IX Encontro Regional
de Topologia
Caderno de resumos
Abstracts
Palestras
Fixed points of selfmaps of 3-manifoldsPeter Wong, D. Gonçalves e X. Zhao
Equivariant functions and vector fields on manifolds with semi–free action of circle
Vladimir V. Sharko
A survey of the cohomological degree of equivariant maps Norbil Cordova, Denise de Mattos e Edivaldo dos Santos Os grupos de tranças do Toro e da Garrafa de Klein
Carolina de Miranda e Pereiro
Spherical Space Forms and the Borsuk-Ulam Theorem
Marjory Del Vecchio dos Santos, Edivaldo Lopes dos Santos e Mauro Flá-vio Spreafico
Aproximações da diagonal e anéis de cohomologia dos grupos fundamentais das superfícies
Sérgio Tadao Martins e Daciberg Lima Gonçalves Artin braid groups and crystallographic groups
Daciberg Gonçalves, John Guaschi e Oscar Ocampo
Free and properly discontinuously actions of groups on homotopy 2n-spheres Marek Golasiński (Toruń), Daciberg Lima Gonçalves (São Paulo) e Ro-lando Jiménez (Oaxaca)
A propriedade de Borsuk-Ulam para classes de homotopia de funções entre superfícies
Vinicius Casteluber Laass
Two-parameter trace and fixed points of fiber-preserving maps Weslem L. Silva
Z22-actions on KdP (2n) ∪ KeP (2m + 1)
Sergio Tsuyoshi Ura, Allan Edley Ramos de Andrade e Pedro Luiz Queiroz Pergher
Realization of Fixed Point Set in a Prescribed Equivariant Homotopy Class of Maps
Rafael Moreira de Souza e Peter Ngai-Sing Wong
Classificação de Z22-ações fixando KdP2m+1∪ KeP2n, d < e
Allan Edley Ramos de Andrade e Pedro Luiz Queiroz Pergher Cyclic and cocyclic maps and generalized Whitehead products
Thiago de Melo e Marek Golasiński
A special case of involution - a phenomenon of low codimension Patricia E. Desideri e Pedro L. Q. Pergher
Painéis
Sequências Exatas e o Lema dos Cinco Alex Melges Barbosa e João Peres Vieira
Preliminares Sobre Coincidências de Aplicações definidas em Varieda-des com Bordo
Alice Kimie Miwa Libardi, Daciberg Lima Gonçalves e Thais Fer-nanda Mendes Monis
Homological Invariants and Duality
Amanda Ferreira de Lima e Maria Gorete Carreira Andrade A generalization of the Borsuk-Ulam theorem
Ana Maria Mathias Morita e Maria Gorete Carreira Andrade Diagonal involution on products of spaces and the Borsuk-Ulam theo-rem
Anderson Paião dos Santos
A proof of the Jordan Separation Theorem
Bruno Caldeira Carlotti de Souza e Maria Gorete Carreira Andrade Grupo Fundamental e o Teorema de van Kampen
Givanildo Donizeti de Melo e Thiago de Melo
O Teorema Fundamental da Álgebra demonstrado através do grau de aplicações entre esferas
Guilherme Vituri Fernandes Pinto e Thiago de Melo
A Wecken type theorem for the Absolute Degree and Proper Maps Jean Cerqueira Berni e Oziride Manzoli Neto
Homotopic and Topological invariance of singular homology groups Jessica Cristina Rossinati Rodrigues da Costa e Maria Gorete Car-reira Andrade
Some Properties of E(G, W, FTG) and an Application in the Theory of
Splitting of Groups
Letícia Sanches Silva e Ermínia de Lourdes Campello Fanti The Reidemeister torsion of metacyclic spherical space forms
Lígia Laís Fêmina, Ana Paula Tremura Galves e Oziride Manzoli Neto
FIXED POINTS OF SELFMAPS OF 3-MANIFOLDS
PETER WONG, D. GONÇALVES, AND X. ZHAO
Let f : M → M be a selfmap on a closed n-manifold M where n ≥ 3. The Nielsen number N (f ) yields the minimal number of fixed points of maps in the homotopy class of f . In this talk, I will discuss the computation of N (f ) when M is a 3-manifold with (i) flat, (ii) S2× R, or (iii) S3 - geometry. As a byproduct, we classify all such selfmaps up to homotopy.
Bates College, U.S.A. E-mail address: pwong@bates.edu
EQUIVARIANT FUNCTIONS AND VECTOR FIELDS ON MANIFOLDS WITH SEMI–FREE ACTION OF CIRCLE
V. SHARKO
The report will present new results on the structure of equivariant Bott functions and vector fields, similar to the vector fields of Morse-Smale on manifolds with semi-free circle action. Will use the results of the work
References
[1] V.Sharko and D. Repovs, S1-Bott functions on manifolds. Ukrainian Math. Journal, vol.64, N12, pp. 1685-1698.
A SURVEY OF THE COHOMOLOGICAL DEGREE OF EQUIVARIANT MAPSI
NORBIL CORDOVA, DENISE DE MATTOS, AND EDIVALDO DOS SANTOS
The first result in degree theory for equivariant maps was the famous Borsuk-Ulam theorem which states that the degree of an odd map of a finite-dimensional sphere into itself is odd [1]. The oddness presents the simplest example of equivariance with respect to Z2 = {I, −I}. Since the
degree theory only depends of the (co)homology theory, we are interested in extending these results to G-spaces which reflect many of the local and global (co)homology properties of the topological manifolds. Namely, generalized manifolds.
References
[1] Lamport, L. Uber Zerlegung einer euklidischen n-dimensionalen Vollkugeln in n Men-gen, In: Verhandlungen des Internationalen Mathematiker Kongresses, Zürich, 1932; II. Band: Sekt.Vortaäge, Orel Fusli, Zürich, 1932, 142–198.
(Norbil Cordova) ICMC-USP E-mail address: norbil@icmc.usp.brl (Denise de Mattos) ICMC-USP E-mail address: deniseml@icmc.usp.br (Edivaldo dos Santos) DM-UFSCar E-mail address: edivaldo@dm.ufscar.br
OS GRUPOS DE TRANÇAS DO TORO E DA GARRAFA DE KLEIN
CAROLINA DE MIRANDA E PEREIRO
O objetivo principal deste trabalho é estudar as propriedades topológicas e algébricas dos grupos de tranças do toro T e da garrafa de Klein K, desta-cando os pontos comuns e as diferenças entre eles e entre o grupo de tranças de Artin.
Temos as seguintes definições: o grupo de n-tranças puras Pn(M ) =
π1(Fn(M )), o grupo de n-tranças Bn(M ) = π1(Fn(M )/Sn) e o grupo de
tranças mistas como Bn,m(M ) = π1(Fn+m(M )/(Sn× Sm)) onde Fn(M ) é o
espaço de configuração. Seja M = T ou M = K, calculamos explicitamente a secção da seguinte sequência exata curta:
1 → π1(M \ {x1, . . . , xn}) → Pn+1(M ) → Pn(M ) → 1.
e após isso, foi possível descobrir propriedades algébricas destes grupos, como por exemplo o cálculo do centro. Mostramos também que a sequência exata curta abaixo cinde se, e somente se, m é múltiplo de n:
1 → Bm(M \ {x1, . . . , xn}) → Bn,m(M ) → Bn(M ) → 1.
Estudamos as séries centrais descendentes e as séries derivadas destes gru-pos. Por enquanto, obtemos os seguintes resultados:
(1) Bn(T) é residualmente solúvel se, e somente se, n < 5.
(2) Bn(K) é residualmente solúvel se n = 2 e não é residualmente solúvel para n ≥ 5.
(3) Bn(K) não é residualmente nilpotente se n ≥ 3. References
[1] P. Bellingeri, S. Gervais and J. Guaschi, Lower central series of Artin-Tits and surface braid groups, J. Algebra 319 (2008), 1409-1427.
[2] E. Fadell and L. Neuwirth, Configuration spaces, Math. Scandinavica 10 (1962), 111-118.
SPHERICAL SPACE FORMS AND THE BORSUK-ULAM THEOREM
MARJORY DEL VECCHIO DOS SANTOS AND EDIVALDO LOPES DOS SANTOS AND MAURO FLÁVIO SPREAFICO
Consider the following general formulation of the Borsuk-Ulam Theorem: Let X, Y be topological spaces, G a finite group acting freely on X and f : X → Y be a continuous map. We say that a point x ∈ X is a point of G-coincidence of f if the map f takes each orbit Gx at a single point. The set of all points of G-coincidence of f is denoted by A(f ).
An n-spherical space form is actually a completely connected Riemannian manifold of positive constant curvature, ie, is exactly quotient Sn/Γ of the sphere by the action of a subgroup Γ ⊂ O(n + 1, R) of fixed point free isometries. A complete classification of spherical space forms was made by J. Wolf in [4].
In our work we consider the domain X be a n-spherical space form which admits a free action of Zp, with p > 2 prime and f : X → Rkbe a continuous
map and we show that, under certain relations between the numbers n and k, the set A(f ) is not empty, that is, A(f ) 6= ∅. Also, we study when such spaces admits a free action of the cyclic group G = Zp, with p > 2 prime.
We note that the case G = Z2 has been studied in [2].
References
[1] ADEM, A.; MILGRAM, R.J. Cohomology of finite groups. Springer-Verlag, 1994. [2] GONÇALVES D.L.; MANZOLI NETO O.; SPREAFICO M. The Borsuk-Ulam
The-orem for homotopy spherical space forms. Journal of Fixed Point Theory and Appli-cations 9, pp. 285-294, 2011.
[3] TURYGIN, Y.A. Borsuk-Ulam Proprety of Finite Group Actions on Manifolds and Applications. PhD thesis - University of Florida, 2007.
[4] WOLF, J. A. Spaces of constant curvature. MacGraw-Hill, Inc.,1967. (Marjory Del Vecchio dos Santos) ICMC-USP
E-mail address: marjory@icmc.usp.br (Edivaldo Lopes dos Santos) DM-UFSCar E-mail address: edivaldo@dm.ufscar.br (Mauro Flávio Spreafico) ICMC-USP E-mail address: mauros@icmc.usp.br
APROXIMAÇÕES DA DIAGONAL E ANÉIS DE COHOMOLOGIA DOS GRUPOS FUNDAMENTAIS DAS
SUPERFÍCIES
SÉRGIO TADAO MARTINS AND DACIBERG LIMA GONÇALVES
As superfícies fechadas distintas de S2 e RP2 são espaços K(G, 1), logo o anel de cohomologia de uma tal superfície coincide com o anel de cohomologia de seu grupo fundamental.
Considere então o grupo fundamental G de uma superfície fechada dis-tinta de S2 e RP2 (orientável ou não). Apresentaremos uma resolução livre P de G e, para esta resolução, apresentaremos também uma aproximação da diagonal ∆ : P → P ⊗ P . O conhecimento da resolução livre P e da aproximação da diagonal ∆ nos permite então calcular de forma eficiente os produtos
Hp(G, M ) ⊗ Hq(G, N )→ H^ p+q(G, M ⊗ N )
para G-módulos M e N . Mostraremos exemplos de tais cálculos em alguns casos relevantes.
References
[1] Brown, K. Cohomology of groups, Springer, 1982.
[2] Fadell, E. and Husseini, S. The Nielsen number on surfaces. Topological methods in nonlinear functional analysis, Contemporary Mathematics, 21:59–98, 1983.
[3] Gonçalves, D. L. and Oliveira, E. The Lefschetz coincidence number for maps among compact surfaces. Far East Journal of Math Science, 2:147–166, 1997.
[4] Tomoda, S. and Zvengrowski, P. Remarks on the cohomology of finite fundamental groups of 3-manifolds. Geometry & Topology Monographs, 14:519–556, 2008. (Sérgio Tadao Martins) Depto. de Matemática — IME-USP, Caixa Postal 66.281 — CEP 05314–970, São Paulo — SP
E-mail address: sergiotm@ime.usp.br
(Daciberg Lima Gonçalves) Depto. de Matemática — IME-USP, Caixa Postal 66.281 — CEP 05314–970, São Paulo — SP
ARTIN BRAID GROUPS AND CRYSTALLOGRAPHIC GROUPS
DACIBERG GONÇALVES, JOHN GUASCHI, AND OSCAR OCAMPO
A uniform discrete subgroup Π of Rno O(n) ⊆ Aff(Rn) is called a crystal-lographic group of dimension n. Let Bn (resp. Pn) denote the Artin braid
group (resp. the Artin pure braid group) with n strings and let n ≥ 3. In this work we will show that the quotient Bn
[Pn, Pn]
is a crystallographic group, where [Pn, Pn] means the commutator subgroup of Pn, and that has
no 2-torsion. As a consequence, for a 2-group H contained in the symmetric group Σn we have that
σ−1(H) [Pn, Pn]
is a Bieberbach group (torsion-free crystal-lographic group), where σn: Bn → Σn is the canonical projection. Also we
will present previous results about the odd torsion of Bn [Pn, Pn]
as well as the state of the art of the study of finite groups contained in that quotient.
References
[1] L. Charlap, Bieberbach groups and flat manifolds, Springer-Verlag, New York, (1986). [2] K. Dekimpe, Almost-Bieberbach groups: affine and polynomial structures, Lecture
Notes in Mathematics - Springer 1639, Berlin, (1996).
[3] O. Ocampo, Grupos de tranças Brunnianas e grupos de homotopia da esfera S2, PhD thesis, Universidade de São Paulo, Brasil, 2013.
[4] J. A. Wolf, Spaces of constant curvature, 6 ed. AMS Chelsea Publishing, Providence, (2011).
(Daciberg Gonçalves) Departamento de Matemática - IME-USP E-mail address: dlgoncal@ime.usp.br
(John Guaschi) Laboratoire de Mathématiques Nicolas Oresme - Université de Caen
E-mail address: guaschi@math.unicaen.fr
(Oscar Ocampo) Departamento de Matemática - IME-USP E-mail address: oeocampo@ime.usp.br
FREE AND PROPERLY DISCONTINUOUSLY ACTIONS OF GROUPS ON HOMOTOPY 2n-SPHERES
MAREK GOLASIŃSKI (TORUŃ), DACIBERG LIMA GONÇALVES (SÃO PAULO) AND ROLANDO JIMÉNEZ (OAXACA)
Let G be a group acting freely, properly discontinuously and cellularly on a finite dimensional CW-complex Σ(2n) which has the homotopy type of the 2n-sphere S2n.
Under the hypothesis that dim Σ(2n) ≤ 2n + 1 we study the groups with the virtual cohomological dimension vcd G < ∞ which act as above on Σ(2n). It turns out that they consist of free groups and certain semi-direct products F o Z2 with F an arbitrary free group. Given a free group F , we present
an algebraic criterion equivalent to the criterion of realizability of an action of F on Σ(2n). For the special case where the group G is virtually cyclic, we classify all groups which act on Σ(2n) and also the homotopy type of all possible orbit spaces. In case the group F is free of a finite rank, we classify all groups which act on Σ(2n). The classification takes into account the action of the group on the top cohomology class of the 2n-homotopy sphere.
(Marek Golasiński) Faculty of Mathematics and Computer Science, Univer-sity of Warmia and Mazury, Słoneczna 54, 10-710 Olsztyn,Poland
A PROPRIEDADE DE BORSUK-ULAM PARA CLASSES DE HOMOTOPIA DE FUNÇÕES ENTRE SUPERFÍCIES
VINICIUS CASTELUBER LAASS
Seja τ : X → X uma involução livre de pontos fixos. Dizemos que uma tripla (X, τ, S) tem a propriedade de Borsuk-Ulam, se dada f : X → S contínua, existe x ∈ X tal que f (τ (x)) = f (x). Essa definição surgiu a partir do teorema clássico de Borsuk-Ulam que afirma o seguinte: se f : Sn → Rn
é uma aplicação contínua, então existe x ∈ Sn tal que f (x) = f (−x). Em [1], os autores classificaram se (X, τ, S) tem, ou não, a P.B.U., no caso em que X e S são superfícies. Quando (X, τ, S) não tem a P.B.U., significa que existe uma função f : X → S tal que f (τ (x)) 6= f (x) para todo x ∈ X. Isto não dá uma boa medida da quantidade de funções que colapsam uma órbita.
A partir deste trabalho uma questão natural que surge é: dado uma tripla (X, τ, S), classificar as classes de homotopia de [X, S] com a propriedade que ela contém um representante f tal que f (τ (x)) 6= f (x) para todo x ∈ X.
Trataremos sobre a abordagem geral deste problema, inclusive no caso onde temos um grupo cíclico finito atuando e, como aplicação, daremos ên-fase ao caso específico em que S = S2.
References
[1] D. L. Gonçalves and J. Guaschi The Borsuk-Ulam theorem for maps into a surface, Topology and its Applications 157 (2010) 1742-1759
(Vinicius Casteluber Laass) IME - USP E-mail address: vinicl@ime.usp.br
TWO-PARAMETER TRACE AND FIXED POINTS OF FIBER-PRESERVING MAPS
WESLEM L. SILVA
Let F : X × Ir → X be a cellular map where X is a finite connected CW complex and Ir = I × ... × I
| {z }
r−times
. R. Geoghegan and A. Nicas in [1] defined a r−parameter trace, R(F ), for F using the Hochschild homology. If r = 1 and F ix(F |X×∂I) = ∅ then R(F ) 6= 0 implies that the “minimal” fixed point
set of F is nonempty. When r > 1 the r−parameter trace captures less of the geometric obstruction, developed in [2], to homotoping F relative to X × ∂Ir to a map F0 : X × Ir → X which has no fixed points in classes other than those which meet X × ∂Ir than in the one-parameter case. In this case we need further hypotheses on X or invariants beyond the r−parameter trace that still are unknown. In this work we consider X = T , torus, and we tried to investigate who is the minimal fixed point set of a fiber-preserving map f : M → M , where M is the trivial bundle T × T , using the parameterized fixed point theory. In this situation we can write f like f (x, y) = (F (x, y), y), where F : T × I2 → T . Some examples will be presented.
References
[1] R. Geoghegan and A. Nicas, Parametrized Lefschetz-Nielsen fixed point theory and Hochschild homology traces, Amer. J. Math. 116 (1994), 397-446.
[2] A. Hatcher and F. Quinn, Bordism invariants of intersections of submanifolds, Trans. Amer. Math. Soc. 200 (1974), 327-344.
[3] Weslem L. Silva, Conjuntos minimais de ponto fixo e coincidência de aplicações fibradas, Tese de doutorado, UFSCar, 2012.
(Weslem L. Silva)
Z22-ACTIONS ON KdP (2n) ∪ KeP (2m + 1)
SERGIO TSUYOSHI URA, ALLAN EDLEY RAMOS DE ANDRADE AND PEDRO LUIZ QUEIROZ PERGHER
In this work, we classify the bordism classes of spaces with par of com-muting involutions fixing KdP (2n) ∪ KeP (2m + 1), where d < e ∈ {1, 2, 4}
and KiP (j) is the projective space RP (j), CP (j) or HP (j), for i = 1, 2 or 4, respectively.
References
[1] Conner, P.E. Differentiable Periodic Maps- Second Edition , Springer- Verlag, 1979. [2] de Oliveira, R. Involuções Comutantes Fixando dois Espaços Projetivos Pares, Tese
de doutorado, UFSCar 2002.
[3] Pergher, P. L. Q. Bordism of Two Commuting Involutions, Proceedings of the AMS, 1998 vol.126, number 7, pages 2141-2149.
[4] Ramos, A. Involuções Fixando Espaços projetivos, Tese de doutorado, UFSCar 2007. [5] Stong, R. E. Equivariant bordism and Z2k-action, Duke Math. J. 17(1970), 779-785.
(Sergio Tsuyoshi Ura) UFSCar - Universidade Federal de São Carlos E-mail address: stura@dm.ufscar.br
(Allan Edley Ramos de Andrade) UFSCar - Universidade Federal de São Car-los
E-mail address: aandrade@dm.ufscar.br
(Pedro Luiz Queiroz Pergher) UFSCar - Universidade Federal de São Carlos E-mail address: pergher@dm.ufscar.br
REALIZATION OF FIXED POINT SET IN A PRESCRIBED EQUIVARIANT HOMOTOPY CLASS OF MAPS
RAFAEL MOREIRA DE SOUZA AND PETER NGAI-SING WONG
In 1990, H. Shirmer made use of the relative Nielsen fixed point theory [5] to give necessary and sufficient conditions for nonempty subpolyhedra A to be realized as the fixed point set of a map in the homotopy class of a given self map [3]. In fact, the paper [3] corrected an incorrect assumption previously thought to be sufficient as claimed by P. Strantzalos in 1977.
In other hand, many applications involve symmetries as a result of a group action. In the equivariant setting, we are concerned with a group G acting on a space X together with a G-map f : X → x which respect the group action, that is, for all α ∈ G, f (αx) = αf (x) for all x ∈ X. In this case, the fixed point set F ix(f ) is a priori a G-invariant subset of X.
The study of topological fixed point theory for equivariant maps `a la Nielsen began with [6] and subsequently in [7, 8, 9]. Combining the setting of [3] together with the equivariant setting of [9], we determined necessary and sufficient conditions for a nonempty closed G-invariant subset A to be the fixed point set of a equivariant map in a given G-homotopy class of a fixed map.
References
[1] Brown,R. F. The Lefschetz Fixed Point Theorem, Scott, Foresman and Co., Glenview, I11., 1971.
[2] Bredon,G. E. Introduction to Compact Transformation Groups, Academic Press, New York, 1972.
[3] Schirmer,H. Fixed point sets in a prescribed homotopy class, Topology Appl. 37 (1990), 153-162.
[4] —————- Fixed point sets of deformations of pairs of spaces, Topology Appl. 23 (1986), 193-205.
[5] —————- A relative Nielsen number, Pacific J. Math. 122 (1986), 459-473. [6] Fadell, E. and Wong, P. On deforming G-maps to be fixed point free, Pacific J. Math.
132 (1988), 277-281.
[7] Wong, P. Equivariant Nielsen fixed point theory for G-maps, Pacific J. Math. 150 (1991), 179-200.
CLASSIFICAÇÃO DE Z22-AÇÕES FIXANDO KdP2m+1∪ KeP2n, d < e
ALLAN EDLEY RAMOS DE ANDRADE AND PEDRO LUIZ QUEIROZ PERGHER
Classificamos a menos de bordismo equivariante, as Z22-ações (M, Φ) cujo conjunto de pontos fixos é KdP2m+1∪ KeP2n, onde d < e, ou seja, faremos
uma classificação de Z22-açções (M, Φ) onde FΦ é um dos seguintes tipos: RP2m+1∪ CP2n, RP2m+1∪ QP2n e CP2m+1∪ QP2n.
Nesta análise, aparecerá ações exóticas, ou seja, que não são equivari-antemente cobordantes à uma ação do tipo Γ2j(M, T ), onde (M, T ) é uma involução fixando KdP2m+1∪KeP2n. Este fato torna a classificação para este caso, um dos resultados mais inesperado e interessante do nosso trabalho.
References
[1] Conner, P.E. Differentiable Periodic Maps- Second Edition , Springer- Verlag, 1979. [2] A. Ramos, Involuções Fixando Espaços Projetivos, Tese de
Doutorado-DM-UFSCar(2007).
[3] R. de oliveira, Involuções Comutantes Fixando Dois Espaços Projetivos Pares, Tese de Doutorado-DM-UFSCar(2002).
[4] R. E. Stong, Equivariant bordism and Z2k-action, Duke Math. J. 17(1970), 779-785.
(Departamento de Matemática - Caixa Postal 676, Rodovia Washington Luiz, Km 235, 13565-905 São Carlos, SP) UFSCar
E-mail address: aandrade@dm.ufscar.br E-mail address: pergher@dm.ufscar.br
CYCLIC AND COCYCLIC MAPS AND GENERALIZED WHITEHEAD PRODUCTS
THIAGO DE MELO AND MAREK GOLASIŃSKI
Given co-H-spaces X and Y , B. Gray [3] has defined a co-H-space X ◦ Y and a natural transformation X ◦ Y → X ∨ Y which leads to a generalized Whitehead product. We make use of that product and sketch ideas on its dual to examine cyclic and cocyclic maps. Given spaces X and Y , some results on Gottlieb sets G(X, Y ) and dual Gottlieb sets DG(X, Y ) are stated.
References
[1] M. Arkowitz, The generalized Whitehead product, Pacific J. Math. 12 (1962), 7–23. [2] D. H. Gottlieb, A certain subgroup of the fundamental group, Amer. J. Math. 87
(1965), 840–856.
[3] B. Gray, On generalized Whitehead products, Trans. Amer. Math. Soc. 11 (2011), 6143–6158.
[4] K. L. Lim, On cyclic maps, J. Austral. Math. Soc. Ser. A 32 (1982), 349–357. [5] K. Varadarajan, Generalized Gottlieb groups, J. Indian Math. Soc. 33 (1969), 141–
164.
(Thiago de Melo) Unesp Rio Claro E-mail address: tmelo@rc.unesp.br
(Marek Golasiński) Faculty of Mathematics and Computer Science, Univer-sity of Warmia and Mazury, Słoneczna 54, 10-710 Olsztyn,Poland
A SPECIAL CASE OF INVOLUTION - A PHENOMENON OF LOW CODIMENSION
PATRICIA E. DESIDERI AND PEDRO L. Q. PERGHER
Let Mm be a closed and smooth manifold and T : Mm → Mm be a
smooth involution. The fixed point set F of T consists of a disjoint union of closed submanifolds with dimensions ranging from 0 to m. We denote F =
n
[
i=0
Fi, n ≤ m, where Fi means the union of i-dimensional components of F . Let us suppose F of the form
F = F1∪ ( n [ j=0 j even Fj).
Then, m ≤ n + 2, and that this bound is best possible; note that this char-acterizes a small codimension phenomenon. We emphasize that the above bound lies in the context of finding better bounds for the Five Halves Theo-rem of J. Boardman when we omit some components of F or specialize the values of n = the greatest in the fixed point set of (Mm; T ).
(Patricia E. Desideri) Instituto de Matemática e Computação, Universidade Federal de Itajubá, Itajubá/MG, Brazil
E-mail address: patricia.desideri@unifei.edu.br
(Pedro L. Q. Pergher) Departamento de Matemática, Universidade Federal de São Carlos, São Carlos/SP, Brazil
THE CONSTRUCTION OF FUNDAMENTAL DOMAIN OF TETRAHEDRAL SPHERICAL SPACE FORMS
ANA PAULA TREMURA GALVES, LÍGIA LAÍS FÊMINA AND OZIRIDE MANZOLI NETO
The topological spherical space forms problem is the study of fixed-point free actions of finite groups on spheres. Equivalently, it is the study of space forms.
The earliest examples of space forms are the Clifford-Klein manifolds. A Clifford-Klein manifold is a complete Riemannian manifold with constant sectional curvature equal to +1. These manifolds are of the form S4n−1/G where G is a finite group acting freely and orthogonally on the sphere 4n − 1-dimensional (S4n−1). The classification of Clifford-Klein manifolds is thus a completely algebraic question in group representation theory. A complete solution was given by J. Wolf [5].
We denote by P24 the binary tetrahedral group of order 24. This is the
group with three generators and presentation
hx, y, z | x2 = (xy)2 = y2, zxz−1 = y, zyz−1= xy, xyx−1 = y−1, z3 = x4= 1i
that acts freely on the odd dimension spheres.
The main purpose of this work is to describe a fundamental domain of the spherical space forms which fundamental group is the binary tetrahe-dral group, that we call tetrahetetrahe-dral spherical space forms and we denote by P = S4n−1/P
24.
References
[1] Cohen, M.M. A course in simple homotopy theory. New York: Springer-Verlag, 1973. [2] Fêmina, L.L.; Galves, A.P.T.; Neto, O.M.; Spreafico, M.F. Cellular decomposition and free resolution for split metacyclic spherical space forms. Homology Homotopy and Applications, v.15, p.257-278, 2013.
[3] Melo, T.; Neto, O.M.; Spreafico, M.F. Cellular decomposition of quaternionic spherical space forms. Geom Dedicata, DOI 10.1007/s10711-012-9714-4, 2012. [4] Tomoda, S.; Zvengrowski, P. Remarks on the cohomology of finite fundamental groups
of 3-manifolds. Geometry and Topology Monographics, v.14, p.519-556, 2008. [5] Wolf, J.A. Spaces of constant curvature. McGraw-Hill Inc., 1967.
SEQUÊNCIAS EXATAS E O LEMA DOS CINCO
ALEX MELGES BARBOSA AND JOÃO PERES VIEIRA
Neste trabalho apresentamos alguns resultados sobre Sequências Exatas e demonstramos o Lema dos Cinco.
References
[1] Vick, J.W. Homology Theory, Academic Press, New York-London, 1973.
[2] Wall, C.T.C. A Geometric Introduction to Topology, Addison-Wesley Publishing Com-pany, 1972.
(Alex Melges Barbosa) Universidade Estadual Paulista "Júlio de Mesquita Filho", Campus de Rio Claro - Instituto de Geociências e Ciências Exatas
E-mail address: 214001101@rc.unesp.br (João Peres Vieira) Unesp Rio Claro E-mail address: jpvieira@rc.unesp.br
PRELIMINARES SOBRE COINCIDÊNCIAS DE APLICAÇÕES DEFINIDAS EM VARIEDADES COM BORDO
ALICE KIMIE MIWA LIBARDI, DACIBERG LIMA GONÇALVES, AND THAIS FERNANDA MENDES MONIS
Seja B a categoria cujos objetos são pares ordenados X e Y de n-variedades compactas, conexas, orientadas com bordo ∂X e ∂Y e cujos morfismos são pares ordenados f, g : X −→ Y de aplicações onde g(∂X) ⊂ ∂Y .
Em [1] Brown e Schirmer definiram o índice de coincidência, o qual é uma versão local de Nakaoka e o número de Lefschetz para variedades com bordo e também definiram um número de coincidência de Nielsen para aplicações entre variedades com bordo e estudaram algumas propriedades.
Em [2] Daciberg e Jezierski deram uma generalização do teorema de coincidência de Lefschetz para variedades não orientadas, usando grupos de (co)homologia com coeficientes locais. Esta generalização requer que uma das aplicações consideradas seja "true oriented". Também estenderam o indice de coincidência e o número de Lefschetz para pares de aplicações f, g : (M, ∂M ) −→ (N, ∂N ), com M e N n-variedades e g "orientation true" e g(∂M ) ⊂ ∂N .
Em [3] Koschorke estendeu a teoria de coincidência de Nielsen considerando f, g : M −→ N entre variedades de dimensões arbitrárias, mas para variedades sem bordo e M compacta.
Consideraremos os correspondentes número mínimo de componentes conexas por caminhos de conjuntos de coincidências definidos em [3] e usaremos uma abordagem através de grupos de bordismo normal para definir um limitante inferior N (f, g) que generaliza o clássico número de Nielsen e o definido por Koschorke.
O invariante para aplicações em variedades com bordo se encaixa em uma se-quencia exata de bordismo normal e é levado pelo operador bordo no invariante para variedades sem bordo, cujo estudo encontra-se na literatura. Pretende-se consid-erar primeiramente variedades cujos bordos são esferas e estudar o nucleo e imagem do operador bordo, além de aplicar os resultados já obtidos particularmente por Koschorke e Gonçalves para aplicaçoes entre esferas.
References
[1] Brown, R. F., Schirmer, H., Nielsen coincidence theory and coincidence-producing maps for manifolds with boundary. Topology Appl. 46, no. 1, 65-79 (1992).
HOMOLOGICAL INVARIANTS AND DUALITY
AMANDA FERREIRA DE LIMA MARIA GORETE CARREIRA ANDRADE
Let G be a group, S = {Si, i ∈ I} a family of subgroups of G with infinite
index and M a Z2G-module. In [3] it was defined a homological invariant
E∗(G, S, M ), which is “dual" to the cohomological invariant defined in [1].
In this work we continue the study of the invariant E∗(G, S, M ) under a
more general point of view, obtaining results and properties which are dual to those obtained by Andrade and Fanti in [2]. The invariant E∗(G, S, M )
provides, for instance, properties in the duality theory of groups and about groups that satisfy certain finitude conditions.
References
[1] M.G.C. Andrade, E.L.C. Fanti, A relative cohomological invariant for pairs of groups, Manuscripta Math.,83 (1994), 1-18.
[2] M.G.C. Andrade, E.L.C. Fanti, J. A. Daccach, On certain relative invariants, Int. Journal of Pure and Appl. Math. 21 , 335-352. (2005).
[3] M.G.C. Andrade; A. B. Gazon, A dual relative homological invariant, Preprint, 2013. (Amanda Ferreira de Lima) Universidade Federal de São Carlos UFSCAR -São Carlos - SP
E-mail address: mandinha−lima10@hotmail.com
(Maria Gorete Carreira Andrade) Universidade Estadual Paulista UNESP -São José do Rio Preto - SP
A GENERALIZATION OF THE BORSUK-ULAM THEOREM
ANA MARIA MATHIAS MORITA AND MARIA GORETE CARREIRA ANDRADE
The Borsuk-Ulam Theorem was first conjectured by the mathematician S. Ulam and was proved later by the mathematician K. Borsuk, in 1933. It states the following: “Given any continuous map f : Sn −→ Rn, there
is a point x ∈ Sn such that f (x) = f (−x)”. Since then, many proofs, generalizations and applications have been presented. In general, if τ is a free Z2-action on a space X and Y is another topological space, we say that
the Borsuk-Ulam Theorem holds for (X, τ ; Y ) if for any continuous map f : X −→ Y there is a point x ∈ X such that f (x) = f (τ (x)). In [1], D. L. Gonçalves studied necessary and sufficient conditions for the validity of the Borsuk-Ulam Theorem for maps f : S −→ R2, in terms of the existence of an equivariant map, where S is a closed surface. In this work we detail the proof of the following result, that is contained in [1]: “If S is an orientable closed surface with Euler characteristic congruent to 2 mod 4 and τ is a free Z2
-action on S, then the Borsuk-Ulam Theorem holds for the triple (S, τ ; R2)”. This work is part of our master dissertation.
References
[1] Gonçalves, D. L. The Borsuk-Ulam theorem for surfaces, Quaestiones Mathematicae, 29:1, p.117-123, 2006.
[2] Massey, W. S. Algebraic Topology: An Introduction, Springer-Verlag, 1967. [3] Munkres, J. R. Elements of Algebraic Topology, Addison-Wesley, 1984.
(Ana Maria Mathias Morita) Master student supported by CAPES - IBILCE-UNESP
E-mail address: ana.morita@hotmail.com
(Maria Gorete Carreira Andrade) Adviser - Department of Mathematics - IBILCE-UNESP
DIAGONAL INVOLUTION ON PRODUCTS OF SPACES AND THE BORSUK-ULAM THEOREM
ANDERSON PAIÃO DOS SANTOS
The Borsuk-Ulam theorem states that for any continuous map f : Sn→ Rn
there exists x ∈ Snsuch that f (x) = f (A(x)), where A denotes the antipodal involution on Sn. A natural generalization of this theorem is replace Sn by a space X with a free involution τ , and we say that the Borsuk-Ulam theorem holds for the triple (X, τ ; Rn) if for any continuous map f : X → Rn there exists x ∈ X such that f (x) = f (τ (x)). In this work we study this generalization when X is a product of spaces with a certain type of involution. Furthermore, when X is a product of surfaces and the involutions are the one refereed above, we completely classify the triples which satisfies the Borsuk-Ulam theorem.
References
[1] A. Bauval, D. L. Gonçalves, C. Hayat, P. Zvengrowski, The Borsuk-Ulam theorem for the manifolds of geometry spherical, S2× R, flat, and Nil: in preparation. [2] K. Borsuk, Drei Sätze über die n-dimensionale Euklidische Sphäre, Fund. Math., 20
(1933), 177-190.
[3] D. L. Gonçalves, The Borsuk-Ulam theorem for surfaces, Quaestiones Mathematicae, 29 (2006), 117-123.
[4] D. L. Gonçalves, J. Guaschi, The Borsuk-Ulam theorem for maps into a surface, Topology and its applications, 157 (2010), 1742-1759.
[5] D. L. Gonçalves, C. Hayat, P. Zvengrowski, The Borsuk-Ulam Theorem for Manifolds, with Applications to Dimensions Two and Three, Proceedings Bratislava Topology Symposium. “Group Actions and Homogeneous Spaces”, (2010), 9-28.
[6] D. L. Gonçalves, O. M. Neto, M. Spreafico, The Borsuk-Ulam theorem for homotopy spherical space forms, Journal of Fixed Point Theory and Applications 9, 2 (2011), 285–294.
[7] A. P. Santos, Involuções e o teorema de Borsuk-Ulam para algumas variedades de dimensão 4, PhD thesis, IME-USP, 2012.
[8] D. Vendrúscolo, P. E. Desideri, P. L. Q. Pergher, Some generalizations of the Borsuk-Ulam theorem, Plublicationes Mathematicae (Debrecen), 78 (2011), 583-593. (Anderson Paião dos Santos)
A PROOF OF THE JORDAN SEPARATION THEOREM
BRUNO CALDEIRA CARLOTTI DE SOUZA AND MARIA GORETE CARREIRA ANDRADE
The Jordan Separation Theorem states a fact that is geometrically quite believable: “a simple closed curve in S2 (or in the plane) separates it into at least two connected components". This result is useful for the proof of the Curve Jordan Theorem which was originally conjectured by Camille Jordan, in 1892, and many incorrect proofs were published, including one by Jordan himself. In 1905, Oswald Veblen published a correct proof. In this work we present a proof based in [1], of the Jordan Separation Theorem using tools of Algebraic Topology and some knowledge of Point-Set topology.
References
[1] Munkres, J. R. Topology: Second Edition, Prentice Hall, 2000.
(Bruno Caldeira Carlotti de Souza) Graduando em Matemática - Bolsista- PET-MEC-SESu - Ibilce - Unesp - São José do Rio Preto - SP
E-mail address: brunobcc2@gmail.com
(Maria Gorete Carreira Andrade) Orientadora - Ibilce - Unesp - São José do Rio Preto - SP
GRUPO FUNDAMENTAL E O TEOREMA DE VAN KAMPEN
GIVANILDO DONIZETI DE MELO AND THIAGO DE MELO
Na Topologia Algébrica, a Teoria de Homotopia é muito importante e o grupo fundamental é um dos invariantes topológicos mais simples desta teoria. Neste trabalho, definiremos este grupo e faremos o cálculo em alguns casos.
O grupo fundamental será definido por meio de classes de homotopia de laços (caminhos fechados) em X, baseados em ∗ ∈ X, e será denotado por π1(X, ∗).
Mostraremos que o grupo π1(X, ∗) não depende do ponto ∗ escolhido, quando X é conexo por caminhos, e também que, se (X, ∗) tem o mesmo tipo de homotopia de (Y, ∗), então π1(X, ∗) é isomorfo π1(Y, ∗), além de
outros resultados imediatos.
Por fim, calcularemos alguns grupos fundamentais através do Teorema de van Kampen.
(Givanildo Donizeti de Melo) Unesp Rio Claro E-mail address: givadonimelo@hotmail.com
O TEOREMA FUNDAMENTAL DA ÁLGEBRA
DEMONSTRADO ATRAVÉS DO GRAU DE APLICAÇÕES ENTRE ESFERAS
GUILHERME VITURI FERNANDES PINTO AND THIAGO DE MELO
O grau de uma aplicação h : S1 → S1 é um número inteiro deg(f )
asso-ciado a esta aplicação através do levantamento p(t) = e2πt. Fazendo uso da teoria de homologia com coeficientes inteiros, podemos levar este conceito a aplicações g : Sn → Sn, em que deg(g) é o inteiro tal que g
?(x) = deg(g)x,
∀x ∈ Hn(Sn), sendo g? : Hn(Sn) → Hn(Sn) a induzida da aplicação g. É
possível provar que, se g1, g2 : Sn→ Sn, então g1 é homotópica à g2 se, e
so-mente se, deg(g1) = deg(g2). Uma aplicação f : Rn→ Rn é dita admissível
se possui uma extensão contínua f : Rn→ Rn, em que Rn= Rn∪ {∞}
defi-nida por f (x) = f (x) se x ∈ Rn e f (∞) = ∞. Definimos deg(f ) = deg(f?), sendo f? : h ◦ f ◦ k : Sn → Sn, k : Sn → Rn um homeomorfismo e h seu
inverso. Demonstraremos que se f : Rn→ Rn satisfaz deg(f ) 6= 0 então f é
sobrejetora, e a partir daí, identificando R2 com o conjunto C dos números complexos, bastará mostrar que um polinômio não-constante qualquer com coeficientes complexos p(x) = a0+ a1x1+ . . . + xn possui grau não nulo.
Referências
[1] Hu, Sze-Tsen Homology Theory: A First Course in Algebraic Topology, Holden–Day, Inc., 1966.
[2] Hu, Sze-Tsen Homotopy Theory, Academic Press, New York, 1959. [3] Hu, Sze-Tsen Elements of General Topology, Holden–Day, Inc., 1964.
(Guilherme Vituri Fernandes Pinto) Unesp - Rio Claro E-mail address: vituri_vituri@hotmail.com
(Thiago de Melo) Unesp - Rio Claro E-mail address: tmelo@rc.unesp.br
A WECKEN TYPE THEOREM FOR THE ABSOLUTE DEGREE AND PROPER MAPS
JEAN CERQUEIRA BERNI AND OZIRIDE MANZOLI NETO
In this work we will present the following result, proved by Brooks [1]: Theorem: Let f : X → Y be a proper map of a connected n−manifold X into a connected, locally path-connected, semilocally simply connected Haus-dorff space that is locally n−Euclidean at the point y0 ∈ Y . Then every
map properly homotopic to f and transverse to y0 has at least A(f, y0) roots, where A(f, y0) denotes the absolute degree of f at y0.
Moreover, if n > 2, then there is a map properly homotopic to f and transverse to y0 that has exactly A(f, y0) roots at y0.
References
[1] Brooks, R. Roots of mappings from manifolds, Fixed Point Theory Appl., no. 4, 273-307, 2004
[2] Brooks, R. Nielsen root theory, Handbook of Topological Fixed Point Theory (L. Gorniewicz, ed.), in press.
(Jean Cerqueira Berni) ICMC - USP - Brazil E-mail address: jeancb@icmc.usp.br
(Oziride Manzoli Neto) ICMC - USP - Brazil E-mail address: ozimneto@icmc.usp.br
HOMOTOPIC AND TOPOLOGICAL INVARIANCE OF SINGULAR HOMOLOGY GROUPS
JESSICA CRISTINA ROSSINATI RODRIGUES DA COSTA AND MARIA GORETE CARREIRA ANDRADE
The homological algebra is a branch oh mathematics rather interesting, which is a tool for many areas of pure mathematics, among which stand out the algebraic topology and commutative algebra. Inside the homological algebra fits the study of complex chains and within this context, the study of the homology of a chain complex and its fundamental properties. The study of the homology of a chain complex is basic pre-requisites for the treatment of singular homology of a topological space X denoted by H∗(X). The original
motivation for defining homology groups is the observation that topological spaces (in particular manifolds) are distinguished by their n-dimensional "holes". In this work we present an introduction to the theory of singular homology and, as the main result, we see that the singular homology groups are invariants under homotopy and consequently are topological invariants. We also present some applications of this result.
References
[1] Munkres, J.R.; Elements of algebraic topology, Addison-Wesley, 1984.
[2] Vick, J.W.;Homology Theory: an Introduction to Algebraic Topology, Academic Press, New York, 1973.
[3] Hatcher, A.; Algebraic Topologic , 2001.
(Jessica Cristina Rossinati Rodrigues da Costa) Graduanda em Matemática -Bolsista FAPESP - IBILCE - UNESP - São José do Rio Preto - SP
E-mail address: jessica−rossinati@hotmail.com
(Maria Gorete Carreira Andrade) Orientadora - IBILCE - UNESP - São José do Rio Preto - SP
SOME PROPERTIES OF E(G, W, FTG) AND AN APPLICATION IN THE THEORY OF SPLITTING OF
GROUPS
LETÍCIA SANCHES SILVA AND ERMÍNIA DE LOURDES CAMPELLO FANTI
In this work we present some properties of the invariant E(G, W, M ), defined by a group G, a G-set W and a Z2G-module M ([2]). We dedicated,
especially, for the case in that M = FTG, where T is a subgroup of G. We
present a result about splittings of group and E(G, W, FTG), and we give an example. Many important results of the theory of splittings of groups are related with the theory of ends of groups. The first result about splittings of group and ends is know as structured theorem Stallings and it is a result involving the number of ends of a group G, e(G), and a condition to G splits over a finite subgroup T ([3]). In our result we analyze splitting of G over a subgroup T, finitely generated and normal in G, not necessarily finite.
References
[1] ANDRADE, M.G.C.; FANTI, E.L.C. A relative cohomological invariant for pairs of groups. Manuscripta Math., v.83, p. 1-18, 1994.
[2] Andrade, M. G. C.; Fanti, E. L. C. The cohomological invariant E’(G,W) and Some properties, International Journal of Applied Mathematics, p. 183-192, 2012. [3] Stallings, J. R.; On torsion free groups with infinitely many ends. Annals of Math.
88(1968), 312 − 334.
(Letícia Sanches Silva) Aluna de Doutorado em Matemática - UNESP - IBILCE - SJRP. Bolsa: CAPES
E-mail address: le-cissinha@hotmail.com
(Ermínia de Lourdes Campello Fanti) Dep. Matemática UNESP IBILCE -SJRP
THE REIDEMEISTER TORSION OF METACYCLIC SPHERICAL SPACE FORMS
LÍGIA LAÍS FÊMINA, ANA PAULA TREMURA GALVES AND OZIRIDE MANZOLI NETO
The Reidemeister torsion (R-torsion) is an important invariant topological originally introduced by Reidemeister [6], Franz [3] and Rham [7] for classifying lens spaces. These invariants were the first examples of manifolds that have the same homotopy type without be homeomorphic. The Reidemeister torsion was the first invariant of a manifold which is not invariant of by homotopy type.
We study the actions of the split metacyclic groups D(2h+1)2t = hx, y|x2h+1= y2
t
= 1, yxy−1= y−1i,
h ≥ 1 e t ≥ 3 on the odd dimensional spheres. The actions on the spheres which gives the quotient spaces it is called Metacyclic Spherical Space Forms. By the chain complex constructed for the Metacyclic Spherical Space Forms, we calculate the R-torsion of these spaces for a given representation of its fundamental group. We obtained also a relation between the different tor-sions found.
References
[1] Cohen, M.M. A course in simple homotopy theory. New York: Springer-Verlag, 1973. [2] Fêmina, L.L.; Galves, A.P.T.; Neto, O.M.; Spreafico, M.F. Cellular decomposition and free resolution for split metacyclic spherical space forms. Homology Homotopy and Applications, v.15, p.257-278, 2013.
[3] Franz, W. Über die Torsion einer Überdeckung, J. Reine Angew. Math, v. 173, p. 245-254, 1935.
[4] Melo, T.; Neto, O.M.; Spreafico, M.F. Cellular decomposition of quaternionic spherical space forms. Geom Dedicata, DOI 10.1007/s10711-012-9714-4, 2012. [5] Milnor, J. Whitehead Torsion, Bulletin of the American Mathematical Society, v.
72, p. 358-426, 1966.
[6] Reidemeister, K. Homotopieringe und Linseräume, Hamburger Abhandl. 11, 102-109. 1935
COBORDISMO DE APLICAÇÕES
PABLO GONZALEZ PAGOTTO AND ALICE KIMIE MIWA LIBARDI
Denotemos por (f, M, N ) a tripla constituída por M e N variedades fechadas, diferenciáveis de dimensão m e n, respectivamente, e f : M −→ N uma aplicação contínua.
Dizemos que (f, M, N ) e (f0, M0, N0) são cobordantes se existe uma tripla (F, V, W ) onde:
a) V e W são variedades compactas, diferenciáveis com bordos, de dimen-são m + 1 e n + 1, respectivamente, com ∂V = M ∪ M0, ∂W = N ∪ N0 e
b) F : V −→ W contínua, cujas restrições a M é f e a N é f0.
O conjunto das classes de equivalências sob esta relação será denotada por N(m, n).
Dada uma tripla (f, M, N ) em N(m, n) tem-se a induzida em cohomologia f∗ : H∗(N ) −→ H∗(M ). Definimos f∗ : H∗(M ) −→ H∗(N ) da seguinte
maneira: para cada α ∈ Hi(M ), considere o homomorfismo: Hm−i(N ) −→ Hm−i(M ) −→ Hm(M ) −→ Z2,
onde a primeira flecha é f∗, a segunda é o produto cup por α e a ultima é a avaliação na classe fundamental de M , obtendo assim um elemento f∗(α) ∈
Hn+i−m(n).
Para qualquer partição {w, w1, · · · , wr} com | w | + | w1 | + · · · | wr |
+r(n − m) = n, define-se o número de Stiefel-Whitney, segundo Stong, por: hvw(N ) ∪ f∗(vw1(M )) ∪ f∗(vwr(M )), [N ]i, onde para µ = (i1, · · · , is), vµ é o
produto das classes de Stiefel-Whitney vi1, · · · vis.
O objetivo do trabalho é demonstrar o teorema:
Teorema: (f, M, N ) e (f0, M0, N0) são cobordantes se, e somente se, eles tem os mesmos números de Stiefel-Whitney definidos acima.
(Pablo Gonzalez Pagotto) UNESP/Rio Claro E-mail address: pgp 2008@hotmail.com
(Alice KImie Miwa Libardi-orientadora) UNESP/Rio Claro E-mail address: alicekml@rc.unesp.br
COHOMOLOGY OF GROUPS AND ITS RELATION WITH WALLPAPER GROUPS
RAFAELLA DE SOUZA MARTINS, ERMÍNIA DE LOURDES CAMPELLO FANTI AND FLÁVIA SOUZA MACHADO DA SILVA
The main goal of this work is to study the relation between the cohomology of groups and the problem of classifying wallpaper groups (or plane groups), which are symmetry groups of wallpaper patterns. This study envolves an interesting mixture of mathematical ideas, from the very simple to complex ones. There are, up to isomorphism, exactly 17 symmetry groups of wall-paper patterns. The beautiful Escher’s works of art illustrate very nicely group-theoric aspects of the different symmetry groups. Given a plane group G, we have an abelian normal subgroup T associated to G (subgroup of the translations) called lattice, a quocient group G0 ' GT called point group, an
action of G0 on T (in such a way that T is a ZG0 - module) and a group extension (of G0 by T ), 0 −→ T −→ G −→ G0−→ 0 ([2] and [3]). Fixing a
ZG0 - module T we define an equivalence relation in the set of extensions of
G0 by T and show that: “There is an one-to-one correspondence between the
elements of the second cohomology group H2(G0, T ) and the set of
equiva-lence classes of the extensions of G0 by T that give rise to the fixed action of G0 in T ” ([1]). So that the number of those equivalence classes is equal
to |H2(G0, T )|. Thus, in the context of wallpaper groups, by analyzing the
possibilities for G0, and considering the previous result, we obtain an
up-per bound for the number of those groups (because there may exist diferent equivalence classes with isomorphic wallpaper groups).
References
[1] Brown, K. S. Cohomology of Groups, Queen Mary College Math. Notes, Londres, 1976.
[2] Hiller, H. Crystallography and Cohomology of Groups. The American Mathematical Monthly, Vol. 93, No. 10, p. 765-779, 1986.
[3] Morandi, P. The Classication of Wallpaper Patterns: From Group Cohomology to Escher.s Tessellations. Notes. New Mexico State University. 2003.
Math-TOPOLOGIA ALGEBRICA NÃO-ABELIANA
RENATO VIEIRA
Em [4] Loday provou que os (n+1)-tipos de homotopia conexos podem ser modelados por catn-grupos, que são codificações algébricas de n-categorias estritas internas aos grupos, da mesma forma que grupos modelam 1-tipos de homotopia conexos. Brown e Loday provaram em [1] que o funtor que associa um catn-grupo a um (n + 1)-tipo de homotopia preserva certos tipos de colimites, generalizando o clássico teorema de Seifert-van Kampen, o que facilita computações dessa invariante topológica em certos casos. Em [3] Ellis e Steiner provaram a equivalência da categoria dos catn-grupos com a catego-ria dos n-cubos cruzados de grupos e apresentam computações de uma classe de colimites que eles denominaram n-cubos cruzados de grupos r-universais. Em [2] Ellis e Mikhailov mostraram como esses resultados podem ser usados para demonstrar a descrição combinatória dos grupos de homotopia da esfera encontrada originalmente por Wu em [5], além de outros resultados de teoria dos grupos. O painel que será apresentado pretende enunciar esses resulta-dos, além de mostrar como essa teoria pode descrever o tipo de homotopia da soma wedge de espaços de Eilenberg-MacLane K(G1, m1) ∨ K(G2, m2).
References
[1] BROWN, Ronald; LODAY, Jean-Louis. Van Kampen theorems for diagrams of spaces. Topology, v. 26, n. 3, p. 311-335, 1987.
[2] ELLIS, Graham; MIKHAILOV, Roman. A colimit of classifying spaces. Advances in Mathematics, v. 223, n. 6, p. 2097-2113, 2010.
[3] ELLIS, Graham; STEINER, Richard. Higher-dimensional crossed modules and the homotopy groups of (n + 1)-ads. Journal of Pure and Applied Algebra, v. 46, n. 2, p. 117-136, 1987.
[4] LODAY, Jean-Louis. Spaces with finitely many non-trivial homotopy groups. Journal of Pure and Applied Algebra, v. 24, n. 2, p. 179-202, 1982.
[5] WU, Jie. Combinatorial descriptions of homotopy groups of certain spaces. Mathe-matical Proceedings of the Cambridge Philosophical Society. [Cambridge, Eng.] Cam-bridge Philosophical Society., 2001. p. 489-513.
(Renato Vieira)
HOMOTOPIA E APLICAÇÕES
TAIS ROBERTA RIBEIRO AND JOÃO PERES VIEIRA
Neste trabalho introduzimos o estudo de Homotopia e apresentamos al-guns exemplos de aplicações homotópicas.
References
[1] Wall, C. T. C. A Geometric Introduction to Topology, Addison-Wesley Publishing Company, 1972.
[2] Lima, E. L. Grupo Fundamental e Espaços de Recobrimento. 11o Colóquio Brasileiro de Matemática, Impa, 1977
(Tais Roberta Ribeiro) Universidade Estadual Paulista "Júlio de Mesquita Filho", Campus de Rio Claro - Instituto de Geociências e Ciências Exatas
E-mail address: 214000903@rc.unesp.br (João Peres Vieira) Unesp Rio Claro E-mail address: jpvieira@rc.unesp.br
