplinary study of a number of relatively new
topological invariants, as well as possible supporting frameworks for them. It was
born in the early 1980s and has established many unexpected, exciting relations
between low-dimensional topology and various areas of mathematics, theoretical
physics, and quantum information processing. A major trend in quantum topology
is the gradual abstraction of its constructions and techniques, often up to the level
of categories with additional structure (tensor product, braiding, duality, trace,
etc.)
Hopf algebras (in particular, quantum groups) and their generalizations (in particular, Hopf-algebra objects) have appeared in quantum topology in two ways.
On the one hand, some of them have categories of representations endowed with
structures suitable for the construction of invariants. In this case, Hopf-algebra
objects have provided examples of those invariants. On the other hand, there are
constructions that use Hopf-algebra objects as essential ingredients. Some of these
have even led to partial algebrizations of the 3-manifolds themselves. We shall focus
on this matter.
In this talk, after a small introduction to Hopf algebras and Hopf-algebra objects, we will explain the involutory Kuperberg invariant, focusing on the interplay
between the axioms for involutory Hopf-algebra axioms and the Reidemeister-Singer
moves on generalized Heegaard diagrams. Then we will mention its counterpart for
framed or combed 3-manifolds. Finally, we will discuss some surgical constructions
of 3-manifold invariants, emphasizing the Kauffman-Radford-Hennings invariant
and the partial algebrization of 3-manifolds started by Crane, Kerler and Yetter.
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Explicit constructions over the exotic 8-sphere
Llohann Dallagnol Sperança, UNICAMP
Abstract: We present an elementary construction of the exotic 8-sphere and some consequences. We exhibit a formula for a diffeomorphism of S 7 non-isotopic to the identity
and an isotopy from its square to idS 7 . In this context we construct exotic Hopf fibrations
(S 7 · · · Σ15 → Σ8 ) over the exotic 8-sphere and prove that these are all such linear bundles.
In fact we prove the following theorems:
Theorem 1. The total space of the non-trivial S 3 -principal bundle over S 8 admits an S 3
action wich is free with quotient space the exotic 8 sphere.
Theorem 2. An homotopy sphere can be realized as the total space of the sphere bundle of
a linear bundle over the exotic 8-sphere if and only if it can be realized as the sphere bundle
over the standard 8-sphere.
This is a joint work with C. Durán and A. Rigas.
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AN EQUIVALENCE TO THE BORSUK-ULAM THEOREM ON Rn
PEDRO L. Q. PERGHER AND DANIEL VENDRÚSCOLO
In some recent works the classical Borsuk-Ulam theorem has been generalized
in different directions. One possible approuch is to consider triples (X, τ ; Y ) where
X and Y are topological spaces and τ is a free involution on X, we say that
(X, τ ; Y ) satisfies the Borsuk-Ulam theorem (shortly (X, τ ; Y ) satisfies BUT) if for
any continuous map f : X → Y there exists a point X ∈ X such that f (x) =
f (τ (x)).
In [2] Gonçalves studies the triples (S, τ ; R2 ) where S is a closed surface. They
prove that in many cases (S, τ ; R2 ) satisfies BUT and, surpriselly, for some surfaces
S the result depends on the involution τ .
In [1] we find results for (X, τ ; R2 ) where X is a topological space with finite
fundamental group and in [3] they study (Sp , τ ; R3 ) where Sp is a 3-dimensional
homotopy spherical space form.
In [1] one can find a formulation for a Weak Borsuk-Ulam theorem for the 2dimensional torus: we say that the triple (X, τ ; T 2 ) satisfies the weak Borsuk-Ulam
theorem (WBUT) when for any continuos map f : X → T 2 there exists a point
x ∈ X such that f (x) − f (τ (x)) = f (τ (x)) − f (x).
In this paper we propose a more general formulation for weak Borsuk-Ulam property (WBUP) and we prove that (X, τ ; Rn ) satisfies BUT if and only if (X, τ ; T n , −1)
has the WBUP.
References
[1] P. E. Desideri; P. L. Q. Pergher; D. Vendrúscolo, Some generalizations of the Borsuk-Ulam
Theorem, submited.
[2] D. L. Gonçalves , The Borsuk-Ulam theorem for surfaces, Quaestiones Mathematicae 29,
(2006), no. 1, 117-123.
[3] D. L. Gonçalves; O. Manzoli Neto; M. Spreafico, The Borsuk-Ulam Theorem for 3dimensional homotopy spherical space forms, preprint.
Departamento de Matemática - UFSCar, São Carlos, SP - Brazil.
E-mail address: [email protected]
(Daniel Vendrúscolo) DM - UFSCar
E-mail address: [email protected]
1
TORUS HOMOTOPY GROUPS AND GROUP OF HOMOTOPY
GROUPS
PETER WONG
Let J(S 1 ) be the James construction of the unit sphere S 1 . For any space
X, the group [J(S 1 ), ΩX] of homotopy classes of maps from J(S 1 ) to the loop
space ΩX encodes the information of all higherQ
homotopy groups π1 X, i ≥ 2. In
fact, [J(S 1 ), ΩX] is isomorphic to the product i≥2 πi X as a set but NOT as a
group. On the other hand, the torus homotopy groups τi (X), first introduced by
Fox, provide a rich structure involving the homotopy groups πi X as well as their
Whitehead products. We use the structure of the torus homotopy groups to study
the group structure of [J(S 1 ), ΩX] and we show that [J(S 1 ), ΩX] can be embedded
as a subgroup of the infinite torus homotopy group τ∞ (X).
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THREE-DIMENSIONAL SPHERICAL SPACE FORMS
PETER ZVENGROWSKI
This talk will start with a little history on the subject of space forms, which are
defined as Riemannian manifolds having constant curvature. The two dimensional
case goes back to Euclid, and the development of non-euclidean geometries in the
nineteenth century. The three dimensional case was solved in the period 1925-1933
thanks to work of Hopf followed by two papers of Threlfall and Seifert. The two
Threlfall-Seifert papers total 110 pages and are difficult reading, but in 1961 Hattori
published a 3 page paper in Sugaku which gives a new (and obviously simpler and
shorter) proof of these results. Thanks to work by Luciana Martins, Sadao Massago,
Mamoru Mimura, and the author, this paper is now translated into English, and
will be the subject of this talk. The main ingredients of the proof are the use of
quaternions, Lie groups, and some group theory.
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