XV Encontro Brasileiro de
Topologia
10 a 14 de julho de 2006
Unesp – Rio Claro – SP – Brasil
SELF-COINCIDENCES OF MAPPINGS BETWEEN SPHERES
DUANE RANDALL
Abstract. This joint work with Daciberg studied the relationship between homotopy disjointness and homotopy disjointness by small deformation. In particular, we determine the minimal dimensions for which a mapping between spheres is homotopy disjoint from itself, but not by small deformation. Closely related are recent results of Ulrich Koschorke.
Duane Randall
Loyola University-USA E-mail: randall@loyno.edu
ON THE NORMAL BORDISM GROUP AND THE FORGETFUL
MAP
CARLOS BIASI, ALICE K. M. LIBARDI AND ISABEL C. ROSSINI
Abstract. In this work we use an exact sequence of normal bordism groups given in a paper by C.Biasi and A.K.M.Libardi ( Proc. of Edinburgh Math.Soc. 47(2004) 289-296) to give an homological characterization of the kernel of the forgetful map F : Ωn(X × BSO(k), ˜φp+k) −→ Ωn(X). This sequence is a
generalization of a sequence given by Salomonsen (Math.Scand. 32(1973), 87-111). In the particular case when X is an (n + k)- manifold and ˜φp+k=
εp+k− νp
X× ˜γk, p large enough, the kernel of the forgetful map F consists
of elements [(M, f )] null bordant such that f is homotopic to an immersion and the image of F consists of elements [(M, f )] which are bordant to an immersion. There is an unoriented version of the results presented. To give this characterization we prove that Ωi(X, εs− ηs) and Hi(X, Z) are C2,3
-isomorphic for i ≤ 3 and C2-isomorphic for i ≤ 2 where C2,3( resp. C2) is the
class of abelian group having elements of order 2p· 3q( resp. 2p), and ηsis a
stable vector bundle over X.
References
[1] C.Biasi and A.K.M.Libardi. Remarks on immersions in the metastable dimension range.
Proc. of the Edinburgh Math. Soc. 47 (2004) 289-296.
[2] C.Biasi and D.L.Gonc¸alves. Equivalent conditions for n-equivalences modulo a class C of groups. Bol. Soc. Bras. Mat. 27 (1996) 187-198.
[3] P.E. Conner and E.E. Floyd Differentiable periodic maps. Ergenbisse der Mathematik und
ihrer Grenzgebiete, vol.33. (Springer,1964).
[4] M. Hirsch. Immersions of manifolds. Trans. AMS. 93 (1959) 242-276. [5] D. Husemoller. Fibre Bundles. ( Mc.Graw Hill Book Co.,1966).
[6] U. Koschorke. Vector Fields and Other Vector Bundle Morphism - A Singularity Ap-proach.Lecture Notes in Mathematics 847 (Springer,1981).
[7] H.A. Salomonsen. Bordism and Geometric Dimension. Math. Scand. 32 (1973) 87-111. [8] E. Spanier. Algebraic Topology. ( Mc.Graw Hill Book Co.,1966).
[9] G.Whitehead. Elements of Homotopy Theory. Graduate Texts in Mathematics 61 (Springer,1978). Carlos Biasi ICMC-USP-S˜ao Carlos E-mail: biasi@icmc.usp.br Alice K. M. Libardi IGCE-UNESP-Rio Claro E-mail: alicekml@rc.unesp.br
Isabel Cristina Rossini IGCE-UNESP-Rio Claro E-mail: irrosini@rc.unesp.br
CUTTING AND PASTING OF G-MANIFOLDS AND FAMILIES OF
SUBMANIFOLDS
K. KOMIYA
Abstract. Cutting and pasting (Schneiden und Kleben) operations on man-ifolds give rise to a group of manman-ifolds, denoted by SK∗and called the
SK-group. In this talk we will be concerned with the SK-group of G-manifolds, G a finite abelian group, and the SK-group of families of submanifolds(without
G-action). We will give a necessary and sufficient condition for a family of
submanifolds of a manifold to be cut and pasted into a family of fixed point submanifolds of a G-manifold. These conditions are given in terms of the Euler characteristics of submanifolds. As an application, we will observe a condition for a G-manifold M to be equivariantly cut and pasted into a product N × L of two G-manifolds N and L. As one more application, we will obtain a con-dition for a G-manifold to be equivariantly cobordant to the total space of
G-fibration over the circle.
Katsuhiro Komiya
Yamaguchi University-Jap˜ao E-mail: komiya@yamaguchi-u.ac.jp
SOBRE O GRUPO FUNDAMENTAL DE VARIEDADES ABERTAS
FOLHEADAS POR PLANOS
CARLOS BIASI E CARLOS MAQUERA
Abstract. Em [1], Palmeira mostrou que uma C2folhea¸c˜ao por planos
fecha-dos transversalmente orient´avel em uma n-variedade aberta e orient´avel, com grupo fundamental finitamente gerado, ´e topol´ogicamente conjugado ao pro-duto de uma folhea¸c˜ao (por linhas) de uma superf´ıcie aberta por Rn−2.
Con-seq¨uentemente, o grupo fundamental desta variedade ´e ou Z, neste caso a superf´ıcie ´e S1× R, ou livre. Nos mostramos que esta propriedade ´e
preser-vada, mesmo retirando da hip´otese a orientabilidade da variedade e o fato da folhea¸c˜ao ser transversalmente orient´avel.
Por outro lado, usando este resultado mostramos que: Se F ´e uma folhea¸c˜ao por planos em N , uma n-variedade aberta orient´avel e conexa, e se π1(N )
possui um subgrupo Abeliano de posto maior ou igual do que dois, ent˜ao existem folhas n˜ao fechadas.
Finalmente, no caso que N tem dimens˜ao trˆes e π1(N ) = Z ⊕ Z, veremos
que N admite uma decomposi¸c˜ao da forma:
N = N0∪ (∪∞i=1Ni),
onde
(1) Nj∩ Nk= ∅ para j 6= k,
(2) N0 ´e uma variedade aberta saturada por folhas densas em N0 tal que
π1(N0) = Z ⊕ Z,
(3) Ni, i = 1, 2, . . . , ´e homeomorfo a uma semi-bola aberta que ´e saturado
por folhas fechadas.
References
[1] C. F. Palmeira; Open manifolds foliated by planes, Ann. of Math. 107 (1978), 109–131. Carlos Biasi ICMC-USP-SC E-mail: biasi@icmc.usp.br Carlos Maquera ICMC-USP-SC E-mail: cmaquera@icmc.usp.br
GRAFOS NA CLASSIFICAC
¸ ˜
AO GLOBAL DE APLICAC
¸ ˜
OES
EST ´
AVEIS DE SUPERF´ICIES COMPACTA NO PLANO
CATARINA MENDES DE JESUS
Abstract. Dada uma aplica¸c˜ao est´avel f de uma superf´ıcie compacta, M , no plano, o conjunto singular Σf , desta aplica¸c˜ao, ´e formado por curvas fechadas disjuntas em M . Ao par (M, Σf ) associamos um grafo com pesos nos v´ertices. A todo grafo com peso nos v´ertices podemos associar uma superf´ıcie com curvas fechadas disjuntas sobre esta. Neste trabalho, damos condi¸c˜oes necess´arias e suficientes para que dado um grafo com peso nos v´ertices, este possa ser realizado como grafo de aplica¸c˜oes est´aveis de alguma superf´ıcie compacta no plano.
References
[1] D. HACON, C. Mendes de Jesus and M. C. Romero Fuster; Topological invariants of stable
maps from a surface to the plane from a global viewpoint, Lecture Notes in Pure and Applied
Mathematics, vol.232 (2003).
[2] D. HACON, C. Mendes de Jesus and M. C. Romero Fuster; Fold maps from the sphere to
the plane, To apper in Experimental Maths (2006).
[3] D. HACON, C. Mendes de Jesus and M. C. Romero Fuster; Cuspless stable maps from the
sphere to the plane, Experimental Mathematics , 2005 , DMA - 77760.
[4] D. HACON, C. Mendes de Jesus and M. C. Romero Fuster; Stable maps from surfaces to the
plane with prescribed branching data, To apper in Topology and its Applications (2006).
Catarina Mendes de Jesus Universidade Federal de Vic¸osa E-mail: cmendes@ufv.br
COBORDISMO DE AUTOMORFISMOS DE 3-VARIEDADES
LEONARDO N. CARVALHO E ULRICH OERTEL
Abstract. A unicidade da decomposi¸c˜ao de uma 3-variedade em fatores pri-mos [1] reduz o problema da classifica¸c˜ao de 3-variedades compactas ao caso de 3–variedades irredut´ıveis.
O estudo de aplica¸c˜oes entre 3-variedades, em particular o estudo de
au-tomorfismos de uma dada variedade, n˜ao se beneficia t˜ao facilmente dessa simplifica¸c˜ao. O motivo: grosseiramente, se os fatores primos obtidos da de-composi¸c˜ao s˜ao ´unicos, a forma de obtˆe-los n˜ao ´e. De fato, v´arios fenˆomenos ausentes no caso irredut´ıvel s˜ao observados em casos redut´ıveis [3].
Nesta palestra propomos o uso da no¸c˜ao de cobordismo de automorfismos para auxiliar o estudo de automorfismos de 3-variedades. Por exemplo, se pode relacionar o estudo de automorfismos de 3-variedades compactas e orient´aveis em geral ao dos automorfismos de seus fatores irredut´ıveis [2].
References
[1] J. Milnor; A unique decomposition theorem for 3–manifolds, Amer. J. Math, 84 (1962), 1–7. [2] L. Carvalho e U. Oertel; A classification of automorphisms of compact 3–manifolds, 2005,
pr´e-print, arXiv:math.GT/0510610.
[3] D. McCullough; Mappings of reducible 3–manifolds, Geometric and Algebraic Topology, Ba-nach Center Publications, 18 (1986), 61–76.
Leonardo N. Carvalho IM-UFF
E-mail: leonardo carvalho@vm.uff.br
Ulrich Oertel
Rutgers University-Newark
APPLICATIONS OF THE NON-STANDARD VERSION OF THE
BORSUK-ULAM THEOREM
DENISE DE MATTOS
Abstract. Erika Pannwitz showed in [4] that for any n ≥ 2, there exist continuous maps ϕ : Sn→ Snand f : Sn→ R2such that f (x) 6= f (ϕ(x)) for
any x ∈ Sn. In this work, we generalize this result replacing the sphere Snby
more general topological space and we give interesting applications of the non-standard version of the Borsuk-Ulam theorem proved in [1]. More specifically we describe an useful method to verify the hypotheses of main result proved in [1] and we construct examples of 1-dimensional topological spaces for which such results can be applied. In particular, when X is a connected graph, an interesting physical interpretation is obtained.
References
[1] Biasi, C., de Mattos, A non-standard version of the Borsuk-Ulam Theorem, Bull. Pol. Ac. Sc. Math., 53 (1) 111-119, 2005.
[2] Borsuk, K., Drei S¨atze ¨uber die n-dimensionale euklidische Sph¨are, Fund. Math., 20 177-190,
1933.
[3] Hopf, H. Freie ¨U berdeckungen und freie Abbildungen. Fund, Math. 28 33-57, 1937.
[4] Pannwitz, E. Eine freie Abbildung der n-dimensionalen Sph¨are in die Ebene, Math. Nachr.
(7) 183-185, 1952. Denise de Mattos
UNESP-IBILCE-S˜ao Jos´e do Rio Preto E-mail: demattos@ibilce.unesp.br
RECOBRIMENTOS RAMIFICADOS PRIMITIVOS
DECOMPON´IVEIS DE GRAU 4 E 6
NATALIA ANDREA VIANA BEDOYA
Abstract. Estudamos a decomponivilidade de revestimento ramificados prim-itivos de grau 4 e 6 trabalhando com o grupo de permuta¸c˜oes que eles deter-minam
Natalia Andrea Viana Bedoya IME-USP-SP
COMMUTING INVOLUTIONS WHOSE FIXED POINT SET
CONSISTS OF TWO SPECIAL COMPONENTS
PEDRO L. Q. PERGHER
Abstract. Let Fnbe a connected, smooth and closed n-dimensional
mani-fold. We call Fna manifold with property H when it satisfies the following
property: if Nmis any smooth and closed m-dimensional manifold with m > n
and T : Nm→ Nmis a smooth involution whose fixed point set is Fn, then
m = 2n. This definition was inspired in an old result of P. Conner and E. E.
Floyd showing that the even dimensional and real projective spaces RP2n
sat-isfy this property (the same type of argument works to show that the complex and quaternionic projective spaces CP2nand HP2n, and the Cayley
projec-tive plane QP2, have this property). The following facts concerning property
H are true: if Fn has property H, then n is even, Fn is nonbounding, the
tangent bundle over Fndoes not have sections and Fncannot be the total
space of a nontrivial differentiable fibering of closed manifolds. Also, prop-erty H is not a cobordism invariant, but is a homotopy invariant. Besides the above projective spaces, other examples of manifolds with this property are: all nonbounding 2-dimensional manifolds, all simply connected and nonbound-ing 4-dimensional manifolds (for example, the connected sum of CP2and any
number of copies of S2× S2), all nonbounding 8-dimensional manifolds M8
with H1(M8, Z
2) = 0 and H2(M8, Z2) = 0, all nonbounding 16-dimensional
manifolds M16 with Hi(M16, Z
2) = 0 for 1 ≤ i ≤ 4, and the connected sum
RP2n#(Sn× Sn)#...#(Sn× Sn) where n is not a power of 2. In this talk we
describe the equivariant cobordism classification of smooth actions (Mm; Φ) of
the group Zk
2 (considered as the group generated by k commuting involutions
T1, T2, ..., Tk) on closed and smooth m-dimensional manifolds Mmfor which
the fixed point set of the action consists of two components K and L with property H, and where dim(K) < dim(L). The description is given in terms of the set of equivariant cobordism classes of involutions fixing K ∪ L. Pedro L. Q. Pergher
DM-UFSCar
DIMENSIONS OF FIXED POINT SETS OF INVOLUTIONS
PERGHER, PEDRO L. Q.; FIGUEIRA, F ´
ABIO G.
Abstract. Suppose that Fn and F2 are disjoint and closed submanifolds
of a connected, smooth and closed manifold M , with dimensions n and 2, respectively, and with n > 2. Let k be the codimension of Fn. We show that
the possible values of k are very low in the situation where Fn∪ F2is the fixed
point set of a smooth involution T : M → M , n is odd and the normal bundle of F2 in M is not a boundary; specifically, we show that k ≤ 3 in this case.
This is a very special case of the situation covered by the famous 5/2-theorem of J. M. Boardman. In addition, we show that this bound can be improved to k ≤ 1 (hence k = 1) for some specific values of n and some fixed stable cobordism classes of the normal bundle of F2in M ; further, we determine in
these cases the equivariant cobordism class of (M, T ).
References
[1] A. Borel and F. Hirzebruch, On characteristic classes and homogeneous spaces, I, Amer. J. Math. 80, (1958), 458-538.
[2] C. Kosniowski and R. E. Stong, Involutions and characteristic numbers, Topology 17, (1978), 309-330.
[3] D. C. Royster, Involutions fixing the disjoint union of two projective spaces, Indiana Univ. Math. J. 29, (1980), 267-276.
[4] J. M. Boardman, On manifolds with involution, Bulletin Amer. Math. Soc. 73, (1967), 136-138.
[5] P. E. Conner, The bordism class of a bundle space, Michigan Math. J. 14, (1967), 289-303. [6] P. E. Conner and E. E. Floyd, Differentiable periodic maps, Springer-Verlag, Berlin, (1964). [7] P. L. Q. Pergher and R. E. Stong, Involutions fixing {point} ∪ Fn, Transformation Groups
6, (2001), 78-85.
[8] Suzanne M. Kelton, Involutions fixing RPj∪ Fn, Doctoral Thesis, University of Virginia,
(2001).
[9] Suzanne M. Kelton, Involutions fixing RPj∪ Fn, Topology Appl. 142, (2004), 197-203.
Pergher, Pedro L. Q. DM-UFSCar E-mail: pergher@dm.ufscar.br Figueira, F´abio G. DM-UFSCar E-mail: fabio@dm.ufscar.br
A NEW IDEA IN REPRESENTATION THEORY AND ITS
APPLICATION TO THE REPRESENTATION SPACE OF THE
BRAID GROUPS.
JOHN BRYDEN
Abstract. Recent work by Tyler Lawson has led to the construction of a spectral sequence that relates the representation ring of an infinite discrete group G to its deformation K-theory, which is basically the algebraic K-theory of a category obtained from the unitary representations of the group G. If
G = Bn, the n-string braid group, then studying the topological K-theory of
BBngives enough information to apply Lawsons results to the braid groups
and obtain information about the homotopy type of the representation space of Bn.
John Bryden
Nicholls State University-Louisiana-USA E-mail: john.bryden@nicholls.edu
HOMOTOPY IDEMPOTENTS ON SURFACES AND BASS’
CONJECTURES
MICHAEL KELLY
Abstract. An interest in idempotent matrices was motivated by a 1976 con-jecture of H. Bass and is well known in algebraic K-theory as the Strong Bass Conjecture. The conjecture roughly states that for any discrete group G, the Hattori-Stallings trace of a finitlely generated projective module over the group ring Z[G] should be supported on the identity component only.
Under the assumption that the group G is finitely presented it was demon-strated by R. Geoghegan in 1981 that the Strong Bass Conjecture can be reformulated as a question in fixed point theory. In particular, regarding the Nielsen theory of the identity map defined on a class of finitely dominated compact topological spaces. This reformulation conjectures that the Nielsen number of the identity map is either 0 or 1. Recently, Berrick, Chatterji and Mislin have shown that the Strong Bass Conjecture is equivalent to the con-dition that any homotopy idempotent self-mapping on a compact manifold of dimension at least 3 can be deformed to a map which has exactly one fixed point. The authors question as to the validity of this condition in the dimen-sion two.
In this talk we discuss these formuations of the Strong Bass Conjecture and present a proof of the answer to this question as given by the following theorem.
Theorem: Let F be a compact, connected surface with negative Euler char-acteristic. Given a positive integer n there exists a homotopy idempotent fn
on F such that the Nielsen number of fnis 0, but any map in the homotopy
class of fnhas at least n fixed points.
We also discuss the Weak Bass Conjecture and its formulation in fixed point theory.
Michael Kelly
Loyola University-New Orleans-USA E-mail: kelly@loyno.edu
DOM´INIOS FUNDAMENTAIS PARA GRUPOS GERADOS POR
TRˆ
ES TRANSLAC
¸ ˜
OES DE HEISENBERG
SEME GEBARA NETO
Abstract. A fronteira de H2
C, o espa¸co hiperb´olico complexo de dimens˜ao
dois, pode ser identificada com a compatifica¸c˜ao de um ponto do Grupo de Heisenberg H = C × R. De modo an´alogo `a identifica¸c˜ao entre transforma¸c˜oes conformes da esfera de Riemann com isometrias do espa¸co hiperb´olico real de dimens˜ao trˆes, transforma¸c˜oes de Heisenberg d˜ao origem a isometrias hiperb´ oli-cas complexas e vice-versa. Grupos gerados por trˆes transla¸c˜oes em dire¸c˜oes independentes do Grupo de Heisenberg s˜ao automaticamente discretos (e ele-mentares, pois o ´unico ponto limite ´e o infinito). A variedade quociente, uma variedade Nil de dimens˜ao trˆes (segundo Thurston) ´e sempre um fibrado de c´ırculos sobre um toro e, como n˜ao se trata de um quociente do espa¸co euclid-iano, o n´umero de Euler deste fibrado ´e n˜ao nulo; em particular este grupo n˜ao admite um cubo como dom´ınio fundamental. A partir de um exemplo onde o n´umero de Euler do quociente ´e e = 2 e de uma combina¸c˜ao entre o Teorema dos Poliedros de Poincar´e e o Teorema de Giraud para geometria hiperb´olica complexa (interpretando o grupo de transla¸c˜oes como um grupo de isometrias hiperb´olicas complexas), este trabalho descreve completamente a estrutura combinat´oria do dom´ınio fundamental de Ford (identifica¸c˜oes de faces e ciclos de aretas) em fun¸c˜ao do n´umero de Euler do fibrado quociente.
References
[1] W.M. Goldman, Complex Hyperbolic Geometry, Oxford University Press, 1999. [2] P. Scott, The geometries of 3-manifolds, Bull. London Math Soc., 15, 1983, 401-487. Seme Gebara Neto
UFMG-MG
THE UNIVERSAL FUNCTORIAL EQUIVARIANT LEFSCHETZ
INVARIANT
JULIA WEBER
Abstract. The Lefschetz number, an integer associated to an endomorphism
f of a topological space X, is an important classical invariant in algebraic
topology. If L(f ) is non-zero, the endomorphism has a fixed point. There are several generalizations of the Lefschetz number which give more precise fixed point information.
We are interested in the case where the space X has an action of a discrete group G and the endomorphism f is equivariant. We construct the universal functorial equivariant Lefschetz invariant using K0of a certain endomorphism
category. We then derive results about fixed points of equivariant endomor-phisms of cocompact proper smooth G-manifolds.
Julia Weber
Max-Planck-Institut fuer Reine Mathematik-Alemanha E-mail: jweber@mpim-bonn.mpg.de
EQUIVARIANT PATH FIELDS ON TOPOLOGICAL MANIFOLDS
L. BORSARI, F. CARDONA, P. WONG
Abstract. Let M be a closed orientable smooth manifold. A classical the-orem of H. Hopf [4] states that M admits a non-singular vector field if and only if the Euler characteristic χ(M ) of M is zero. R. Brown [4] gave a gen-eralization of Hopf’s theorem for topological manifolds, by replacing vector fields with path fields, a concept first introduced by J. Nash [8]. In [4], R. Brown showed that a compact topological manifold M admits a non-singular path field if and only if χ(M ) = 0. Subsequently, R. Brown and E. Fadell [2] extended [4] to topological manifolds with boundary.
The existence of a path field allows one to show the so-called Complete
Invariance Property (CIP) (see [9] and [5]). Recall that a topological space M is said to have the CIP if for any non-empty closed subset A ⊂ M , there
exists a map f : M → M such that A = F ixf := {x ∈ M |f (x) = x}. Similarly, M possesses the CIP with respect to deformation (denoted by CIPD) if f is homotopic to the identity 1M. The non-singular path field problem is
equivalent to the fixed point free deformation problem. That is, M admits a non-singular path field if and only if 1M is homotopic to a fixed point free
map.
In [6], [7], and [10], equivariant vector fields on compact smooth G-manifolds were studied. In particular, an equivariant analog of Hopf’s theorem was proved in [6] and in [10]. Furthermore, an equivariant analog of path fields on Wecken complexes [3] was given in [11], and necessary and sufficient condi-tions for equivariant CIPD were given for smooth G-manifolds. Similar to the non-equivariant case, the equivariant non-singular path field problem is closely related to finding an equivariant fixed point free deformation.
The objective of our work is to obtain an equivariant analog of Brown’s theorem [4] for topological manifolds with locally smooth action of a finite group G. Moreover, we intend to extend the necessary and sufficient conditions for G-CIPD found in [12] to this category of G-manifolds.
References
[1] R. Brown, Path fields on manifolds, Trans. A.M.S. 118 (1965), 180–191.
[2] R. Brown and E. Fadell, Nonsingular path fields on compact topological manifolds, Proc.
A.M.S. 16 (1965), 1342–1349.
[3] E. Fadell, A remark on simple path fields in polyhedra of characteristic zero, Rocky Mountain
J. Math. 4 (1974), 65–68.
[4] H. Hopf, Vektorfelder in n-dimensionalen Mannigfaltigkeiten, Math. Ann. 96 (1927), 225–250. [5] B. Jiang and H. Schirmer, Fixed point sets of continuous selfmaps on polyhedra, in “Fixed Point Theory” (E. Fadell and G. Fournier, eds.), Lect. Notes Math. 886, Springer-Verlag, New York, 1981, 171–177.
[6] K. Komiya, A necessary and sufficient condition for the existence of non-singular G-vector fields on G-manifolds, Osaka J. Math. 13 (1976), 537–546.
[7] K. Komiya, G-manifolds and G-vector fields with isolated zeros, Proc. Japan Acad. Ser. A
Math. Sci. 54 (1978), 124–127.
[8] J. Nash, A path space and the Stiefel-Whitney classes, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 320–321.
[9] H. Schirmer, Fixed point sets of continuous selfmaps, in “Fixed Point Theory” (E. Fadell and G. Fournier, eds.), Lect. Notes Math. 886, Springer-Verlag, New York, 1981, 417–428. [10] D. Wilczy´nski, Fixed point free equivariant homotopy classes Fund. Math. 123 (1984), 47–60. [11] P. Wong, Equivariant path fields on G-complexes, Rocky Mountain. J. Math. 22 (1992),
1139–1145.
JIANG TYPE SPACES FOR COINCIDENCES
DANIEL VENDR ´
USCOLO AND PETER WONG
Abstract. In recent works one can find results about Jiang-type spaces for fixed point theory and for coincidence theory. We intend to present some of these results and, more specifically, we will talk about Jiang-type properties on homogeneous spaces and on suitable manifolds. Finally we will prove that orientable cosets space of a compact connected Lie group by a closed subgroup (not necessarily finite) is a Jiang-type space for coincidences (in codimension zero).
References
[1] Gon¸calves, Daciberg L.; Wong, Peter, Homogeneous spaces in coincidence theory, In:
Pro-ceedings of the X Encontro Brasileiro de Topologia (S˜ao Carlos, Brasil), (P. Schweitzer, ed.),
Sociedade Brasileira de Matem´atica, Matem´atica Contemporˆanea, 13 (1997), 143–158. [2] Gon¸calves, Daciberg L.; Wong, Peter, Nilmanifolds are Jiang-type spaces for coincidences,
Forum Math. 13 (2001), 133–141.
[3] Wong, Peter, Fixed-point theory for homogeneous spaces, Amer. J. Math. 120(1998), 23–42. [4] Wong, Peter, Coincidence of maps in homogeneous spaces, Manuscripta Math. 98(1999),
243–254.
[5] Wong, Peter, Fixed-point theory for homogeneous spaces, II, Fundamenta Math. 186(2005), 161–175.
Daniel Vendr´uscolo DM-UFSCar
E-mail: daniel@dm.ufscar.br
Peter Wong Bates College
GENERALIZATIONS OF THE IMPLICIT FUNCTION AND
DARBOUX THEOREMS
EDIVALDO L. DOS SANTOS, CARLOS BIASI AND CARLOS GUTIERREZ
Abstract. In [1, 2], we proved a homological version of the implicit function theorem for general topological spaces, which establish the existence of a map given implicitly. However, such map was continuous at only a point. In this work, we show that under certain hypotheses, it is possible to guarantee its continuity at every point. Let us consider X a locally pathwise connected Hausdorff space and let Y, Z oriented connected generalized manifolds of di-mension n. For a continuous map f : X ×Y → Z, denote by z0= f (x0, y0) ∈ Z
and define for each x ∈ X the map fx: Y → Z given by fx(y) = f (x, y). In
this conditions, one has the following
Theorem: Suppose that fxis an open and discrete map, for each x in X. If
|deg(fx0, y0)| = 1, then there exist a neighborhood V of x0 and a continuous
function g : V → Y such that f (x, g(x)) = z0, for each x ∈ V .
Moreover, we proved new versions of the classical Darboux theorem by using the notion of “degree”.
References
[1] Biasi, C., Dos Santos, E.L., A homological version of the implicit function theorem, Semigroup Forum, (72), 353-361, 2006.
[2] Dos Santos, E.L., Sobre teoremas de fun¸c˜oes impl´ıcitas, abertas e suas aplica¸c˜oes, Tese de
Doutorado, ICMC - USP, 2005.
[3] Biasi, C., Gutierrez, C., Finite branched coverings in generalized inverse mapping theorem, pr´e-print.
[4] G.E.Bredon, Sheaf Theory. Springer, 170(1967).
[5] ˘Cernavskii, A. V., Finite-to-one open mappings of manifolds, Mat. Sbornik, (65), 357-369,
1964.
[6] ˘Cernavskii, A. V., Addendum to the paper “Finite-to-one open mappings of manifolds”, Mat.
Sbornik, (66), 471-472, 1965.
[7] J. Bryant, S. Ferry, W. Mio and S. Weinberger, Topology of homology manifolds, Ann. of Math., 143 (1996), 435-467.
[8] V¨ais¨al¨a, J., Discrete open mappings on manifolds, Annales Acad. Scie. Fenn. Ser. A, I n.0,
n. 58392, 10 pgs, 1966. Edivaldo L. dos Santos
UNESP-IBILCE-S˜ao Jos´e do Rio Preto E-mail: elsantos@ibilce.unesp.br Carlos Biasi ICMC-USP-S˜ao Carlos E-mail: biasi@icmc.usp.br Carlos Gutierrez ICMC-USP-S˜ao Carlos E-mail: gutp@icmc.usp.br
CAMPOS DE VETORES TANGENTES A FOLHEAC
¸ ˜
OES
LUCIANA F.M. BRITO E FARID TARI
Abstract. Consideremos o par (ω, X), onde ω ´e um germe de uma 1-forma integr´avel e X um germe de um campo de vetores tangente `as folhas da fo-lhea¸c˜ao induzida por ω, ambos definidos em uma vizinhan¸ca da origem de Rn.
Nosso trabalho trata do estudo da estabilidade estrutural local de tais pares. O caso particular em que ω = df , com f uma fun¸c˜ao com uma singularidade de Morse, foi estudado por G. A. Lara, em [4]. Mostramos que o trabalho de Lara se generaliza para pares (ω, X) quando ω possui uma singularidade n˜ao degenerada na origem, e provamos tamb´em um resultado para germes de pares anal´ıticos. Mais precisamente, se um par anal´ıtico (ω, X) ´e localmente estruturalmente est´avel ent˜ao ω ´e localmente estruturalmente est´avel. Ainda neste caso, caracterizamos todos os campos de vetores X tangentes `a ω em termos de ω. Este resultado ´e usado para mostrar que n˜ao existem pares estruturalmente est´aveis na categoria suave, quando ω ´e anal´ıtica e tem uma singularidade n˜ao-degenerada de ´ındice 2 ou n − 2.
References
[1] J. W. Bruce, P. J. Giblin and F. Tari, Families of surfaces: height functions and projections to plane. Math. Scand. 82 (1998), no. 2, 165–185.
[2] C. Camacho and A. Lins Neto, Geometric theory of foliations. Birkhuser, Boston, Mas-sachusetts, 1985.
[3] S-N. Chow, C. Li and D. Wang, Normal forms and bifurcation of planar vector fields. Cam-bridge University Press, CamCam-bridge, 1994.
[4] G. A. Lara Luna, Estudo local dos campos vetoriais com uma integral primeira de Morse. Doctoral thesis, IMPA, Rio de Janeiro, 1978.
[5] H. Matsumura, Commutative ring theory. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1986.
[6] A. S. de Medeiros, Topological stability of integrable differential forms. Springer Lectures
Notes 597 (1977), 197–225.
[7] R. Moussu, Existence d’int´egrales premi`eres pour un germe de forme de Pfaff C∞, non
d´eg´en´er´e. Bol. Soc. Bras. Mat. 7 (1976), no. 2, 111–120.
[8] G. Reeb, Sur certain propri´et´es topologiques des vari´et´es feuillet´ees. Publ. Inst. Math. Univ.
Strasbourg 11, 5–89, 155–156. Actualit´es Sci. Ind., no. 1183 Hermann & Cie., Paris, 1952.
[9] J. Sotomayor, Generic one-parameter families of vector fields on two-dimensional manifolds.
Bull. Amer. Math. Soc. 74 (1968), 722–726.
[10] M. Zhitomirskii, Singularities of foliations and vector fields. Lecture notes based on a course given ICTP-Trieste, 2003.
[11] M. Zhitomirskii, Typical singularities of differential 1-forms and Pfaffian equations.
Trans-lations of Mathematical Monographs, 113. AMS, 1992.
Luciana F. M. Brito
IBILCE-UNESP-S˜ao Jos´e do Rio Preto E-mail: lfmbrito@ibilce.unesp.br
Farid Tari
Durham University, United Kingdom E-mail: farid.tari@durham.ac.uk
QUATERNIONIC LINE BUNDLES OVER QUATERNIONIC
PROJECTIVE SPACES
DACIBERG L. GONC
¸ ALVES AND MAURO SPREAFICO
Abstract. We study the problem of classifying the S3-principal bundles over
the quaternionic projective spaces HPn. Using an homotopy spectral sequence
constructed by N. Iwase, K. Maruyama and S. Oka, and information on the homotopy groups of the spheres, we solve the problem for n ≤ 3. Moreover, for higher dimensions, we show that the problem only depends on the parity of the degree of the maps classifying the bundles.
Daciberg Lima Gonc¸alves IME-USP-SP
E-mail: dlgoncal@ime.usp.br
Mauro Spre´afico ICMC-USP-SC
H-COBORDISM AND EXOTIC R
4CELSO MELCHIADES DORIA
Abstract. In 1961, Stephen Smale announced his proof of the Poincar´e Con-jecture in dimensions greater or equal to 5. He had the clever idea of proving that a handle decomposition of a h-cobordism between manifolds of dimension greater or equal to 5 can be made, after cancelling pairs of handles, diffeo-morphic to the handle decomposition of the product cobordism. However, his techniques fail when they are restricted to a h-cobordism beteween 4-manifolds. So, the smooth h-cobordism theorem proved by Smale is no longer true in dimension 4 and the Smooth Poincar´e Conjecture is, until now, an open question.
Since the 80’s, it has been known examples of closed 4-manifolds admit-ting a infinte number of smooth structures and also the astonishing fact that there are exotic R4, i.e., manifolds homeomorphic but non-diffeomorphic to
the standard 4-dimensional euclidean space R4= {(x
1, x2, x3, x4| xi, i = 1, 2, 3, 4}.
Let W5 be a smooth 5-dim h-cobordism between two simply connected,
closed and smooth 4-manifolds X0and X1. There are two theorems concerning
the understanding of W5 topological structure;
Theorem A: Then there exists a sub-cobordism V5⊂ W5between V 0⊂ X0
and V1⊂ X1with the following properties:
(1) V is a compact contractible 5-manifold, henceforth are V0 and V1.
(2) the h-cobordism W restricted to W − int(V ) is a product, i.e., it is diffeomorphic to (X0− int(V )) × [0, 1].
(3) V , V0× I and V1× I are, each one, diffeomorphic to D5.
(4) V0is diffeomorphic to V1by a diffeomorphism which, restricted to ∂V0=
∂V1, is an involution.
(5) W − V and W − Vi, i = 0, 1, is simply connected.
Theorem B: There is an open subset U ⊂ W , homeomorphic to I × R4, and a
compact subset K ⊂ U such that the pair (W − K, U − K) is diffeomorphic to a product I × (X0− K, (U ∩ X0) − K). The subsets Ri= U ∩ Xi, i = 0, 1, are
diffeomorphic to open subsets of R4. If X
0and X1are not diffeomorphic, then
there is no smooth 4-ball in Ri, i = 0, 1 containing the compact sets K ∩ Ri,
i = 0, 1, and, hence, both Riare exotic R4.
The theorem A was first announced by Curtis-Hsiang which efforts were improved in a join paper [4] with Freedman and Stong. The theorem A have also been proved by Matveyev [3] and Bi˜zaca. The present exposition follows the lines written by Kirby in [5] and R.Gompf in [1].
References
[1] R. GOMPF, A. STIPSICZ - 4-Manifolds and Kirby Calculus, Graduate Studies in Mathe-matics, vol 20, AMS.
[2] A. SCORPAN - The Wild World of 4-Manifolds, AMS.
[3] MATVEYEV, R. - A Decomposition of Smooth Simply Connected h-Cobordant 4-Manifolds, J. Diff. Geometry, 44 (1996), 267-277.
[4] C. CURTIS, M. FREEDMAN, W. C. HSIANG, R. STONG - A Decomposition Theorem for
h-Cobordant Smooth Simply Connected Compact 4-Manifolds, Invent. Math., 123 (1996),
343-348.
[5] R. KIRBY - Akbulut Corks and h-Cobordism of Smooth, Simply Connected 4-Manifolds, Turkish J. Math., 20 (1996), 85-95.
MODELOS DE SUPERF´ICIES N ˜
AO ORIENT ´
AVEIS NO ESPAC
¸ O
EUCLIDIANO TRIDIMENSIONAL
TON MARAR
Abstract. Uma leitura das p´aginas 52 e 53 do livro Algebraic Topology, an introduction de W. Massey, acess´ıvel a alunos de gradua¸c˜ao.
References
[1] Massey, W.; Algebraic Topology, an introduction, Springer, 1967. [2] Ap´ery, F.; Models of the real projective plane, Vieweg, 1987. Ton Marar
ICMC-USP-SC
A NOTE ON THE THEOREMS OF LUSTERNIK-SCHNIRELMANN
AND BORSUK-ULAM
T. E. BARROS AND C. BIASI
Abstract. Let X be a simply connected CW-complex which supports a free Zpaction (p a prime number) generated by fp: X → X. Given l = (l1, . . . , ln) ∈
Znsuch that for each j = 1, 2, . . . n, p does not divide l
j, consider the free Zp
action on S2n−1 generated by α
(p,l): S2n−1→ S2n−1, αp(z1, z2, . . . , zn) =
(e2πil1p · z1, e2πil2p · z2, . . . , e2πilnp · zn) (i2= −1). Our main theorem is the
following
Theorem 1: Suppose that for each 2 ≤ q < m = 2n − 1 i) πq(X) = p · πq(X), if q is odd, and
ii) πq(X) does not have elements of order p, if q is even.
then there exists an equivariant map F : (Sm, α
(p,l)) → (X, fp).
If p = 2, theorem 1 remains valid for any paracompact, Hausdorff, locally path connected space X and for any m odd or even (α2 is the antipodal map
in any case).
This theorem provides the following versions of Lusternik-Schnirelmann and Borsuk-Ulam theorems.
Corollary: Let X, p and m as in theorem 1. Then for each family F={C0, . . . Ck}
of k + 1 closed sets covering X such that (1) p = 2 and k ≤ m or
(2) p = 3, m is odd and k ≤ m + 1 or
(3) p > 3, m is odd and³p−12 ´(k − 2) + 2 ≤ m
there exists j0∈ {0, 1, . . . , k} such that fp(Cj0) ∩ Cj06= ∅.
The case (1) remains true if X is a paracompact, Hausdorff, path connected and locally path connected space.
Corollary: Let X, p and m satisfying the same conditions of theorem 1. i) If m ≥ (k − 1)(p − 1) + 1, then for each continuous map f : X → Rk
there exists x ∈ X such that f (x) = f ◦ fp(x).
ii) If m ≥ k(p − 1), then for each continuous map f : X → Rk there exists
x ∈ X such that f (x) = f ◦ (fp)j(x) ∀ 1 ≤ j ≤ p − 1.
Corollary: If (X, ι) is a paracompact, Hausdorff, locally path connected and simply connected space with an involution ι satisfying the hypotheses of theorem 1 and Y is a separable metric space with topological dimension, dim(Y ) ≤ m−12 then for any map f : X → Y , there exists x ∈ X such that
f (x) = f ◦ ι(x).
References
[1] Aarts, J. M., Fokkink, R. J., Vermeer, H. Variations on a Theorem of Lusternik and
Schnirelmann, Topology, 35(4)(1996), 1051-1056.
[2] Aarts, J. M. and Nishiura, T. Dimension and Extension, North Holland, Amsterdam, (1992). [3] Biasi, C. and Libardi, A. K. M., Some applications of obstruction theory, Notas do Instituto
de Ciˆencias Exatas de S˜ao Carlos - USP (1988).
[4] Cohen, F. and Connet, J. E. A coincidence theorem related to the Borsuk-Ulam theorem, Proc. Amer. math. Soc., vol. 44(1)(1974), 218-220.
[5] Izydorek, M. and Jaworowski, J., Antipodal coincidence for maps of spheres into complexes, Proc. Amer. Math. Soc. 123(1995), 1947-1950.
[6] Mattos, D., Santos, E. L. and Pergher, P. L. Q., A Borsuk-Ulam theorem for general spaces, Arch. Math. (Basel) 81(2003), 96-102.
[7] Munkholm, Hans J. Borsuk-Ulam type theorems for proper Zp-actions on (mod p homology)
n-spheres, Math. Scand. 24(1969), 167-185.
[8] Olum, P. Obstructions to extensions and homotopies, Ann. of Math. 52(2)(1950), 1-50. [9] Santos, E. E. F. Colora¸c˜ao de fun¸c˜oes em espa¸cos de dimens˜ao topol´ogica finita, Master
Thesis, (2001) (in Portuguese).
[10] Lusk, E. L. The mod p Smith index and a generalized Borsuk-Ulam theorem, Michigan Math. J. 22(1975), 151-160.
[11] Steinlein, H. Some abstract generalizations of the Ljusternik-Schnirelmann-Borsuk covering
theorem, Pacific J. Math. 83(1979), 285-296.
[12] Steinlein, H. On the theorems of Borsuk-Ulam and Ljusternik-Schnirelmann-Borsuk, Canad. Math. Bull. vol. 27(2)(1984), 192-204.
T. E. Barros DM-UFSCar E-mail: dteb@dm.ufscar.br C. Biasi ICMC-USP-SC E-mail: biasi@icmc.sc.usp.br
DIFEOMORFISMOS E CAMPOS VETORIAIS. AXIOMA A SEM
POC
¸ OS E SEM FONTES
REGIS SOARES E ENOCH APAZA
Abstract. Um resultado provado por Plykin afirma que todo difeomorfismo axioma A definido na esfera S2possui um po¸co ou uma fonte. Uma quest˜ao
natural ´e se esta propriedade ainda ´e verdadeira para sistemas axioma A em dimens˜oes maiores. Mostraremos que toda variedade fechada de dimens˜ao
n ≥ 3 suporta um sistema axioma A sem po¸cos nem fontes.
References
[1] Bing, R.H.; Martin, J.M. Cubes with knotted holes. Trans. Amer. Math. Soc. 155 1971 217– 231.
[2] Birman, J.S.; Williams, R.F. Knotted Periodic Orbits in Dynamical System II: Knot Holders
for Fibered knots Contemporary Math. 20 (1983), 1–60.
[3] Bowen, R.; Franks, J. The periodic points of maps of the disk and the interval. Topology 15 (1976), no. 4, 337–342.
[4] Franks, J.; Young, L. S. A C2 Kupka-Smale diffeomorphism of the disk with no sources or
sinks. Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), pp. 90–98,
Lecture Notes in Math., 898, Springer, Berlin-New York, 1981.
[5] Gambaudo, J.-M.; van Strien, S.; Tresser, C. H´enon-like maps with strange attractors: there
exist C∞Kupka-Smale diffeomorphisms on S2 with neither sinks nor sources. Nonlinearity
2 (1989), no. 2, 287–304.
[6] Ghrist, R.W.; Holmes, P.; Sullivan, M. Knots and Links in three-Dimensional Flows Lecture Notes in Mathematics, Vol 1654, Springer-Verlag Berlin, 1997.
[7] Harrison, J.; Yorke, James A.Flows on S3 and R3 without periodic orbits. Geometric
dy-namics (Rio de Janeiro, 1981), 401–407, Lecture Notes in Math., 1007, Springer, Berlin, 1983.
[8] Hayashi, Shuhei Connecting invariant manifolds and the solution of the C1 stability and
Ω-stability conjectures for flows. Ann. of Math. (2) 145 (1997), no. 1, 81–137. [9] Ma˜n´e, R. An ergodic closing lemma. Ann. of Math. (2) 116 (1982), no. 3, 503–540.
[10] Morales, C. A.; Pacifico, M. J. A dichotomy for three-dimensional vector fields.Ergodic The-ory Dynam. Systems 23 (2003), no. 5, 1575–1600.
[11] Palis, J. A note on Ω-stability. 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) pp. 221–222 Amer. Math. Soc., Providence, R.I.
[12] Palis, J. Ω-explosions. Proc. Amer. Math. Soc. 27 1971 85–90.
[13] Palis, J. On the C1Ω-stability conjecture. Inst. Hautes ´Etudes Sci. Publ. Math. No. 66 (1988),
211–215.
[14] Plante, J. F.; Thurston, W. P.Anosov flows and the fundamental group. Topology 11 (1972), 147–150.
[15] Plykin, R. V. Sources and sinks of A-diffeomorphisms of surfaces. (Russian) Mat. Sb. (N.S.) 94(136) (1974), 243–264, 336.
[16] Pugh, C.; Shub, M. The Ω-stability theorem for flows. Invent. Math. 11 1970 150–158. [17] Rolfsen, D. Knots and Links. Publish or Perish Berkely CA, 1977.
[18] Smale, S. Dynamical systems and the topological conjugacy problem for diffeomorphisms. 1963 Proc. Internat. Congr. Mathematicians (Stockholm, 1962) pp. 49–496 Inst. Mittag-Leffler, Djursholm
[19] Smale, S. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 1967 747–817. [20] Smale, S. The Ω-stability theorem. 1970 Global Analysis (Proc. Sympos. Pure Math., Vol.
XIV, Berkeley, Calif., 1968) pp. 289–297 Amer. Math. Soc., Providence, r.I.
[21] Thurston, W. The Geometry and Topology of 3-manifolds Lecture Notes, Princeton Univer-sity.
HOMOTOPY AND COHOMOLOGY OF SPACES OF CURVES OF
BOUNDED CURVATURE IN THE SPHERE
NICOLAU SALDANHA
Abstract. Let Q in SO(3) be an orthogonal matrix and k a real number. Let
XQ,k be the space of smooth regular curves γ : [0, 1] → S2 satisfying γ(0) =
e1, γ0(0) = e2, γ(1) = Qe1, γ0(1) = Qe2 and such that the geodesic curvature
of γ is greater than k for all t in the interval [0,1]. The spaces XQ,k have
a surprisingly rich topological structure: in this talk we shall present several partial results concerning the homotopy and cohomology of these spaces.
This work can be considered continuation of the preprint “Homotopy and cohomology of spaces of locally convex curves in the sphere” and involves collaboration with Boris Shapiro.
Nicolau Saldanha PUC-RJ
SPAN OF PROJECTIVE STIEFEL MANIFOLDS
PETER ZVENGROWSKI
Abstract. In this talk we shall give an introduction to the general topic of the span of a smooth manifold, and then concentrate on the family known as projective Stiefel manifolds. Using results of N. Barufatti and D. Hacon which describe the complex K-theory of these spaces, the author together with P. Sankaran give a method of finding upper bounds for the span, which depends on number theoretic considerations such as quadratic residues. Further results, mostly in work with J. Korbas, will also be described.
Peter Zvengrowski
Calgary University-Canad´a E-mail: zvengrow@ucalgary.ca
A TOPOLOGIA DO UNIVERSO
WLADIMIR SEIXAS
Abstract. A descri¸c˜ao f´ısica do universo utiliza como modelo matem´atico o conceito de variedade diferenci´avel. A geometria do universo ´e descrita intr´ınseca e localmente em termos da distribui¸c˜ao de massa. Os matem´aticos caracterizam variedades em termos de sua geometria e topologia. A topologia caracteriza globalmente o espa¸co. Quest˜oes que descrevem globalmente o uni-verso podem ser colocadas, como por exemplo, o uniuni-verso ´e finito ou infinito? ´
E fechado ou aberto? Neste semin´ario discutiremos os seguintes t´opicos: (1) O que ´e um modelo matem´atico. (2) A formula¸c˜ao matem´atica da relatividade geral. (3) Como os conceitos de topologia aparecem na descri¸c˜ao do universo e quais as poss´ıveis geometrias do universo. (4) ´E poss´ıvel medir ou observar a topologia do universo?
References
[1] R.Geroch and G.T.Horowitz, Global structure of spacetimes, General Relativity: An Einstein centenary survey, ed. by Hawking and Israel, Cambridge Univ. Press, 1979.
[2] Roger Penrose, Techniques of Differential Topology in Relativity, Society for Industrial & Applied Mathematics, 1972.
[3] Jean-Pierre Luminet , Glenn D. Starkman and Jeffrey R. Weeks, Is Space Finite? Scientific American, April 1999
Wladimir Seixas IGCE-UNESP-Rio Claro E-mail: seixas@rc.unesp.br
INTERSEC
¸ ˜
AO DE BISSETORES SIM´
ETRICOS EM RELAC
¸ ˜
AO A
UM R-PLANO NO ESPAC
¸ O HIPERB ´
OLICO COMPLEXO
FRANCISCO DUTENHEFNER
Abstract. Na literatura existem poucos resultados publicados a respeito da caracteriza¸c˜ao da interse¸c˜ao de dois bissetores no espa¸co hiperb´olico complexo de dimens˜ao dois. Nesse trabalho, apresentaremos uma caracteriza¸c˜ao de tal interse¸c˜ao supondo que exista um R-c´ırculo cuja invers˜ao permuta os dois bissetores em quest˜ao. Demonstraremos que se B1e B2s˜ao bissetores e R ´e um
R-c´ırculo no espa¸co hiperb´olico complexo de dimens˜ao dois, tal que a invers˜ao em R permuta B1e B2, ent˜ao B1e B2s˜ao bissetores disjuntos se, e somente se,
R e B1s˜ao disjuntos. Mais ainda, se π1´e a proje¸c˜ao ortogonal sobre a espinha
complexa Σ1de B1e se σ1denota a espinha real de B1, demonstraremos que
π1(R) ´e uma elipse contida em Σ1 e que R e B1 s˜ao disjuntos se, e somente
se, a elipse π1(R) ´e disjunta da circunferˆencia σ1. Isso reduz a dificuldade da
an´alise da posi¸c˜ao relativa de duas variedades de dimens˜ao trˆes num espa¸co de dimens˜ao quatro, para a an´alise da posi¸c˜ao relativa de duas curvas planas: uma elipse e uma circunferˆencia.
References
[1] G.D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pac. J. Math. 86 (1980), 171-276.
[2] A.F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, 91, Springer-Verlag, New York, 1983.
[3] B. Maskit, Kleinian Groups, Grundlehren der mathematishen Wiβenschaften 287, Springer-Verlag, New York, 1988.
[4] W.M. Goldman, Complex hyperbolic Kleinian groups, Complex geometry (Osaka, 1990), 31-52, Lecture Notes in Pure and Appl. Math., 143, Dekker, New York, 1993.
Francisco Dutenhefner UFMG-MG
ON COHOMOLOGICALLY RIGID DIFFEOMORPHISMS OF TORI
NATHAN MOREIRA DOS SANTOS
Abstract. A diffeomorphism f of a manifold M is cohomologically rigid (C.R) if every smooth function is cohomologous to a constant i.e the cohomological equation
h − h ◦ f = g − c
has a smooth solution h.
Anatole Katok proposes in ([1], Problem 17) the following question: for a given manifold M what are the C.R diffeomorphisms? R.U. Luz and N.M. dos Santos proved in ([2]): the C.R diffeomorphisms of the tori Tn, n < 4
are smoothly conjugate to Diophantine translations. It is also true for n = 4 provided that the diffeomorphism preserves orientation.
A.Katok has claimed that all the eigenvalues of the mapping induced on
H1(M, R) by a C.R diffeomorphisms are roots of unity. Unfortunately A.Katok
did not find a proof for his claim and we have used this in our article [2]. The
aim our talk is to give a proof for Katok’s claim when the manifold is a torus Tn, n < 5. For n > 5 it remains open as a difficult problem.
References
[1] S. Hurder. Problems on rigidity of group actions and cocycles. Ergod. Th. & Dynam. Sys. 5 (1985), 473-484.
[2] R.U. Luz and N.M. dos Santos. Cohomology-free diffeomorphisms of low-dimension tori. Ergod. Th. & Dynam. Sys. 18 (1998), 985-1006.
Nathan Moreira dos Santos UFF-Niter´oi-RJ
ESPAC
¸ OS DE CONFIGURAC
¸ ˜
OES OCHA (OPEN-CLOSED
HOMOTOPY ALGEBRAS)
EDUARDO HOEFEL
Abstract. Open-Closed Homotopy Algebras (OCHA) are a new kind of SH (Strongly Homotopic) algebraic structure introduced by Kajiura and Stasheff in 2004. In this work we show how OCHA appears on the stratification of a certain manifold with corners. That manifold is the Fulton MacPherson (FM) compactification of the configuration space of points on the closed upper half-plane, with points both on the interior and on the real line. The FM compactification will be denoted C(p, q), where p is the number of interior points and q is the number of points on the real line. We first observe that OCHA are algebras over a certain operad and, second, that the operad is given by the first row of the spectral sequence of C(p, q), viewed as a manifold with corners.
Eduardo Hoefel DM-UFSCar
PROBLEMA DE BURNSIDE NO GRUPO DE
SIMPLECTOMORFISMOS DE UMA VARIEDADE M -SY M P (M )
ANA L ´
UCIA DA SILVA
Abstract. Este trabalho ´e um an´alogo n˜ao linear do Teorema de Schur que afirma que um subgrupo finitamente gerado de um grupo linear, cujos elemen-tos s˜ao todos de ordem finita ´e finito.
Defini¸c˜ao: Chamaremos Grupo de Burnside, todo grupo de tor¸c˜ao finita-mente gerado que seja finito.
Teorema A: Seja M uma 4-variedade compacta e simpl´etica, denote Symp(M ) o grupo de simplectomorfismos de M (classe C2). Suponha que a classe
fun-damental em H4(M, Z) seja um produto de classes em H1(M, Z). Ent˜ao todo
subgrupo G ⊂ Symp(M ) finitamente gerado cujos elementos s˜ao todos de ordem finita, ´e, de fato, finito.
Ana Lucia da Silva UEL-Londrina
LOWER CENTRAL AND DERIVED SERIES OF SURFACE BRAID
GROUPS
JOHN GUASCHI
Abstract. We study the lower central and derived series of the m-string braid group Bm(M ) of a compact, orientable surface M . In most cases, we are able
to show that if m ≥ 3 (resp. m ≥ 5) then the lower central series (resp. derived series) of Bm(M ) is eventually constant, and thus Bm(M ) is not residually
nilpotent. If m = 2, the commutator structure of B2(M ) is somewhat
com-plicated, but if M is either the annulus or the torus, we are able to determine both the lower central series of B2(M ) and its successive quotient groups. This
constitutes joint work on the one hand with D.L.Gon¸calves (USP), and on the other with P.Bellingeri (Nantes, France) and S.Gervais (Nantes, France). John Guaschi
Universit´e Paul Sabatier-Toulouse-Franc¸a E-mail: guaschi@picard.ups-tlse.fr
PONTOS FIXOS EM FIBRADOS SOBRE O CIRCULO E FIBRA
GARRAFA DE KLEIN
D.L. GONC
¸ ALVES, D. PENTEADO AND J.P. VIEIRA
Abstract. O objetivo deste trabalho e classificar , em termos das induzidas no grupo fundamental, as fun¸c˜oes fibradas sobre o c´ırculo S1 com fibra garrafa de Klein, que s˜ao ou n˜ao livres de pontos fixos por homotopias que preservam a fibra. O problema e equivalente a um problema de levantamento geom´etrico numa fibra¸c˜ao te´orica (ver [2]). Como o grupo fundamental da fibra te´orica ´e n˜ao comutativo, a teoria cl´assica de obstru¸c˜ao n˜ao se aplica. Por se tratarem de espa¸cos K(π; 1), a existˆencia do levantamento geom´etrico te´orico ´e equiv-alente a um levantamento alg´ebrico a n´ıvel de grupo fundamental (ver [1]), cuja existˆencia nos leva a discutir a solu¸c˜ao de um sistema de equa¸c˜oes num grupo livre de infinitos geradores n˜ao comutativo. A condi¸c˜ao do numero de Nielsen de tais fun¸c˜oes restrita a fibra ser zero, juntamente com a condi¸c˜ao de preservar a fibra reduzem muito tais fun¸c˜oes a n´ıvel de grupo fundamen-tal. Outras redu¸c˜oes s˜ao consideradas de modo a classificar os sistemas que possuem solu¸c˜ao, explicitando-as nestes casos. Nos casos em que o sistema n˜ao tem solu¸c˜ao, prova-se que o sistema n˜ao tem solu¸c˜ao na abelianiza¸c˜ao do grupo fundamental da fibra te´orica. Em ambos caso, as condi¸c˜oes s˜ao traduzi-das em termos traduzi-das dos grupos fundamentais envolvidos e traduzi-das induzitraduzi-das de tais fun¸c˜oes.
References
[1] H.J. Baues; Obstruction Theory- Lecture Notes, Lecture Notes , vol. 628, Springer-Verlag, 1977.
[2] E. Fadell and S. Husseini; A fixed point theory for fiber preserving maps , Lecture Notes in Mathematics, vol. 886, Springer Verlag, (1981), 49-72.
[3] E. Fadell and S. Husseini; The Nielsen number on surfaces, Contemporary Mathematics, vol. 21, Topological methods in non linear functional analysis (1982), .
[4] D. L. Gon¸calves, D. Penteado and J. P. Vieira; Fixed Points on Torus Fiber Bundles over the Circle, Fundamenta Mathematicae, vol.183(1)(2004), 1-38.
[5] B. Jiang; Fixed Points and Braids II, Math. Ann. 272 (1985), 249-256. Daciberg Lima Gonc¸alves
IME-USP-SP
E-mail: dlgoncal@ime.usp.br
Dirceu Penteado DM-UFSCar
E-mail: dirceu@dm.ufscar.br
Jo˜ao Peres Vieira IGCE-UNESP-Rio Claro E-mail: jpvieira@rc.unesp.br
PLANAR EMBEDDINGS WITH A GLOBALLY ATTRACTING
FIXED POINT
BEGO ˜
NA ALARC ´
ON COTILLAS
Abstract. We study the appearing of a globally attracting fixed point for planar embeddings using techniques of topological dynamic. The motivation of our work is the Discrete Markus-Yamabe Conjecture.
This subject of global attractor of a discrete dynamical system generated by a C1−map has been studied for many authors because the Markus-Yamabe
Conjecture (see [1]). The papers [3] and [4] develop a theory that provides us with the means for considering more conditions that make possible the study of the globally attracting fixed point for certain maps of class C1. Though
these maps are not necessarily polynomial, they somehow contain the results in [1]. Moreover we also use strongly the theorem of injectivity of C1−map in
[2] to guarantee the uniqueness of the fixed point of the map.
References
[1] A. Cima, A. Gasull, and F. Ma˜nosas. The discrete markus-yamabe problem. Nonlinear
Anal-ysis, 35:343-354, 1999.
[2] A. Fernandez, C. Gutierrez, and R. Rabanal. Global asymptotic stability for differentiable vector fields of R2. J. Differential Equations, 2004.
[3] P. Murthy. Periodic solutions of two-dimensional forced systems: The massera theorem and its extension. Journal of Dynamics and Differential Equations, 10(2):275–302, 1998. [4] F. L. Roux. Migration des points errants d’hom´eomorphisme de surface. C. R. Acad. Sci.
Paris, S´erie I(t. 330):225–230, 2000.
Bego˜na Alarc´on Cotillas
Departamento de Matem´atica Aplicada, Universidade de Valencia, Espanha E-mail: bego.alarcon@uv.es
COMPLEXIFICATION, (1,2)-SYMPLECTIZATION AND TRIPLE
OF ROOTS ON FLAGS
CAIO J.C. NEGREIROS
Abstract. In this talk we will discuss about a generating set for the second homotopy group of flag manifolds. This homotopy group consisting of a special class of maps minimizing area in their homotopy classes. These maps are usually called harmonic maps. It is similar to the Hodge Theorem which states that given M compact, Hp(M, R) is generating by harmonic p-forms.
We will in this talk discuss the possible candidates for this generating set and discuss some facts on the rich invariant Hermitian geometry of flags. Caio J. C. Negreiros
IMECC-UNICAMP-Campinas E-mail: caione@ime.unicamp.br
NUMERICAL CONTROL OF EQUISINGULARITY FOR MAP
GERMS FROM C
NTO C
3, N ≥ 4
V. H. JORGE P´
EREZ, E. C. RIZZIOLLI AND M. J. SAIA
Abstract. We give the minimal number of invariants that is sufficient for the Whitney equisingularity of a one parameter deformation of corank one finitely determined holomorphic germ f : Cn, 0 → C3, 0, with n ≥ 4. We see
from the work of Gaffney that the invariants needed to control the Whitney equisingularity are the 0-stable invariants and the polar multiplicities which appear in the stable types of a stable deformation of the germ. First describe all stable types which appear in these dimensions, then we use the relationship between the invariants in the stable types in the source and the target to reduce this number. V. H. Jorge P´erez ICMC-USP-SC E-mail: vhjperez@icmc.usp.br E. C. Rizziolli IGCE-UNESP-Rio Claro E-mail: eliris@rc.unesp.br M. J. Saia ICMC-USP-SC E-mail: mjsaia@icmc.usp.br
ON SIMPLIFYING SEIFERT SURFACES
PEDRO LOPES
Abstract. Given a knot, the characterization of one of its Seifert surfaces is a step towards the calculation of the Alexander invariant of this knot. A simplification of the Seifert surface could conceivably lead to a more straight-forward calculation of the invariant. We speculate on the possibility of listing a finite number of moves on Seifert surfaces that would allow us to achieve this goal. We support our ideas with calculations involving classes of Pretzel and Rational knots.
Pedro Lopes
IST, Portugal; IMPA, Brasil E-mail: pelopes@impa.br
FINITENESS OF THE REIDEMEISTER CLASS OF
AUTOMORPHISM OF GROUPS
DACIBERG LIMA GONC
¸ ALVES
Abstract. This is joint work with Peter Wong. We will presents several new example of group which have the property that all automorphisms have Reidemeister number infinity. They include some finitely generated nilpotent groups which in some sense it is a surprise. From this we can obtain a proof that free group on two generators have the R−∞ property (i.e. every automorphism has Reidemeister number infinite). Finally we will provide an example of a manifold which has the property that every homeomorphism can be deformed to a fixed point free map. A small survey presenting the state of art of this problem will be provided.
Daciberg Lima Gonc¸alves IME-USP-SP
OS TEOREMAS DAS PANQUECAS
ADRIANA CRISTIANE RUY
Abstract. Os Problemas de Panquecas constituem, na topologia, teoremas de existˆencia. O Teorema 1 enuncia que dadas duas panquecas, disjuntas ou n˜ao, ´e poss´ıvel dividi-las em duas metades de ´areas iguais, com apenas uma linha de corte. J´a o Teorema 2 enuncia que dada uma panqueca, podemos dividi-la em quatro partes de ´areas iguais, com apenas dois cortes perpendicu-lares. Por panquecas entendemos regi˜oes limitadas do plano euclidiano.
References
[1] C. Kosniowski; A First Course in Algebraic Topology, Cambridge [Eng.], New York: Cam-bridge University Press, 1980.
Adriana Cristiane Ruy DM-UFSCar
SOBRE UM TEOREMA DE SWARUP
ANDERSON PAI ˜
AO DOS SANTOS E ERM´INIA DE LOURDES CAMPELLO FANTI
Abstract. A teoria de ends de espa¸cos surgiu com Hopf e Freudenthal. In-tuitivamente, o n´umero de ends, e(X ), de um espa¸co X , mede de quantas maneiras X pode “ir para o infinito”. Posteriormente, motivados na teoria de ends de espa¸cos, conceitos como o n´umero de ends de um grupo G e de pares de grupos (G, H), onde G ´e um grupo e H um subgrupo de G, e mais geralmente, de pares (G, F), onde F ´e uma fam´ılia de subgrupos de G foram definidos, e tˆem sido objetos de estudos recentes ([8], [1], [2], [6], [4]).
O principal objetivo deste trabalho ´e apresentar a prova alg´ebrica do Teo-rema de Swarup, que sob certas hip´oteses, d´a uma f´ormula cohomol´ogica 1-dimensional para o invariante alg´ebrico e(G, H), definido por Scott ([8]) e Houghton ([5]), mais precisamente, considerando G um grupo finitamente gerado e H um subgrupo de ´ındice infinito em G, se e(G, N ) = 1, para todo
N C H, comH N ' Z, ent˜ao e(G, H) = 1 + rankZH1 ¡ G, I(Z)¢= 1 + rankZH1 ¡ G, Z(G/H)¢.
Para tanto, o conceito de subconjunto H-quase invariante de G e resulta-dos como a interpreta¸c˜ao do grupo de cohomologia H1(G, M ) em termos de
deriva¸c˜oes (`a direita), onde M ´e um ZG-m´odulo, e o Lema de Shapiro, s˜ao resultados imprescind´ıveis. Algumas rela¸c˜oes do invariante e(G, H) com ends de espa¸cos s˜ao tamb´em apresentadas.
References
[1] M.G.C. Andrade; E.L.C. Fanti; A Relative Cohomological Invariant for Group Pairs, Manuscripta Math., 83(1994), 1-18.
[2] M.G.C. Andrade; E.L.C. Fanti; J.A. Daccach; On Certain Relative Cohomological Invariants, International Journal of Pure and Applied Mathematics, 21(2005), 335-351.
[3] K.S. Brown; Cohomology of Groups, G. T. M. 87, New York, Springer Verlag, 1982. [4] R. Geoghegan; Topological Methods in Groups Theory. (to appear)
[5] C.H. Houghton; Ends of locally compact groups and their quotient spaces, J. Aust. Math. Soc., 17(1974), 274-284.
[6] P.H. Kropholler; M.A. Roller; Remarks on the theorem of Swarup on ends of pairs of groups, Journal of Pure and Applied Algebra, 109(1996), 107-110.
[7] A.P. Santos; Cohomologia de Grupos e Invariantes Alg´ebricos, Disserta¸c˜ao (Mestrado em Matem´atica) - IBILCE-UNESP, S˜ao Jos´e do Rio Preto, 2006.
[8] G.P. SCOTT; Ends of pairs of groups, Journal of Pure and Applied Algebra, 11(1977), 179-198.
[9] G.A. SWARUP; On the ends of pairs of groups, Journal of Pure and Applied Algebra, 87(1993), 93-96.
Anderson Pai˜ao dos Santos
IBILCE-UNESP-S˜ao Jos´e do Rio Preto E-mail: andersonpaiao@hotmail.com
Erm´ınia de Lourdes Campello Fanti IBILCE-UNESP-S˜ao Jos´e do Rio Preto E-mail: fanti@ibilce.unesp.br
ALGUNS TEOREMAS DO TIPO BORSUK-ULAM
CIBELE CRISTINA TRINCA E MARIA GORETE CARREIRA ANDRADE
Abstract. Este trabalho trata de estudos realizados no desenvolvimento de nossa disserta¸c˜ao de mestrado. Primeiramente enunciaremos o Teorema Cl´assico de Borsuk-Ulam para aplica¸c˜oes cont´ınuas f : Sn → Rn e o
demonstraremos para n ≤ 2. Seja Z2 = {1, t} o grupo c´ıclico de ordem 2.
Considere em (Rn+1− {0}) a Z
2-a¸c˜ao definida por 1.x = x e t.x = −x, para
x ∈ Rn+1− {0}. Como (Rn+1− {0}) tem o mesmo tipo de homotopia que
Sn, podemos enunciar o Teorema Cl´assico de Borsuk-Ulam da seguinte forma:
“Se f : (Rn+1− {0}) → Rn´e uma aplica¸c˜ao cont´ınua, ent˜ao alguma ´orbita da
Z2-a¸c˜ao em (Rn+1− {0}) ´e aplicada em um ´unico ponto de Rn”. Mais
geral-mente, estudaremos outros Teoremas do Tipo Borsuk-Ulam. Para isso con-sideraremos aplica¸c˜oes cont´ınuas f : (Cn+1− {0}) → Cn. Uma ra´ız primitiva
k-´esima da unidade ξ nos fornece uma Zk-a¸c˜ao livre sobre Cn. Um teorema nos
diz que a equa¸c˜ao
k
X
i=1
ξif (ξix) = 0 sempre tem uma solu¸c˜ao x ∈ Cn+1− {0}.
Com esse resultado vemos algumas situa¸c˜oes em que uma ´orbita da Zk-a¸c˜ao
em Cn+1− {0} ´e aplicada por f em um ´unico ponto.
References
[1] D. H. Gottilieb, The Lefschetz number and Borsuk-Ulam Theorems, Pacific Journal of Math-ematics, vol. 103, 1, (1982), 29-37.
[2] A. Hatcher, Algebraic Topology, Cambridge University Press,2002. Cibele Cristina Trinca
IBILCE-UNESP-S˜ao Jos´e do Rio Preto E-mail: cibtrinca@yahoo.com.br
Maria Gorete Carreira Andrade IBILCE-UNESP-S˜ao Jos´e do Rio Preto E-mail: gorete@ibilce.unesp.br
UMA GENERALIZAC
¸ ˜
AO DO TEOREMA DE LJUSTERNIK
-SCHNIRELMANN
EDUARDO S. PALMEIRA E TOMAS EDSON BARROS
Abstract. Consideremos o teorema cl´assico de Ljusternik-Schnirelmann: se-jam H1, H2, . . . , Hksubconjuntos fechados da esfera Sntais que ∪ki=1Hi= Sn
e Hi∩ (−Hi) = ∅ para i = 1, . . . , k. Ent˜ao k ≥ n + 2.
Nesse trabalho apresentamos uma vers˜ao generalizada, segundo H. Stein-lein, do teorema acima, usando o conceito de genus segundo A. S. ˇSvarc, onde substituimos a Snpor um espa¸co M normal e a aplica¸c˜ao antipodal por uma
fun¸c˜ao f : M −→ M que gera uma Zpa¸c˜ao livre sobre M, de forma que ainda
se tenha Hi∩ f (Hi) = ∅.
Al´em disso, exibimos uma estimativa para o genus de Lk,p, espa¸co esse
especialmente construido para majorar o genus de (M, f ).
References
[1] Palmeira, Eduardo S., Uma Generaliza¸c˜ao do Teorema de Ljusternik-Schnirelmann,
Dis-serta¸c˜ao de Mestrado, Departamento de Matem´atica-UFSCar, S˜ao Carlos, (2005).
[2] Steinlein, H., Some abstract generalizations of the Ljusternik-Schnirelmann-Borsuk covering
theorem, Pacific J. Math. 83 (1979), 285-296.
Eduardo S. Palmeira DM-UFSCar
E-mail: eduardo@dm.ufscar.br
Tomas Edson Barros DM-UFSCar
