Relatório Científico

Campus de São José do Rio Preto

RELATÓRIO CIENTÍFICO

Trata o presente do relatório científico final referente ao Edital 10/2014 – PROPG – Programa de Apoio a Organização de Eventos pelos Programas de Pós-graduação (PAOE), para a realização do “XIX Encontro Brasileiro de Topologia (EBT)”.

1. Descrição do evento O evento ocorreu no Ipê Park Hotel, em São José do Rio Preto-SP, tendo sido organizado pelo Ibilce/UNESP por meio de seus docentes participantes dos grupos de pesquisa em Topologia Algébrica e Teoria de Singularidades do nosso Programa de Pós-graduação em Matemática, no período de 03 a 09 de agosto de 2014. O Encontro Brasileiro de Topologia (EBT) é um congresso internacional de grande prestígio, sendo o evento nacional mais importante para atualizar e dinamizar os pesquisadores brasileiros na área da Topologia e outras áreas afins. Neste ano foi realizada a 19 a edição do evento, a qual deu ênfase a tópicos na fronteira da pesquisa atual, tais como aplicações de sequências espectrais, teoria de homotopia e o invariante de Kervaire (que inclusive foram objetos de minicursos avançados), além de tópicos mais tradicionais da área, como: classes características, folheações, teoria de ponto fixo e coincidência de aplicações; ações de grupos e grafos de aplicações estáveis entre superficies fechadas, grupos de bordismo, singularidades, entre outros. O Encontro Brasileiro de Topologia foi iniciado em 1979 e desde 1980 é realizado a cada 2 anos. Por ter se firmado como uma tradição de bom nível para a Topologia brasileira, este congresso tem atraído a presença de pesquisadores brasileiros e estrangeiros de reputação internacional. Sendo assim, os Encontros Brasileiros de Topologia têm sua principal importância na tradição de ser uma oportunidade de interação entre topólogos brasileiros e estrangeiros, e de divulgação de novos trabalhos e linhas de pesquisa dentro da Topologia. Para os grupos de pesquisa de nosso Programa de Pós-graduação em Matemática foi uma excelente oportunidade de interação e contato. O EBT ofereceu uma rica programação de atividades científicas, entre conferências, minicursos e comunicações, tanto para os pesquisadores quanto para os alunos de pós-graduação. A programação completa deste ano encontra-se no Anexo 1. As instuições brasileiras que têm participado com assiduidade deste congresso são: USP-SP, USP - S. Carlos, IMPA, PUC-RJ, UFF, UFSCar, UNICAMP, UFPe, UFC, UNESP-Rio Claro, UNESP-S.J.Rio Preto, UFSC e UFMG.

2. Programação do evento

A programação do evento está disponível na página do evento, no endereço http://www.mat.ibilce.unesp.br/EBT2014/

Basta clicar no ícone Programação/Programme. No Anexo 1 encontra-se uma cópia da Programação do XIX EBT.

Vale destacar que foram oferecidos dois minicursos em nível avançado:

INSTITUTO DE BIOCIÊNCIAS, LETRAS E CIÊNCIAS EXATAS – DEPARTAMENTO DE MATEMÁTICA Rua Cristóvão Colombo, 2265 CEP 15054-000 S. J. Rio Preto SP Brasil Tel 17 3221 2330 fax 17 3221 2340

Campus de São José do Rio Preto

“Lectures on Applications of Spectral Sequences ” ministrado pelo Prof. Dr. John McCleary, do Vassar College-USA;

“The Kervaire invariant ” ministrado pelo Prof. Dr. Michael J. Hopkins, da Universidade de Harvard- USA. É importante ressaltar que o Prof. Hopkins recentemente foi o ganhador de um prêmio de Matemática, o Frederic Esser Nemmers Prize in Mathematics 2014 , devido sua contribuição no tema objeto de seu minicurso.

Foi também oferecido um minicurso elementar, “ An introduction to the study of graphs of stable maps between closed surfaces ” ministrado pela Profa. Dra. Catarina Mendes de Jesus, da Universidade Federal de Viçosa.

Além dos minicursos, foram oferecidas palestras, comunicações e duas sessões de pôsteres.

O caderno de resumos/abstracts contendo todas as palestras, minicursos, comunicações e pôsteres pode ser encontrado no Anexo 2 deste documento.

3. Avaliação do evento: resultados alcançados e descrição da contribuição do evento para a Pós- graduação

O evento foi um sucesso. Tivemos a participação de mais de 90 pesquisadores, sendo muitos deles estrangeiros. Na página do evento encontra-se a lista dos participantes, a qual também pode ser encontrada no Anexo 3 a este documento. Para nosso Programa de Pós-Graduação em Matemática do Ibilce/UNESP foi uma oportunidade excelente de fortalecer os grupos e as linhas de pesquisa em Topologia Algébrica e Teoria de Singularidades. Estes grupos ainda estão em fase de consolidação e contam com a participação de jovens pesquisadores. Neste sentido, o apoio do Programa de Pós-Graduação local e da PROPG ao evento foi de grande valia e de extrema importância para o fortalecimento destes grupos e linhas de pesquisa. Vale ressaltar que recentemente nosso Programa de Pós-Graduação teve a inclusão de quatro docentes da UNESP-Rio Claro, da área de Topologia Algébrica, o que veio a fortalecer o grupo do Ibilce de Topologia. Estes docentes também estavam presentes no evento e foi uma oportunidade de reunir estes jovens pesquisadores de Rio Preto e Rio Claro, juntamente com renomados pesquisadores do Brasil e do exterior. Acreditamos que muitas parcerias científicas foram iniciadas com este evento, e esperamos com isso um aumento na quantidade e qualidade da produção científica e acadêmica de nosso Programa de Pós-Graduação. Acreditamos também que a presença de ilustres matemáticos, como é o caso do Prof. Hopkins, deu visibilidade à UNESP como organizadora do evento, trazendo benefícios no sentido de divulgação dos seus trabalhos científicos e internacionalização de sua pesquisa.

4. Divulgação do evento na mídia e Publicações

O evento foi manchete na página do Ibilce no seguintes link: http://www.ibilce.unesp.br/#!/noticia/499/

Também foi enviada para publicação uma matéria do evento para os periódicos que circulam em nossa comunidade/cidade “Notícias do Ibilce” e “Notícias FAPERP”, que serão publicadas nos próximos dias.

INSTITUTO DE BIOCIÊNCIAS, LETRAS E CIÊNCIAS EXATAS – DEPARTAMENTO DE MATEMÁTICA Rua Cristóvão Colombo, 2265 CEP 15054-000 S. J. Rio Preto SP Brasil Tel 17 3221 2330 fax 17 3221 2340

Campus de São José do Rio Preto

Foi publicado os Anais “Caderno de Resumos e Abstracts do XIX EBT”, o qual encontra-se no Anexo 2. Além disso, os textos dos minicursos oferecidos durante o XIX EBT estão disponíveis na página do evento.

5. Relatório financeiro

Através deste Programa de Apoio a Eventos da PROPG, recebemos um total de recursos no vlaor de R$ 8.000,00. As despesas que tivemos foram concentradas em Ajudas de Custo para pesquisadores de diversas universidades do país, os quais mantém ou iniciarão contato científico com os pesquisadores do nosso Programa de Pós-graduação.. Estas despesas foram os seguintes pagamentos de Ajuda de Custo:

R$ 1. 326,00 - Prof. Dr. Peter Wong – Battes College-USA R$ 1. 370,00 - Prof. Dr. Paul Alexander Schweitzer – PUC-Rio de Janeiro-RJ R$ 1. 326,00 - Prof. Dr. Alexandre Paiva Barreto – UFSCar – São Carlos-SP R$ 1. 326,00 - Prof. Dr. Kisnney Emiliano de Almeida – UEFS – Feira de Santana-BA R$ 1. 326,00 - Prof. Dr. Anderson Paião dos Santos – UTFPR- Cornélio Procópio-PR R$ 1. 326,00 - Prof. Dr. Allan Edley Ramos de Andrade – UFMS- Três Lagoas-MS Os recibos foram devidamente assinados e entregues na Seção de Finanças do Ibilce/UNESP. Uma cópia dos recibos encontra-se no Anexo 4. Vale ressaltar que o evento contou também com apoio financeiro de outras instituições e agências de fomento, como é o caso da FAPESP, CAPES, FAPERP, INCTMat, CNPq, UNESP, USP e UFSCAR.

O Comitê Científico do evento foi formado pelos seguintes pesquisadores:

Daciberg Lima Gonçalves (USP São Paulo - Brasil) (coordenador) Claude Hayat (Université Toulouse III - França) Eduardo Hoefel (UFPR Curitiba - PR - Brasil) John Guaschi (Université de Caen - França) Marek Golasi ński (Kazimierz Wielki University - Polônia) Pedro Luiz Queiroz Pergher (UFSCar São Carlos - SP - Brasil)

O Comitê Organizador do evento foi formado pelos seguintes pesquisadores:

Ermínia de Loudes Campello Fanti (UNESP SJRP - SP) (coordenadora) Alice Kimie Miwa Libardi (UNESP Rio Claro - SP) Daniel Vendrúscolo (UFSCar São Carlos - SP) Darlan Rabelo Girao (UFC Fortaleza - CE) Denise de Mattos (USP São Carlos - SP)

INSTITUTO DE BIOCIÊNCIAS, LETRAS E CIÊNCIAS EXATAS – DEPARTAMENTO DE MATEMÁTICA Rua Cristóvão Colombo, 2265 CEP 15054-000 S. J. Rio Preto SP Brasil Tel 17 3221 2330 fax 17 3221 2340

Campus de São José do Rio Preto

Evelin Meneguesso Barbaresco (UNESP SJRP - SP) Flávia Souza Machado da Silva (UNESP SJRP - SP) João Carlos Ferreira Costa (UNESP SJRP - SP) Leonardo Navarro De Carvalho (UFF Niterói - RJ) Ligia Laís Fêmina (UFU Uberlândia - MG) Luciana de Fátima Martins (UNESP SJRP - SP) Lucilia Daruiz Borsari (USP São Paulo - SP) Luiz Roberto Hartmann Junior (UFSCar São Carlos - SP) Maria Gorete Carreira Andrade (UNESP SJRP - SP) Michelle Ferreira Zanchetta Morgado (UNESP SJRP - SP) Thiago de Melo (UNESP Rio Claro - SP)

A página do evento na internet está disponível no seguinte endereço: http://www.mat.ibilce.unesp.br/EBT2014/

Gostaria de agradecer, em nome da organização do evento, o apoio financeiro da PROPG e o apoio da coordenação do Programa de Pós-Graduação em Matemática do IBILCE/UNESP, na pessoa do Prof. Dr. Paulo Ricardo da Silva (coordenador), que apoiou a realização deste grande evento científico.

Prof. Dr. João Carlos Ferreira Costa Beneficiário

INSTITUTO DE BIOCIÊNCIAS, LETRAS E CIÊNCIAS EXATAS – DEPARTAMENTO DE MATEMÁTICA Rua Cristóvão Colombo, 2265 CEP 15054-000 S. J. Rio Preto SP Brasil Tel 17 3221 2330 fax 17 3221 2340

Campus de São José do Rio Preto

ANEXOS

1. Programação do evento

2. Caderno de Resumos/Abstracts

3. Lista de participantes

4. Recibos das Ajudas de Custo (Prestação de Contas)

INSTITUTO DE BIOCIÊNCIAS, LETRAS E CIÊNCIAS EXATAS – DEPARTAMENTO DE MATEMÁTICA Rua Cristóvão Colombo, 2265 CEP 15054-000 S. J. Rio Preto SP Brasil Tel 17 3221 2330 fax 17 3221 2340 Programação [Programme] SUNDAY 03/08

17:00 ARRIVAL

17:00 18:30 REGISTRATION

18:30 19:00 OPENING CEREMONY

Michael R. Kelly 19:10 20:00 Fixed point index bounds and a class of negatively curved 2-complexes

20:20 22:00 DINNER MONDAY 04/08 Catarina Mendes de Jesus David Herrera-Carrasco 9:00 9:20 (09:00 - 09:50) Continua and hyperspaces

Elementary minicourse: An introduction to Oscar Ocampo the study of graphs of stable maps 9:25 9:55 between closed surfaces Conjugacy classes of torsion elements in the crystallographic group Bn /[Pn ,Pn]

Peter Wong 10:00 10:50 Geometric invariants for group extensions with applications to twisted conjugacy classes 11:00 11:20 BREAK John McCleary 11:20 12:10 Advanced minicourse: Lectures on Applications of Spectral Sequences 12:20 14:00 LUNCH Michael J. Hopkins 14:00 14:50 Advanced minicourse: The Kervaire Invariant Elizabeth Gasparim 15:00 15:30 Lefschetz fibrations on adjoint orbits A. P. Barreto 15:35 16:05 Involutions on closed Sol 3-manifolds and the Borsuk-Ulam theorem for maps into ℝn 16:10 16:40 BREAK Daniel Vendrúscolo 16:40 17:10 The minimal number of Borsuk-Ulam coincidences on surfaces José Gregorio Rodríguez Nieto 17:15 17:45 Virtual knot theory Fernando Macías-Romero 17:50 18:10 Uniqueness of hyperspaces in a continuum Brian Callander 18:15 18:35 Compactifications of adjoint orbits and their Hodge diamonds 20:00 22:00 DINNER TUESDAY 05/08 Catarina Mendes de Jesus (09:00 - 09:50) 9:00 9:30

Elementary minicourse: An introduction to the study of graphs of stable maps Northon Canevari Leme Penteado between closed surfaces 9:30 9:50 Representing homotopy classes by maps with certain minimality root properties P. Sankaran 10:00 10:50 Twisted conjugacy in certain PL-homeomorphism groups of the reals 11:00 11:20 BREAK John McCleary 11:20 12:10 Advanced minicourse: Lectures on Applications of Spectral Sequences 12:20 14:00 LUNCH Michael J. Hopkins 14:00 14:50 Advanced minicourse: The Kervaire Invariant João Miguel Nogueira 15:00 15:30 The number of strings on essential tangle decompositions of a knot can be unbounded Vinicius Casteluber Laass 15:35 16:05 The Borsuk–Ulam property for homotopy classes of functions between surfaces 16:10 17:00 BREAK & POSTER SESSION 1 Claude Hayat 17:00 17:30 Using Reidemeister–Schreier algorithm

Erica Boizan Batista 17:35 17:55 The Reeb graph of a map germ from ℝ3 to ℝ2

18:00 18:50 POSTER SESSION 2

20:00 22:00 DINNER WEDNESDAY 06/08 Gabriel Calsamiglia 9:00 9:20 Realizing covering maps as holonomies

Kisnney Almeida

9:25 9:55 Σ1-invariant for Artin groups of circuit rank 1 and 2

Jérôme Los 10:00 10:50 Volume entropy for minimal presentations of surfaces groups in all ranks 11:00 11:20 BREAK Thaís F. M. Moniz 11:20 11:50 Lefschetz coincidence class for several maps 12:00 14:00 LUNCH

FREE AFTERNOON

20:00 22:00 DINNER THURSDAY 07/08 Catarina Mendes de Jesus Severin Barmeier 9:00 9:20 (09:00 - 09:50) Deformations of the discrete Heisenberg group

Elementary minicourse: An introduction to António Salgueiro the study of graphs of stable maps 9:25 9:55 between closed surfaces Fiber surfaces from alternating states

Osamu Saeki 10:00 10:50 Connected components of regular fibers of differentiable maps 11:00 11:20 BREAK John McCleary 11:20 12:10 Advanced minicourse: Lectures on Applications of Spectral Sequences 12:20 14:00 LUNCH Michael J. Hopkins 14:00 14:50 Advanced minicourse: The Kervaire Invariant Mariana Silveira 15:00 15:30 Bifurcations associated to spectral sequences Nelson Silva 15:35 16:05 On representation of the Reeb graph as a sub-complex of a manifold 16:10 16:40 BREAK Paul A. Schweitzer, S.J. 16:40 17:10 A generalization of Novikov’s Theorem on the existence of Reeb components in codimension one foliations

Taciana O. Souza 17:15 17:45 New examples of Neuwirth–Stallings pairs and non-trivial real Milnor fibrations

Paulo Gusmão 17:50 18:10 Typical foliations

20:00 22:00 DINNER FRIDAY 08/08 Catarina Mendes de Jesus Juan Valentín Mendoza 9:00 9:20 (09:00 - 09:50) The Boyland order for maximal horseshoe braid types

Elementary minicourse: An introduction to Rafael Moreira de Souza the study of graphs of stable maps 9:25 9:55 between closed surfaces Fixed point sets of equivariant fiber-preserving maps

Marek Golasinski 10:00 10:50 On the group structure of [Ω핊2, ΩY] 11:00 11:20 BREAK John McCleary 11:20 12:10 Advanced minicourse: Lectures on Applications of Spectral Sequences 12:20 14:00 LUNCH Michael J. Hopkins 14:00 14:50 Advanced minicourse: The Kervaire Invariant

Vladimir V. Sharko

15:00 15:50 Functions and vector fields on C(ℂ Pn)-singular manifolds

16:00 16:30 BREAK Carolina de Miranda e Pereiro 16:30 17:00 The lower central and derived series of the braid groups of the torus and of the Klein bottle Luiz Hartmann 17:05 17:35 Reidemeister metric and Ray–Singer metric for pseudomanifolds with conical singularity Claudemir Aniz

17:40 18:00 Minimizing the Nielsen root classes for maps between CW-complexes and manifolds of the same dimension ≥ 3

20:00 22:00 DINNER SATURDAY 09/08 Catarina Mendes de Jesus Guido Ledesma 9:00 9:20 (09:00 - 09:50) Smale flows on 핊2 × 핊1

Elementary minicourse: An introduction to Gustavo de Lima Prado the study of graphs of stable maps 9:25 9:55 between closed surfaces Bordism and coincidences in codimension one

Wacław Marzantowicz 10:00 10:50 Bourgin-Yang version of the Borsuk-Ulam theorem and related topics 11:00 11:20 BREAK Daciberg Lima Gonçalves 11:20 11:50 Actions of infinite groups on homotopy even spheres 12:00 13:00 LUNCH

13:00 DEPARTURE

th 19 Brazilian Topology Meeting

Caderno de Resumos

(Abstracts)

03 a 09 de agosto de 2014 (August 03-09, 2014) { W w t {t

Comitê Científico (Scientific Committee) Daciberg Lima Gonçalves (USP São Paulo - Brasil) (coord.) Claude Hayat (Université Toulouse III - França) Eduardo Hoefel (UFPR Curitiba - PR - Brasil) John Guaschi (Université de Caen–Basse Normandie - França) Marek Golasiński (Kazimierz Wielki University - Polônia) Pedro Luiz Queiroz Pergher (UFSCar São Carlos - SP - Brasil)

Comissão Organizadora (Organizing Committee) Ermínia de Lourdes Campello Fanti (UNESP SJRP–SP) (coord.)

Alice Kimie Miwa Libardi (UNESP Rio Claro–SP) Daniel Vendrúscolo (UFSCar São Carlos–SP) Darlan Rabelo Girao (UFC Fortaleza–CE)

Denise de Mattos (USP São Carlos–SP) Evelin Meneguesso Barbaresco (UNESP SJRP–SP) Flávia Souza Machado da Silva (UNESP SJRP–SP)

João Carlos Ferreira Costa (UNESP SJRP–SP) Leonardo Navarro De Carvalho (UFF Niterói–RJ) Ligia Laís Fêmina (UFU Uberlândia–MG)

Luciana de Fátima Martins (UNESP SJRP–SP) Lucilia Daruiz Borsari (USP São Paulo–SP) Luiz Roberto Hartmann Junior (UFSCar São Carlos–SP)

Maria Gorete Carreira Andrade (UNESP SJRP–SP) Michelle Ferreira Zanchetta Morgado (UNESP SJRP–SP) Thiago de Melo (UNESP Rio Claro–SP)

Realização (Hosted by) o:

Apoio (Support)

PRÓ-REITORIA DE PÓS-GRADUAÇÃO XIX Encontro Brasileiro de Topologia

(19th Brazilian Topology Meeting)

Caderno de resumos

(Abstracts)

de 03 a 09 de agosto de 2014

(August 03–09, 2014)

São José do Rio Preto/SP – Brasil List of abstracts

MINICURSOS (MINI COURSES) 7

PALESTRAS (TALKS) 8

A. P. Bar reto, D. L. Gonçalves, and D. Vendrúscolo Involutions on closed Sol 3-manifolds and the Borsuk-Ulam theorem for maps into Rn 8

Darlan Girão, João Nogueira, and An tónio Salgueiro Fiber surfaces from alternating states 8

Brian Callan der and Elizabeth Gasparim Compactiﬁcations of adjoint orbits and their Hodge diamonds 9

Carolina de Miranda e Pereiro The lower central and derived series of the braid groups of the torus and of the Klein bottle 10

Claudemir Aniz Minimizing the Nielsen root classes for maps between CW-complexes and manifolds of the same dimension ≥ 3 11

Anne Bauval and Claude Hayat Using Reidemeister–Schreier algorithm 11

Daciberg Lima Gonçalves and Sérgio Martins Actions of inﬁnite groups on homotopy even spheres 12

John Guaschi and Daniel Ven drús colo The minimal number of Borsuk-Ulam coincidences on surfaces 13

Luis A. Guerrero-Méndez, David Her rera -Car rasco, and Fernando Macías-Romero Continua and hyperspaces 14

Elizabeth Gasparim Lefschetz ﬁbrations on adjoint orbits 14

Er ica Boizan Batista, João Carlos Ferreira Costa, and Juan José Nuño Ballesteros The Reeb graph of a map germ from R3 to R2 15

Luis A. Guerrero-Méndez and David Herrera-Carrasco and Fer nando Macías -Romero Uniqueness of hyperspaces in a continuum 16

Gabriel Calsamiglia Realizing covering maps as holonomies 17

Ketty de Rezende and Guido Ledesma Smale ﬂows on S2 × S1 18

Gustavo de Lima Prado Bordism and coincidences in codimension one 19

Luis Alseda, David Juher, Jérôme Los, and Francesc Manosas Volume entropy for minimal presentations of surface groups in all ranks 20

João Miguel Nogueira The number of strings on essential tangle decompositions of a knot can be unbounded 20

3 José Gregorio Rodríguez Nieto Virtual knot theory 21

Juan Valentín Mendoza The Boyland order for maximal horseshoe braid types 22

Kisnney Almeida Σ1-invariant for Artin groups of circuit rank 1 and 2 23

Leonardo N. Car valho and Ulrich Oertel The complete sphere complex of an irreducible 3-manifold and Heegaard splittings: the case of a connected sum of S2 × S1’s and handlebodies 23

Llohann D. Sperança Morita equivalences, equivariant bundles and exotic spheres 24

Luiz Hartmann Reidemeister metric and Ray–Singer metric for pseudomanifolds with conical singularity 24

Marek Go lasiński, Daciberg L. Gonçalves, and Peter Wong On the group structure of [Ω S2, ΩY ] 25

Dahisy Lima, Mar iana Sil veira, and Ketty de Rezende Bifurcations associated to spectral sequences 26

Daciberg L. Gonçalves and Michael R. Kelly Fixed point index bounds and a class of negatively curved 2-complexes 27

Marek Kaluba, Wacław Marzantowicz, and Nel son Silva On representation of the Reeb graph as a sub-complex of a manifold 28

Oziride Manzoli Neto and Northon Canevari Leme Pen teado Representing homotopy classes by maps with certain minimality root properties 29

Jorge T. Hiratuka and Os amu Saeki Connected components of regular ﬁbers of diﬀerentiable maps 29

Daciberg Lima Gonçalves, John Guaschi, and Os car Ocampo Conjugacy classes of torsion elements in the crystallographic group Bn/[Pn, P n] 30

D. L. Gonçalves, P. Sankaran, and R. Strebel Twisted conjugacy in certain PL-homeomorphism groups of the reals 30

Paulo Gusmão Typical foliations 31

Fernando Alcalde Cuesta, Gilbert Hector, and Paul A. Schweitzer, S.J. A generalization of Novikov’s Theorem on the existence of Reeb components in codimension one foliations 31

Nic Koban and Pe ter Wong Geometric invariants for group extensions with applications to twisted conjugacy classes 32

Rafael Moreira de Souza and Peter Ngai-Sing Wong Fixed point sets of equivariant ﬁber-preserving maps 33

Severin Barmeier Deformations of the discrete Heisenberg group 34

4 Taciana O. Souza, R. Araújo dos Santos, M. A. B. Hohlenwerger, and O. Saeki New examples of Neuwirth–Stallings pairs and non-trivial real Milnor ﬁbrations 34

Thaís F. M. Mo nis and Stanisłlaw Spiez Lefschetz coincidence class for several maps 35

Vinicius Casteluber Laass The Borsuk–Ulam property for homotopy classes of functions between surfaces 36

Alice K.M. Libardi and Vladimir V. Sharko Functions and vector ﬁelds on C(CP n)-singular manifolds 36

Wacław Marzan tow icz, Denise de Mattos, and Edivaldo dos Santos Bourgin-Yang version of the Borsuk-Ulam theorem and related topics 37

PAINÉIS (POSTERS) 38

Al lan Ed ley Ramos de An drade, Pedro L. Q. Pergher, and Sergio Tsuyoshi Ura Zk 2-actions ﬁxing KP (2 m + 1) t KP (2 n + 1) 38

Amanda Fer reira de Lima and Maria Gorete Carreira Andrade Some properties of the invariant E∗(G, S) 38

Ana Maria Math ias Morita and Maria Gorete Carreira Andrade The Borsuk-Ulam property for closed surfaces 39

Ana Paula Tremura Galves, Ligia Laís Fêmina, and Oziride Manzoli Neto The homology groups of tetrahedral spherical space forms 40

José Carlos Cifuentes Vasquez and An der son Luis Gama A connected compact hyperreal line 40

Anderson Paião dos Santos Some results about the Borsuk–Ulam theorem for double coverings of surface bundles 41

Dahisy Lima, Mariana Silveira, and Ketty de Rezende Conley index theory and spectral sequences in the Morse–Bott setting 42

Évelin Me negesso Bar baresco and Flávia Souza Machado da Silva A notion of asymptotic dimension and ﬁnite decomposition complexity 43

Gi vanildo Donizeti de Melo and Thiago de Melo On covering maps and deck transformations 43

Guil herme Vi turi and Thiago de Melo H-spaces and cyclic maps 44

Gustavo C. I. Figueiredo The nonabelian tensor product of groups 45

Soraya Rosana Torres Kudri and Izael do Nasci mento Separation axioms in approach spaces 46

Jes sica Cristina Rossi nati Ro drigues da Costa and Maria Gorete Carreira Andrade Spaces with operators and cohomology of groups under a topological viewpoint 47

João Carlos Ferreira Costa A note on topological classiﬁcation of singularities 48

5 D. L. Gonçalves, A. K. M. Libardi, D. Penteado, and J. P. Vieira Fibre maps and ﬁxed points on certain surface bundles 49

Letí cia Sanches Silva and Ermínia de Lourdes Campello Fanti Some results about a certain cohomological invariant 50

Ligia Laís Fêmina, Ana Paula Tremura Galves, and Oziride Manzoli Neto Contracting homotopy and diagonal maps for binary tetrahedral groups 51

José L. Arraut, Lu ciana F. Mar tins, and Dirk Schütz On singular foliations on the solid torus 52

Marcelo José Saia Minimal Whitney stratiﬁcation and Euler obstruction of discriminants 52

Marcio de Jesus Soares Cohomology of the ﬁxed point sets of semifree actions on a space X of type (a, b ) 53

Neemias Silva Mar tins and Simone Maria de Moraes Introduction to the theory of knots and polynomial invariants 54

Pablo Gonza lez Pagotto and Alistair Savage Graphical calculus for the Hecke algebra 54

Renato Vieira Nonabelian algebraic topology 55

S. To moda, O. Manzoli Neto, M. Spreaﬁco, L.L. Fêmina, and A.P.T. Galves A periodic resolution for the binary icosahedral group 56

Allan Edley Ramos de Andrade, Pedro L. Q. Pergher, and Ser gio Tsuyoshi Ura Zk 2-actions ﬁxing KP (2 m + 1) t KP (2 n + 1) 57

Thi ago de Melo and Marek Golasiński Dual higher order Whitehead products 57

Thuy Nguyen Thi Bich Study of certain singularity sets associated to a polynomial map 58

6 MINICURSOS (MINI COURSES)

Catarina Mendes de Jesus (Depto. de Matemática, Universidade Federal de Viçosa, Brasil) Introdução ao estudo de grafos de aplicações estáveis entre superfícies fechadas An introduction to the study of graphs of stable maps between closed surfaces – elementar, 5 aulas [elementary, 5 lectures]

O objetivo deste é apresentar como podemos associar grafos com pesos nos vértices às aplicações estáveis entre superfícies fechadas e mostrar como construir aplicações estáveis sobre o plano e esfera, com o conjunto singular e grau pré-determinado, a partir de um grafo dado. Trataremos das aplicações com grau zero como aplicações no plano onde faremos uma relação com o Teorema de Borsuk–Ulam.

John McCleary (Vassar College, USA) Lectures on applications of spectral sequences – [4 lectures]

Spectral sequences have long become standard and valuable tools in algebraic topology and other ﬁelds in which homological methods are used. There are plenty of areas of study where these methods can deliver some rich results and valuable orientation. In this series of lectures I will review the origins of spectral sequences and then show how they can be applied to computations of geometric, algebraic, or combinatorial interest, often in unexpected ways. In the lectures I expect to emphasize how a spectral sequence was found to be of use, and to discuss what this tells us about related problems. My goal is to reveal some new tools, known and perhaps not yet known, and the problems to which they may be applied and to identify how these tools can be used to explore some seemingly unrelated areas of study. The central topics will be chosen from the following: • Spectral sequences in diﬀerential geometry • Spectral sequences in algebra and analysis • Spectral sequences in combinatorics • Spectral sequences in equivariant theory • Other spectral sequences

Michael J. Hopkins (Dept. of Mathematics, Harvard University, USA) The Kervaire invariant – [4 lectures]

In this short course we develop stable homotopy theory, mainly the part which is closely related to the Kervaire invariant one problem, including the Freudenthal suspension theorem and the stable homotopy groups of the spheres. Then we state the Kervaire problem and its equivalent version in terms of stable homotopy groups. This is followed by the reduction of the problem to the case where the homotopy group is in dimension 2k − 2 , due to W. Browder. We exhibit the solution for k ≤ 6, where 32 the answer is positive, using explicit calculations up to π62 (S ). Then we develop the background to show the main result, which is that there is no element of Kervaire invariant one for k ≥ 8. The case 64 π126 (S ) remains open.

7 PALESTRAS (TALKS)

F 8 f

Involutions on closed Sol 3-manifolds and the Borsuk-Ulam theorem for maps into Rn

Alexan dre P. Bar reto, 1 D. L. Gonçalves, 2 and D. Vendrúscolo 3

Abstract For each Sol 3-manifold, we discuss the classiﬁcation of free involutions. For each triple (M, τ ; Rn) where M is a Sol 3-manifold and τ is a free involution, we determine whether (M, τ ; Rn) has the Borsuk-Ulam property or not. It is known that for n > 3 the Borsuk-Ulam property does not hold for any involution, so we provide a classiﬁcation when n = 1 , n = 2 and n = 3 . References

[1] C. Hayat, D. L. Gonçalves and P. Zvengrowski, The Borsuk-Ulam theorem for manifolds, with applications to dimensions two and three , Proceedings of the International Conference Bratislava Topology Symposium. "Group Actions and Homogeneous Spaces" (2009), 09–28. [2] D. Gonçalves and P. Wong, Nielsen numbers of self maps of Sol 3-manifolds , Topology and its Applications, Proceedings of the Nielsen ﬁxed point conference- 2011 Beijing June 13-17, 159 Issue 18 (2012), 3729–3737. 3 [3] J. Hillman, The F2-cohomology rings of Sol -manifolds , Bulletin of the Australian Mathematical Society, 89 (2014), 191–201. [4] K. Morimoto, Some orientable 3-manifolds containing Klein bottles , Kobe J. Math. 2 (1985), 37–44. [5] M. Sakuma, Involutions on torus bundles over S1, Osaka J. Math. 22 (1985), 163–185.

1 DM-UFSCar [email protected] 2 IME-USP [email protected] 3 DM-UFSCar [email protected]

F 8 f

Fiber surfaces from alternating states

Darlan Girão 1, João Nogueira 2, and An tónio Salgueiro 3

Abstract We define alternating Kauffman states of a link L and characterize when the corresponding state surface is a fiber of L. References

1 Universidade Federal do Ceará. [email protected] 2 Universidade de Coimbra. [email protected] 3 Universidade de Coimbra. [email protected]

8 Compactifications of adjoint orbits and their Hodge diamonds

Brian Callan der 1 and Elizabeth Gasprim 2

Abstract A recent theorem of [GGS2] showed that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We investigate the behaviour of their fibrewise compactifications. Expressing adjoint orbits and fibres as affine varieties in their Lie algebra, we compactify them to projective varieties via homogenisation of the defining ideals. We find that their Hodge diamonds vary wildly according to the choice of homogenisation, and that extensions of the potential to the compactification must acquire degenerate singularities. References

[C] B. Callander, Lefschetz Fibrations , Master’s Thesis, Universidade Estadual de Campinas (2013). [CG] B. Callander; E. Gasparim Hodge diamonds and adjoint orbits , arXiv:1311.1265. [Cy] Cynk, S. Euler characteristic of a complete intersection [CR] Cynk, S.; Rams, S. Invariants of hypersurfaces and logarithmic diﬀerential forms [GGS1] E. Gasparim, L. Grama, L. A. B. San Martin, Lefschetz ﬁbrations on adjoint orbits , arXiv:1309.4418. [GGS2] E. Gasparim, L. Grama, L. A. B. San Martin, Adjoint orbits of semisimple Lie groups and Lagrangian submanifolds , arXiv:1401.2418. [M2] D. Grayson, M. Stillman, Macaulay2 , a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/. [Ko] M. Kontsevich, Homological algebra of Mirror Symmetry , Proc. International Congress of Mathematicians (Zurich, 1994) Birkhäuser, Basel (1995) 120–139. [Se] P. Seidel, Fukaya categories and Picard-Lefschetz theory , Zurich Lectures in Advanced Mathematics, Eu- ropean Math. Soc., Zurich (2008).

1 Universidade Estadual de Campinas. [email protected] 2 Universidade Estadual de Campinas. [email protected]

9 The lower central and derived series of the braid groups of the torus and of the Klein bottle

Carolina de Miranda e Pereiro 1

Abstract

Let G be a group. The Lower Central Series {Γi(G)}i∈N is deﬁned inductively by: Γ1(G) = G, and Γi+1 (G) = [ G, Γi(G)] , for all i ∈ N, Derived Series (i) and the G i∈N∪{ 0} is deﬁned inductively by: G(0) = G and G(i) = [ G(i−1) , G (i−1) ] for all i ∈ N. It is well known that a group G is residually nilpotent (resp. residually soluble ) if and only if (i) Ti≥1 Γi(G) = {1} (resp. Ti≥0 G = {1}). We are interested in studying these series in the case that G is the braid group (resp. pure braid group) of the torus, Bn(T) (resp. Pn(T)), or of the Klein bottle, Bn(K) (resp. Pn(K)). For the braid groups of surfaces, these series have been studied in the case of the disc, sphere and the projective plane. Further, the lower central series of Bn(T) was studied in [1] where the authors show that Bn(T) is residually nilpotent if and only if n ≤ 2, and Pn(T) is residually nilpotent for all n. For K we have the same result, that Bn(K) is residually nilpotent if and only if n ≤ 2. As in the case of the torus, we conjecture that Pn(K) is residually nilpotent for all n, unfortunately, we have not been able to prove this conjecture, but we have been able to show a slightly weaker property, that Pn(K) is residually soluble for all n. We also show that Bn(T) and Bn(K) are residually soluble if and only if n ≤ 4. 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] D. L. Gonçalves and J. Guaschi, The lower central and derived series of the braid groups of the sphere, Trans. Amer. Math. Soc. 361 (2009), 3375-3399. [3] D. L. Gonçalves and J. Guaschi, The lower central and derived series of the braid groups of the projective plane, Journal of Algebra 331 (2011), 96-129. [4] K. W. Gruenberg, Residual properties of inﬁnite soluble groups. Proc. London Math. Soc. 7 (1957), 29–62.

1 Universidade Federal de São Carlos - Université de Caen Basse-Normandie, Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139. Supported by FAPESP - 2010/18930-6. [email protected]

10 Minimizing the Nielsen root classes for maps between CW-complexes and manifolds of the same dimension ≥ 3 Claudemir Aniz 1 Abstract

Given a map f : K → M, where K is a CW -complex and M a manifold, both of the same dimension n ≥ 3, and a Nielsen root class, there is a number associated to this root class, which is the minimum number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We discuss the following question: Is there a map g homotopic to f in which all classes have cardinality equal to the minimal number? We show that the question has a positive answer if f is homotopic to a map that has a Nielsen class with minimum number of points contained in the interiors of n-cells. In the particular case where K is a simplicial complex, we give a suﬃcient condition on K so that the question has a positive answer.

References

[1] D.L. Gonçalves , Coincidence theory for maps from a complex into a manifold , Topology and its Applications 92 , (1999) 63-77. [2] D.L. Gonçalves , Coincidence Theory , in Handbook of Topological Fixed Point Theory, (2005). [3] D. L. Gonçalves and C. Aniz , The minimizing of the Nielsen root classes , Central European Journal of Mathematics, (2004) 112-122. [4] H. Schirmer , Mindestzahlen von Koinzidenzpunkte , J. Reine Angew. Math. 194, (1955) 21-39. [5] L.D. Borsari and D.L. Gonçalves , A Van Kampen type theorem for coincidences , Topology and its Appli- cations 101, (2000) 149-160. [6] L.D. Borsari and D.L. Gonçalves ,Obstruction theory and minimal number of coincidences for maps from a complex into a manifold , Topological Methods in Nonlinear Analysis 21 , (2003) 115-130. [7] M.C. Fenille and O. M. Neto ,Minimal Nielsen root classes and roots of liftings , Fixed Point Theory and Applications 2009, (2009) 1-17. [8] R. Brooks , On the sharpness of the ∆2 and ∆1 Nielsen numbers , J. Reine Angew. Math. 259, (1973) 101-108. [9] R. Brooks , On Removing Coincidences of two maps when only one, rather than both, of them may be deformed by a homotopy , Paciﬁc Journal of Mathematics 39 No 3, (1971) 45-52. [10] Tsai-han Kiang ,The Theory of Fixed Point Classes , Springer-Verlag Berlin Heidelberg and Science Press Beijing, (1989).

1 Universidade Federal de Mato Grosso do Sul. [email protected]

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Using Reidemeister–Schreier algorithm Anne Bauval 1 and Claude Hayat 2

Abstract

We propose to present in a short talk a table giving Seifert invariants of ker ϕ (double covering of a Seifert manifold M deﬁned by ϕ a surjection from π1(M) to Z/2Z) in terms of the Seifert invariants of M. It is a synthesization of the Reidemeister–Schreier algorithm.

1 Institut de Mathématiques de Toulouse. [email protected] 2 Institut de Mathématiques de Toulouse. [email protected]

11 Actions of infinite groups on homotopy even spheres Sergio Martins 1 and Daciberg Lima Gonçalves 2

Abstract

In this talk we present some results about action of discrete infinite groups on homotopy even dimensional spheres. We show that any group of the form G0 o Z2, for G0 having cohomological dimension finite, has periodic Farell cohomology of period two. This is a necessary condition that a group must satisfy in order to act freely on an even homotopy sphere. Then we describe a necessary cohomological condition that a pair (G, φ ) must satisfy, where φ : G → Aut (Hn(Σ)) is the orientation action induced by the action of G, in order to have a free action of G with the prescribed orientation action φ. We show that for some pairs (G0 o Z2, φ ), where G0 is a surface group, this necessary condition is satisfied. The case where G0 has cohomological dimension one has been completely solved in [5]. We will present a brief exposition of the state of art of the problem in question, i.e. action of infinite groups on even homotopy spheres. References

[1] A. Adem, J.H. Smith , Periodic complexes and group actions , Ann. of Math. 154 (2001), 407-435. [2] K.S. Brown , Cohomology of groups , Springer-Verlag, New York-Heidelberg-Berlin, 1982. [3] ———– , The homology of Richard Thompson’s group F. Topological and asymptotic aspects of group theory , 47-59, Contemp. Math., 394, Amer. Math. Soc., Providence, RI, 2006. m n− [4] F.X. Connolly, S. Prassidis , Groups which act freely on R × S 1. Topology 28 , no. 2 (1989), 133-148. [5] M. Golasiński, D.L. Gonçalves, R. Jiménez , Free and properly discontinuous actions of groups on homotopy 2n-spheres (submitted). n m [6] J.B. Lee , Transformtion groups on S × R , Topology Appl. 53 (1993), 187–204. [7] C.T.C. Wall , Resolutions for extensions of groups , Proc. Cambridge Philos. Soc. 57 (1961), 251-255

1 Institute of Mathematic and Statistic- University of São Paulo. [email protected] 2 Institute of Mathematics and Statistic- University of São Paulo. [email protected]

12 The minimal number of Borsuk-Ulam coincidences on surfaces John Guaschi 1 and Daniel Ven drús colo 2

Abstract

Given two topological spaces X and Y such that X admits a free involution τ, we say the triple (X, τ, Y ) satisﬁes the Borsuk-Ulam property if for any continuous f : X → Y there exists a point x ∈ X such that f(x) = f(τ(x)) . In [4] such questions were studied when Y is a compact, connected surface S without boundary and the conditions obtained are presented in terms of some homomorphism from π1(X/τ ) to B2(S) (the 2-string braid group of the surface S). In [3] Gonçalves wrote: Remark: [3, 8.5] (...) perhaps the “correct” formulation of the Borsuk-Ulam question is the following. Given a triple (M, τ, Y ), which homotopy classes of maps M → Y satisfy the Borsuk-Ulam property? Observe that if Y is contractible, this is exactly the same as the original Borsuk-Ulam question. In this paper we use braid groups to describe the minimal number of Borsuk-Ulam coincidence points for maps f : M → N between surfaces (with π2(N) = 0 ). The analogous question for ﬁxed points was studied by B. Jiang ([6]) and for coincidences by J. Jezierski ([5]). The approach presented here follows these works. References

[1] D. H. Gottlieb , A certain subgroup of the fundamental group , Amer. J. Math. 87 (1965), 840–856. [2] Gonçalves, D. L.; The Borsuk-Ulam theorem for surfaces , Quaestiones Mathematicae 29, (2006), no. 1, 117-123. [3] Gonçalves, D. L.; Braid groups of surfaces and one application to a Borsuk-Ulam type theorem , Toruń, 2011 (available at http://ssdnm.mimuw.edu.pl/pliki/wyklady/Braids_FINAL.pdf; acessed at April 14, 2013). [4] Gonçalves, D. L.; Guaschi, J.; The Borsuk-Ulam theorem for maps into a surface . Topology Appl. 157 (2010), no. 10-11, 1742–1759, [5] Jezierski, J.; The least number of coincidence points on surfaces , J. Austral. Math. Soc. (Series A) 58 (1995), 27–38. [6] Jiang, B.; Surface maps and braids equations, I , Lectures Notes in Math. 139 (Springer, New York, 1989), pp. 125–141.

1 Normandie Université, UNICAEN, Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139, 14032 Caen Cedex, France [email protected] 2 Departamento de Mateática - UFSCar [email protected]

13 Continua and hyperspaces Luis Alberto Guerrero-Méndez, 1 David Her rera -Car rasco, 2 and Fernando Macías-Romero 3

Abstract

This talk will discuss topological spaces that are metric, non-empty, compact and connected, we will call such continuum spaces. We will consider some properties of continuum; continuum path connected, locally connected, unicoherence, etc. We define some families decomposable continua: finite graphs, dendrites, dendroides and indecomposable continua. Let X be a continuum, a hyperspace of X is a specified collection of subsets of X with the Hausdorff metric. We define some hyperspaces, and look at properties of the hyperspaces well as models of them. References

[1] R. Escobedo, S. Macías, H. Méndez , Invitación a la Teoría de los Continuos y sus Hiperespacios , Aporta- ciones Matemáticas, Serie Textos N. 31, Sociedad Matemática Mexicana, ISBN: 970-32-3872-6, 2006. [2] A. Illanes , Hiperespacios de continuos , Aportaciones Matemáticas, Serie Textos N. 28, Sociedad Matemática Mexicana, ISBN: 968-36-3594-6, 2004. [3] A. Illanes, S. B. Nadler Jr. , Hyperspaces Fundamentals and Recent Advances, Monographs and Textbooks in Pure and Applied Math., Vol. 216, Marcel Dekker, Inc., New York, 1999. [4] F. Leon Jones , Historia y Desarrollo de la Teoría de los Continuos Indescomponibles , Aportaciones Matemáti- cas, Serie Textos N. 27, Sociedad Matemática Mexicana, ISBN: 970-32-2130-0, 2004. [5] S. B. Nadler, Jr. , Continuum Theory: An Introduction , Monographs and Text books in Pure and Applied Math., Vol. 158, Marcel Dekker, Inc., New York, 1992. [6] S. B. Nadler, Jr. , Hyperspaces of Sets , Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, Inc., New York, 1978.

1 Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla. [email protected] 2 Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla. [email protected] 3 Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla. [email protected]

F 8 f

Lefschetz fibrations on adjoint orbits Elizabeth Gasparim 1

Abstract

We will show that adjoint orbits of semisimple Lie algebras admit the structure of symplectic Lefschetz fibrations. We will describe the topology of the regular and singular fibres in terms of Lie theory. This is joint work with L. Grama and Luiz A. B. San Martin. Then we will discuss some delicate aspects of Hodge diamonds of fibrewise compactifications of these fibrations. This second part is joint work with Brian Callander. References

[1] E. Gasparim, L. Grama, L. A. B. San Martin , Lefschetz ﬁbrations on adjoint orbits , arXiv:1309.4418. [2] B. Callander, E. Gasparim , Hodge diamonds and adjoint orbits , arXiv:1311.1265.

1 Departamento de Matemática, Imecc Unicamp. [email protected]

14 The Reeb graph of a map germ from R3 to R2 Er ica Boizan Batista, 1 João Carlos Ferreira Costa, 2 and Juan José Nuño Ballesteros 3

Abstract

The topological type of a finitely determined map germ f : ( R3, 0) → (R2, 0) is given by the so- called link of f. The link of f is obtained by taking a small enough representative f : U ⊂ R3 → R2 1 R2 and the intersection of its image with a small enough sphere Sδ centered at the origin in . As a consequence of Fukuda’s theorem, two finitely determined map germs f, g : ( R3, 0) → (R2, 0) are topologically equivalent if their associated links are topologically equivalent. Inspired by the works of Arnold, Maksymenko and Prishlyak, we introduce an adapted version of the Reeb graph that turns out to be a complete topological invariant for the links. We give special attention to the case where f has corank 1. Moreover, we give a complete description of those map germs with Boardman symbol Σ2,1 and we provide a complete topological classification of this type of map germs up to multiplicity 6. References

[1] E. B. Batista , J. C. F. Costa , J. J. Nuño-Ballesteros , The Reeb graph of a map germ from R3 to R2 with isolated zeros , preprint. [2] E. B. Batista , J. C. F. Costa , J. J. Nuño-Ballesteros , The Reeb graph of a map germ from R3 to R2 with non isolated zeros , preprint. [3] J.A. Moya-Pérez, J.J. Nuño-Ballesteros, The link of finitely determined map germ from R2 to R2, J. Math. Soc. Japan 62, no. 4 (2010) 1069–1092. [4] J.A. Moya-Pérez, J.J. Nuño-Ballesteros, Gauss words and the topology of map germs from R3 to R3, Preprint (2013), available at http://www.uv.es/nuno [5] J.A. Moya-Pérez, J.J. Nuño-Ballesteros, Topological classification of corank 1 map germs from R3 to R3, to appear in Rev. Mat. Complut. (2013), doi:10.1007/s13163-013-0137-z [6] S.A. Izar, Funções de Morse e topologia das superfícies I: O grafo de Reeb de f : M → R, Métrica no. 31, Estudo e Pesquisas em Matemática, IBILCE, Brazil, 1988. (Available at http://www.ibilce.unesp.br/Home/ Departamentos/Matematica/metrica-31.pdf ) [7] S.A. Izar, Funções de Morse e topologia das superfícies II: Classificação das funções de Morse estáveis sobre superfícies , Métrica no. 35, Estudo e Pesquisas em Matemática, IBILCE, Brazil, 1989. (Available at http: //www.ibilce.unesp.br/Home/Departamentos/Matematica/metrica-35.pdf ) [8] S.A. Izar, Funções de Morse e topologia das superfícies III: Campos pseudo-gradientes de uma função de Morse sobre uma superfície , Métrica no. 44, Estudo e Pesquisas em Matemática, IBILCE, Brazil, 1992. (Available at http://www.ibilce.unesp.br/Home/Departamentos/Matematica/metrica-44.pdf )

1 Universidade Estadual Paulista. [email protected] 2 Universidade Estadual Paulista. [email protected] 3 Universitat de València [email protected]

15 Uniqueness of hyperspaces in a continuum Luis Alberto Guerrero-Méndez, 1 David Herrera-Carrasco, 2 and Fer nando Macías -Romero 3

Abstract

As we see in this talk, the theory of uniqueness of hyperspaces in a continuum is extensively researched, the n-th symmetric product one such example. See also Palestra/Talk 9 for some of the basic definitions. A continuum is a compact connected metric space, with more than one point. The set of positive integers is denoted by N. Given a continuum X and n ∈ N, we consider the following hyperspaces of X: 2X = {A ⊂ X : A is a nonempty closed subset of X}, X Cn(X) = {A ∈ 2 : A has at most n components }, and X Fn(X) = {A ∈ 2 : A has at most n points }; all these hyperspaces are metrized by the Hausdorff metric. The hyperspaces Fn(X) and Cn(X) are called the n-th symmetric product of X and the n-fold hyperspace of X, respectively. The n-fold hyperspace suspension of a continuum X is defined as the quotient space Cn(X)/F n(X) which is obtained from Cn(X) by identifying Fn(X) into a one-point set, denoted by HS n(X). Let H(X) be any one of the hyperspaces defined above. We say that a continuum X has unique hyperspace H(X) provided that the following implication holds: if Y is a continuum and H(X) is homeomorphic to H(Y ), then X is homeomorphic to Y. Problem. Find conditions, on the continuum X, in order that X has unique hyperspace H(X). We shall study this problem, and obtain full or partial solutions, in various special cases of continua such as finite graphs, dendrites, almost framed and framed continua, and wires. References

[1] G. Acosta, D. Herrera-Carrasco, F. Macías-Romero , Local dendrites with unique hyperspace C(X), Topology Appl. 157 (2010), 2069–2085. [2] L. A. Guerrero-Méndez, D. Herrera-Carrasco, María de J. López, F. Macías-Romero , Meshed continua have unique second and third symmetric products, sent to Topology Appl. [3] D. Herrera-Carrasco, A. Illanes, M. de J. López, F. Macías-Romero , Dendrites with unique hyperspace C2(X), Topology Appl. 156 (2009), 549–557. [4] D. Herrera-Carrasco, A. Illanes, F. Macías-Romero, F. Vázquez-Juárez , Finite graphs have unique hyperspace HS n(X), Topology Proc. 44 (2014), 75-95. [5] D. Herrera-Carrasco, M. de J. López, F. Macías-Romero , Dendrites with unique symmetric products, Topology Proc. 34 (2009), 175–190. [6] D. Herrera-Carrasco, M. de J. López, F. Macías-Romero , Framed continua have unique n-fold hyperspace suspension , sent to Topology and Its Applications. [7] D. Herrera-Carrasco, F. Macías-Romero , Dendrites with unique n-fold hyperspace, Topology Proc. 32 (2008), 321–337. [8] D. Herrera-Carrasco, F. Macías-Romero , Local dendrites with unique n-fold hyperspace, Topology Appl. 158 (2011), 244–251. [9] D. Herrera-Carrasco, F. Macías-Romero, F. Vázquez-Juárez , Peano continua with unique symmetric products, Journal of Mathematics Research 4(4) (2012), 1–9.

16 Realizing covering maps as holonomies Gabriel Calsamiglia 1

Abstract

We will discuss the problem of extending holonomy maps associated to a foliation induced by a complex polynomial plane vector ﬁeld

P (x, y )∂x + Q(x, y )∂y in a continuous or analytic way. A topological construction will be given that allows one to ﬁnd some covering maps among the holonomy maps for a generic vector ﬁeld. From the construction we will deduce that these holonomy maps cannot be extended continuously on an uncountable number of points, thus giving a negative answer to a question posed by Ilyashenko [2] and F. Loray [3]. References

[1] G. Calsamiglia, B.Deroin, S. Frankel and A. Guillot , Singular sets of holonomy maps for algebraic foliations. , J. Eur. Math. Soc. (JEMS) 15 (2013), no. 3, 1067?1099. [2] Ilyashenko, Yu. , Centennial history of Hilbert’s 16th problem. , Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, p.301-354 [3] Loray, F. , Sur les théorèmes I et II de Painlevé. , Contemporary Mathematics 389 (2005) p. 165-190.

1 IME, Universidade Federal Fluminense [email protected]

17 Smale flows on S2 × S1 Ketty de Rezende 1 and Guido Ledesma 2

Abstract

In this conference the Lyapunov graphs are used as a combinatorial tool in order to obtain a complete classification of Smale flows on S2 × S1. This classification consists in determining necessary and sufficient conditions that must be satisfied by an abstract Lyapunov graph so that it is associated to a Smale flow on S2 × S1. In summary we got the following results: (1) The local conditions that must be satisfied by each vertex on a Lyapunov graph is determined as well as the global conditions on the graph in order for it to be associated to a Smale flow on S2 × S1. (2) The realization of these graphs subject to the conditions found above, as Smale flows on S2 × S1, is obtained. References

[1] Ch. Bonatti and V. Grines , Knots as topological invariants for gradient-like diffeomorphisms of the sphere S3, Journal of Dynamical and Control Systems 6 (2000), 579–602. [2] C. Conley , Isolated invariant sets and the Morse index , AMS Bookstore 38 (1978). [3] C. Cruz and K. de Rezende , Cycle rank of Lyapunov graphs and the genera of manifolds , Proceeding of the American Mathematical Society 126 (1998), 3715–3720. [4] K. A. de Rezende , Gradient-like flows on 3-manifolds , Ergodic Theory and Dynamical Systems. 303 (1987), 557–580. [5] K. A. de Rezende , Smale flows on the three-sphere , Transactions of the American Mathematical Society. 303 (1987), 557–580. [6] J. Franks , Homology and dynamical systems , AMS Bookstore 49 (1982). [7] J. Franks , Nonsingular Smale flows on S3, Topology 24 (1985), 265–282. [8] B. Yu , Lyapunov graphs of nonsingular Smale flows on S2 × S1, Transactions of the American Mathematical Society. 365 (2012), 767–783.

1 Universidade Estadual de Campinas. [email protected] 2 Universidade Estadual de Campinas. [email protected]

18 Bordism and coincidences in codimension one Gustavo de Lima Prado 1 Abstract

n+1 n In this work, we study coincidences between two maps f1, f 2 : X → Y , where X, Y are smooth manifolds, connected, being X closed (compact and without boundary) and Y without boundary. We say that x ∈ X is a coincidence of (f1, f 2) if f1(x) = f2(x). In codimension one we have that the coincidence set Coin (f1, f 2) is generically a submanifold of X of dimension one. We 0 0 0 0 0 say that (f1, f 2) is loose if there exists (f1, f 2) such that fi ' fi and Coin (f1, f 2) = ∅. In [4] and [5], U. Koschorke introduces the normal bordism invariant ω˜(f1, f 2) ∈ Ω1(E(f1, f 2);ϕ ˜ ), involving the submanifold Coin (f1, f 2), that, in particular, is null when the pair of maps, (f1, f 2), is loose. In n+1 the codimension one case, when the domain is the sphere ( X = S ), and when (f1, f 2) = ( y0, f ) (where y0 denotes a constant map), we can decompose ω˜(y0, f ) into two parts involving two new 0 00 invariants ω (f) ∈ Z2 and ω (f) ∈ π2(Y ) which together determine ω˜(y0, f ). Also because the n+1 00 domain is the sphere S , we can see ω as a homomorphism from πn+1 (Y ) to π2(Y ). In this 00 work, we calculate ω : πn+1 (Y ) → π2(Y ) for several spaces and, in particular, for sphere bundles over spheres we obtain that ω00 = 0 if and only if Y is trivial or Y is not an S2-bundle over S4. Now, a Wecken type result is a result which asserts that, in some dimensions, the minimum number of components of the coincidence set is equal to the Nielsen number, that is, MCC (f1, f 2) = N˜(f1, f 2) for all pairs of maps. In [5], U. Koschorke obtains Wecken type results for n = 1 and for the stable dimension (which, in the codimension one case, means n > 4). In this work, in codimension one, we obtain, when the domain is the sphere ( X = Sn+1 ), that there exists a Wecken type result for n = 2 , 3. Hence, in codimension one, when the domain is the sphere, there exists a Wecken type result for every dimension. References

[1] R. F. Brown , The Lefschetz fixed point theorem , Scott, Foresman and Co. (1971). [2] A. Dold and D. L. Gonçalves , Self-coincidence of fibre maps , Osaka J. Math. 42 (2005), 291–307. [3] U. Koschorke , Vector fields and other vector bundle morphisms. A singularity approach , Lecture Notes in Math. 847 (1981). [4] U. Koschorke , Selfcoincidences in higher codimensions , J. Reine Angew. Math. 576 (2004), 1–10. [5] U. Koschorke , Nielsen coincidence theory in arbitrary codimensions , J. Reine Angew. Math. 598 (2006), 211–236. [6] H. Toda , Composition methods in homotopy groups of spheres , Annals of Mathematics Studies, Princeton University Press 49 (1962).

1 University of São Paulo. [email protected]

19 Volume entropy for minimal presentations of surface groups in all ranks Luis Alseda, David Juher, Jérôme Los, 1 and Francesc Manosas

Abstract

We study the volume entropy for classical presentations of all surface groups and we rediscover a formula obtained ﬁrst by Cannon and Wagreich, the computation being in an unpublished manuscript by Cannon. The result is surprising: an explicit polynomial of degree r, the rank of the group, encodes the volume entropy of all classical presentations of surface groups. The approach we use is completely diﬀerent, it is based on a dynamical system construction following an idea due to Bowen and Series and extended recently to all geometric presentations. The result is an explicit formula for the volume entropy of classical presentations for all surface groups, showing a polynomial dependence in the rank r > 2. We prove that for a surface group Gr of rank r with a classical presentation Pr the volume entropy is log( λr), where λr is the unique real root larger than one of the polynomial : r r−1 j Qr(x) := x − 2( r − 1) Pj=1 x + 1 .

1 Universitat Autonoma de Barcelona, Aix Marseille University, CNRS. [email protected]

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The number of strings on essential tangle decompositions of a knot can be unbounded João Miguel Nogueira 1

Abstract

Tangle decompositions of knots have been shown to be relevant in knot theory, 3-manifold topology, and their applications. In particular, essential tangle decompositions extend the concept of connected sum and were used, for instance, to prove that all knots are concordant to a prime knot. In this talk we show the existence of inﬁnitely many (prime) knots, each of which has n-string essential tangle decompositions for any given n (resp., n>1). References

[1] J. M. Nogueira , The number of strings on essential tangle decompositions of a knot can be unbounded , DMUC 13-45 Preprint.

1 University of Coimbra [email protected]

20 Virtual knot theory José Gregorio Rodríguez Nieto 1

Abstract

Classical knot theory studies embeddings of S1 into S3 modulo ambient isotopy. Many generalizations of this area have been proposed but one which attracted the most attention was the virtual knots introduced by Kauffman in 1999 [7]. This “new” theory also relies on planar diagrams (like knot projections) but interpretations as knots in thickened surface have been done, see for example [4], [6] and [3]. As soon as virtual knot theory was proposed several questions arose, one important one is, when a virtual knot is a classical knot or not , this question is known in the literature as the recognition problem of virtual knots . It has been studied in [5], [9], [10] and [11] In this talk we give a short introduction to the subject of virtual knots through the concept of diagrams on S2, also, for a virtual knot diagram K I use the Carter algorithm [2] and abstract 3 knots diagrams [8] to construct a closed, connected, compact and oriented surface ΣK ⊂ R , of minimal genus, in which K may be realized as a kind of diagram on ΣK . After that, we use a generating set of the first homology group of ΣK , defined by Cairns and Elton [1], and homology intersection to give conditions to determine when a virtual knot is a classical knot or not. These conditions, modulo a special equivalence relation, define a powerful invariant, similar to that of Turaev in [12] of virtual knots that is able to determine, in many cases, when a virtual knot is not a classical knot. References

[1] Cairns, G. and Elton, D. The Planarity Problem for Signed Gauss Words , J. Knot Theory Ramifications , 2, No. 4 (1993), 359-367. [2] Carter, J. Classifying Immersed Curves, Proc. Amer. Math. Soc. 111, No. 1 (1991), 281-287. [3] Carter, S. Kamada, S. and Saito, M. Stable Equivalence of Knots on surfaces and Virtual Knots Cobordism, J. Knot Theory Ramifications, 11 , No. 6 (2002), 311-320. [4] Cotta-Ramusino, P. y Rinaldi, M. On the Algebraic Structure of Link-Diagrams on a 2-Dimensional Surface , Communications in MAthematical Physics, 131, 137-173 (1991). [5] Dye, H. A., Detection and Characterization of Virtual Knot Diagrams , Ph. D. Thesis. University of Illinois at Chicago, 2003. [6] Dye, H. and Kauffman, L. Minimal Surface representations of virtual Knots and Links, Algebraic and Geometric Topology, Vol. 5 (2005), 509-535. [7] Kauffman, L. “Virtual Knot theory”, Europ. J. Combinatorics ,Vol. 20 (1999), 663-691. [8] Kamada, N. and Kamada, S. Abstract Links Diagrams and Virtual Knots, J. Knot Theory Ramifications, Vol. 9, No. 1 (2000), 93-106. [9] Rodríguez, J. Nudos Virtuales , Tesis Doctoral, Universidad Nacional de Colombia. 2011. [10] Rodríguez, J. and Toro, M., “Virtual Knot Groups and Combinatorial Knots”, Sao Paulo Journal of Mathe- matical Sciences 3, 1 (2009), 297–314. [11] Toro, M. and Rodríguez, J., Tripletas Asociadas a Diagramas de Nudos Virtuales , Revista Integración, Vol. 29 , N ° 2, (2011) 97-108. [12] Turaev, V, “Cobordism of knots on surfaces”, Journal of Topology , 1, No 2 (2008), 285-305.

1 National University of Colombia, Medellín. [email protected]

21 The Boyland order for maximal horseshoe braid types Juan Valentín Mendoza 1 Abstract

In one-dimensional dynamics, the Sharkovsky order determines which periodic orbits have a unimodal map f: if m is a period for f then n is also a period of f provided that m is greater than n in the Sharkosky order. For a surface diffeomorphism f, the period is not enough to decide the existence of certain orbits. In this case we must look at the braid type bt( P ) of the periodic orbit P , which is given by the conjugacy class of f by isotopies relative to P . So we say that a braid type β forces a braid type γ, denoted by β >2 γ, if every homeomorphism f containing a periodic orbit with braid type β, must contain a periodic orbit Q with braid type γ. In [1] Boyland proved that >2 is a partial order. This is the context where our work begins. So we will study how the Boyland order organizes the braid types of certain Smale horseshoe orbits. We will define the notion of decoration , introduced by T. Hall in [3], and it will be proved that, for braid types with maximal decorations, the Boyland order coincides with the unimodal order. Our main tool is differentiable pruning as in [2]. References

[1] P. Boyland , Topological methods in surface dynamics . Topology and its Applications, 54 , 223–298 (1994). [2] A. de Carvalho and V. Mendoza , Diﬀerentiable pruning and the hyperbolic pruning front conjecture . Pre-print. (2013) [3] T. Hall , The creation of horseshoes . Nonlinearity. 7, 861–924 (1994). [4] A. Sharkovsky , Coexistence of cycles of a continuous mapping of the line into itself . Ukrain. Mat. Z. , 16 , 61–71 (1964).

1 Universidade Federal de Viçosa [email protected]

22 Σ1-invariant for Artin groups of circuit rank 1 and 2 Kisnney Almeida 1

Abstract

The Σ1-invariant is a geometric invariant of ﬁnitely generated groups that can be used to decide if a given subgroup containing the commutator subgroup is ﬁnitely generated [2]. Artin groups are a group-theoretical generalization of braid groups which are combinatorially constructed from underlying graphs. They form a large class of groups, and are a source of interesting examples in geometry and group theory. Meier, Meinert and VanWyk have obtained a partial description of Σ1 of Artin groups [4]. Let the circuit rank of an Artin group be the free rank of the fundamental group of its underlying graph. Meier, in a previous work, obtained a complete description for Artin groups of circuit rank 0, i.e., whose underlying graphs are trees. We will talk about some ideas we have used, in joint work with Kochloukova, to prove the same description to be true for Artin groups of circuit rank 1 [1] and about my current research that attempts to prove this for Artin groups of circuit rank 2. References

1 [1] Almeida, K.; Kochloukova, D. The Σ -invariant for Artin groups of circuit rank 1 , Forum Mathematicum (2013). [2] Bieri, R.; Neumann, W. D.; Strebel, R. A geometric invariant of discrete groups , Invent. Math 90 (1987), 451-477. [3] Meier, J. Geometric invariants for Artin groups , Proc. London Math. Soc. (3) 74 (1997) 151-173. [4] Meier, J.; Meinert, H.; VanWyk, L. On the Σ-invariants of Artin Groups , Topology and its Applications, 110 (2001), 71-81.

1 Universidade Estadual de Feira de Santana. [email protected]

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The complete sphere complex of an irreducible 3-manifold and Heegaard splittings: the case of a connected sum of S2 × S1’s and handlebodies Ulrich Oertel 1 and Leonardo N. Car valho 2

Abstract

A standard strategy for studying a 3–manifold consists of splitting it into simpler pieces. A Heegaard splitting gives such an example, splitting the manifold into handlebodies (and compression bodies, in the case with boundary). Another is the prime decomposition , which splits the manifold into (holed) irreducible pieces and S2×S1’s. We consider and relate both for the case of a connected sum of copies of S2 ×S1’s and handlebodies. In fact by going further on in the prime decomposition one can consider complete systems of spheres and to these associate the complete sphere complex of the manifold. This encodes a notion of proximity for systems. By regarding Heegaard splittings as somehow dual to sphere systems we can obtain information about one through the other. We illustrate it with an uniqueness result on symmetric Heegaard splittings. References

[1] L. Carvalho, U. Oertel , A classiﬁcation of automorphisms of compact 3-manifolds , arXiv:math/0510610. [2] A. Hatcher , Homological stability for automorphism groups of free groups , Comment. Math. Helv. 70 (1995), 39–62.

1 Rutgers University–Newark [email protected] 2 UFF–Niterói. [email protected]

23 Morita equivalences, equivariant bundles and exotic spheres Llohann D. Sperança 1

Abstract

We present a relation between G-manifolds which we call Morita equivalence, with respect to the homonymous equivalence between transformation groupoids. In practice, M and M 0 are Morita equivalent if and only if there is a ‘biprincipal’ bundle P ﬁtting into the following diagram: G

• ? π0 G P / M 0

π M Interesting geometrical properties of M 0 can be described through M. Here we calculate some speciﬁc Morita classes by looking at π as an equivariant principal bundle and show diﬀerential-topological implications on M 0 by imposing natural restrictions on P . References

[1] L. D. Sperança , Pulling back the Gromoll-Meyer construction and models of exotic spheres , arXiv:1010.6039

1 Universidade Federal do Paraná. [email protected]

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Reidemeister metric and Ray–Singer metric for pseudomanifolds with conical singularity Luiz Hartmann 1 Abstract

The Reidemeister metric was defined for a smooth manifold by J.-M. Bismut and W. Zhang [1]. With this definition, they generalized the Cheeger–Müller theorem. We will present the motivation of the Reidemeister Metric and the Ray–Singer Metric, and a possible extension to the singular case. This is a joint work with Mauro Spreafico. References

[1] J.-M. Bismut and W. Zhang, An extension of a theorem by Cheeger and Müller , Astérisque 2005, 1992.

1 Universidade Federal de São Carlos, UFSCar, Brasil. [email protected]

24 On the group structure of [Ω S2, ΩY ] Marek Go lasiński, 1 Daciberg L. Gonçalves, 2 and Peter Wong 3

Abstract

Let J(X) denote the James construction on a space X and Jn(X) be the n-th stage of the ∼ James ﬁltration of J(X). It is known that [J(X), ΩY ] = lim [Jn(X), ΩY ] for any space Y . When ← X = S1, the circle, J(S1) = ΩΣ S1 = Ω S2. Furthermore, there is a bijection between [J(S1), ΩY ] and the product Qi≥2 πi(Y ), as sets. Since the groups [Jn(X), ΩY ] can be embedded as subgroups of the Fox torus homotopy groups τn+1 (Y ), the group structure of [Jn(X), ΩY ] is induced by that of τn+1 (Y ) which is described by means of the classical Whitehead products. We obtain the group 1 1 structure of [Jn(S ), ΩY ] by determining the co-multiplication structure on the suspension ΣJn(S ) and then via the torus homotopy group structures together with a recent result by Arkowitz and Lee on the co- H structures of a wedge of spheres. To describe the coeﬃcients of Whitehead products, we make use of the combinatorial argument of Fox to determine whether certain Whitehead products are trivial.

1 Toruń, Poland. [email protected] 2 São Paulo, Brazil. [email protected] 3 Lewiston, USA. [email protected]

25 Bifurcations associated to spectral sequences Dahisy Lima, 1 Ketty de Rezende, 2 and Mar iana Sil veira 3

Abstract

We are interested to extract dynamical information from algebraic topological tools found in Morse theory and Conley’s index theory [1, 5]. The problem of providing a qualitative description of a flow in a compact manifold can be divided in two parts: the description of the invariant sets and the description of how these sets are connected to each other. In this sense, the Conley index provides a topological description of the local dynamics around the invariant sets. In order to describe the connections between the invariant sets, we can use a function to construct a filtration and the connections will be reflected on the relative topology of the sets determined by the filtration. Important tools for such description are connection matrices [3] and the spectral sequences. In this talk we consider a chain complex (C, ∆) generated by the critical points of a Morse function f on a compact manifold M endowed with an increasing filtration. Its differential, which is a particular case of connection matrix, determines an associated spectral sequence (Er, d r). In this context, we introduce a sweeping algorithm which codifies, in the connection matrix ∆, the information of the associated spectral sequence [2]. The purpose of this algebraic procedure is to obtain dynamical results about the gradient flow of f exploring the algebraic information of the spectral sequence (Er, d r). The sweeping algorithm produces a family of similar connection matrices and associated transition matrices which codifies the information given by the spectral sequence and recovers dynamical information of the initial flow. For example, in [CdRS] one shows the existence of certain paths in the flow associated to the nonzero differentials of the spectral sequence. In this talk we present another dynamical meaning of the algebraic information codified by the sweeping method. We introduce in [4] directed graphs called schematics that depict the bifurcation that could occur if the sequence of connection matrices and transition matrices were realized in a flow continuation. In this way, a sequence of schematics can be seen as a continuation where the transition matrices give the information about the bifurcation behavior. References

[1] C. Conley , Isolated Invariant Sets and the Morse Index. CBMS Regional Conference Series in Math. AMS, Providence, RI 38 (1978). [2] O. Cornea, K.A. de Rezende and M. R. da Silveira , Spectral sequences in Conley’s theory. Ergodic Theory and Dynamical Systems 30 (4) (2010) 1009–1054 . [3] R. Franzosa , The Connection Matrix Theory for Morse Decompositions, Transactions of the American Math- ematical Society 311 (1989) 561-592. [4] R. D. Franzosa, K. A. de Rezende and M. R. da Silveira, Continuation and bifurcation associated to the dynamical spectral sequence. To appear in Ergodic Theory Dynamical Systems. [5] D. A. Salamon , The Morse theory, the Conley index and the Floer homology, Bull. London Math. Soc. 22 (1990) 113–240.

1 Universidade Estadual de Campinas. [email protected] 2 Universidade Estadual de Campinas. [email protected] 3 Universidade Federal do ABC. [email protected]

26 Fixed point index bounds and a class of negatively curved 2-complexes Daciberg L. Gonçalves 1 and Michael Kelly 2

Abstract

Given a self-map of a compact, connected 2-complex we consider the problem of determining upper and lower bounds for fixed point indices. One can not expect to have bounds in general, so we restrict attention to two natural cases; (1) the indices for the Nielsen fixed point classes, and (2) the indices of isolated fixed points where the self-map is assumed to be fixed point minimal among maps in its homotopy class. The solution to this problem in the case where the complex is a compact surface with nonempty boundary is known: In either case, the upper bound is +1 and the lower bound for the sum of all fixed points of index less than -1 is given by P(index( C)+1) ≥ − 2χ , where χ denotes the Euler characteristic of the surface. The same Nielsen class bounds are also known to hold for closed surfaces with non-positive Euler characteristic. In this talk we consider some constructions on the spaces involved and compute the effect on the fixed point indicies. In joint work with Daciberg L. Gonçalves we produce a relative version of the methods used in the surface case to extend results to a class of 2-complexes for which the exact same index bounds hold. This includes wedges of surfaces and the identification of surfaces along a simple closed curve. References

1 USP - São Paulo [email protected] 2 Loyola University–New Orleans [email protected]

27 On representation of the Reeb graph as a sub-complex of a manifold Marek Kaluba, 1 Wacław Marzantowicz, 2 and Nel son Silva 3

Abstract

The Reeb graph R(f) is one of the fundamental invariants of a smooth function f : M → R with isolated critical points. It is deﬁned as the quotient space M/ ∼ of the closed manifold M by a relation that depends on f. Here we construct a 1-dimensional complex Γ( f) embedded into M which is homotopy equivalent to R(f). As a consequence we show that for every function f on a manifold with ﬁnite fundamental group, the Reeb graph of f is a tree. If π1(M) is an abelian group, or more generally, a discrete amenable group, then R(f) contains at most one loop. Finally we prove that the number of loops in the Reeb graph of every function on a surface Mg is estimated from above by g, the genus of Mg. References

[1] Benedetti, B. and Lutz F.H. Random Discrete Morse Theory and a New Library of Triangulations arXiv:1303.6422, 2013. [2] Biasotti, S., Giorgi, D., Spagnuolo, M. and Falcidieno, B. Reeb graphs for shape analysis and applications , Theoret. Comput. Sci. 392 (2008), 5-22. [3] Bollobás, B. and Riordan, O. A polynomial invariant of graphs on orientable surfaces , Proc. London Math. Soc. 83 (2001), no. 3, 513-531. [4] Cole-McLaughlin, K., Edelsbrunner, H., Harer, J., Natarajan, V. and Pascucci, V. Loops in Reeb graphs of 2-manifolds , Discrete Comput. Geom. 32 (2004), 231-244. [5] Dey, T. K. and Wang, Y. Reeb graphs: Approximation and persistence , Discr. & Comput. Geom. 2013 Volume 49 , Issue 1, pp 46-73. [6] Katok, A. and Hasselblatt, B. Introduction to the Modern Theory of Dynamical Systems , Encyclopedia of Mathematics and its Applications, vol. 54 , Cambridge University Press, 1995. [7] Kronrod, A.S. On functions of two variables , Uspekhi Mat. Nauk 5 (1950), no. 1, 24-134. [8] Milnor, J. Morse Theory . Princeton University Press, (1963). [9] Nitecki, Z. Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms , The M.I.T. Press, Cambridge, Mass.-London, 1971. [10] Nowak, P.W. and Yu, G. Large scale geometry , European Mathematical Society (EMS), Zürich, 2012. [11] Masumoto, Y. and Saeki, O. Smooth function on a manifold with given Reeb graph , Kyushu J. of Math. 65 no. 1, 75-84, (2011). [12] Ore, O. Theory of graphs , Amer. Math. Soc. Colloq. Publ., Vol. 38 , 1962. [13] Prishlyak, A. Topological equivalence of smooth functions with isolated critical points on a closed surface , Topology Appl. 119 (2002), no. 3, 257-267. [14] Reeb, G. Sur les points singuliers d’une forme de Pfaff complétement intégrable ou d’une fonction numerique , C. R. Acad. Sci. Paris 222, 847-849, (1946). [15] Seifert, H. and Threlfall, W. Variationsrechnung im Grossen, (Theorie von Marston Morse) , Leipzig and Berlin, Teubner, 1938. pp. 115 [16] Sharko, V.V. About Kronrod-Reeb Graph of a function on a manifold , Methods of Functional Analysis and Topology Vol. 12 (2006), no. 4, pp. 389-396.

1 WMI - UAM - Poznań/Poland. [email protected] 2 WMI - UAM - Poznań/Poland. [email protected] 3 ICMC - USP - São Carlos/SP. [email protected]

28 Representing homotopy classes by maps with certain minimality root properties Oziride Manzoli Neto 1 and Northon Canevari Leme Pen teado 2

Abstract

1 2 In this work we show how to represent the elements of π2(S ∨ S ) by special kinds of maps which have some minimality root properties. References

[1] R. Brooks , Roots of mappings from manifolds , Fixed Point Theory Appl. 4 (2004), 273–307. [2] F. J. Davis, P. Kirk , Lectures Notes in Algebraic Topology , American Mathematical Society. 55-XX , (1991). [3] J. R. Munkres , Elementary Diﬀerential Topology , Princeton University Press. Study 54 (1966). [4] J. R. Munkres , Elements of Algebraic Topology , Westview Press. (1984). [5] G. W. Whitehead , Elements of Homotopy Theory , Springer-Verlag Berlin Heidelberg New York. (1978).

1 Universidade de São Paulo–ICMC São Carlos. [email protected] 2 Universidade de São Paulo–ICMC São Carlos. [email protected]

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Connected components of regular fibers of differentiable maps Jorge T. Hiratuka 1 and Os amu Saeki 2

Abstract

For a smooth map between smooth manifolds, the space of the connected components of its ﬁbers is called the Stein factorization. In our previous work, we showed that for generic smooth maps, the Stein factorizations are triangulable. As an application, we show that every connected component of a regular ﬁber is null-cobordant if the top dimensional homology of the Stein factorization vanishes. References

[1] J.T. Hiratuka and O. Saeki, Connected components of regular fibers of differentiable maps , Topics on Real and Complex Singularities, Proceedings of the 4th Japanese-Australian Workshop (JARCS4), Kobe 2011, pp. 61–73, World Scientific, 2014.

1 Universidade de São Paulo. [email protected] 2 Kyushu University. [email protected]

29 Conjugacy classes of torsion elements in the crystallographic group Bn/[Pn, P n] Daciberg Lima Gonçalves, 1 John Guaschi, 2 and Os car Ocampo 3

Abstract

Let Bn (resp. Pn) denote the Artin braid group (resp. the Artin pure braid group) with n B strings and let n ≥ 3. We show that the quotient n is a crystallographic group, where [Pn, P n] [Pn, P n] means the commutator subgroup of Pn. This quotient has torsion elements in contrast to the (pure) braid groups Pn and Bn. We classify the torsion elements and its conjugacy classes B B in the crystallographic group n . Finally, for n ≤ 7 we show that n does not have [Pn, P n] [Pn, P n] non-abelian ﬁnite subgroups. The case n > 7 seems to be an open question, or possibly the classiﬁcation of all non-abelian subgroups of Bn/[Pn, P n] for n > 7 can be more general.

1 Universidade de São Paulo. [email protected] 2 Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139, Université de Caen. [email protected] 3 Universidade Federal da Bahia. [email protected]

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Twisted conjugacy in certain PL-homeomorphism groups of the reals D. L. Gonçalves, 1 P. Sankaran, 2 and R. Strebel 3

Abstract

Let φ : G → G be any endomorphism group. One has an equivalence relation ∼φ, called the −1 φ-twisted conjugacy, defined as x ∼φ y if y = zyφ (z ). The equivalence classes are called the φ-twisted conjugacy classes. One says that G has the R∞-property if there are infinitely many φ-twisted conjugacy classes for all φ ∈ Aut (G). Let G be a finitely generated group of PL-homeomorphisms of an interval [0 , ` ]. One says that G is irreducible if there is no G-fixed point in (0 , ` ). One has the characters χj : G → R, j = 0 , `, which sends f ∈ G to the logarithm of the slope of f near j. One says that χ0 and χ` are independent if χ0(ker( χ`)) equals the group χ0(G) and similarly χ`(ker( χ0)) equals χ`(G). One says that χ0 and χ` are unrelated if χ0(ker( χ`)) and χl(ker( χ0)) are finite index subgroups of χ0(G) and χ`(G) respectively. We introduce the notion of transcendence of a character of a finitely generated group. Using this notion, and basic facts concerning the sigma-theory due to Bieri, Neumann, and Strebel, we obtain a criterion for a finitely generated irreducible group G of PL-homeomorphisms of an interval [0 , ` ] to have the R∞ property. Using this criterion we shall see that a large class groups, which include any group commensurable to a generalized Thompson group of type F, have the R∞-property. This talk is based on joint work with Daciberg Gonçalves and Ralph Strebel.

1 University of São Paulo, São Paulo. [email protected] 2 Institute of Mathematical Sciences, Chennai. [email protected] 3 University of Freiburg, Switzerland. strebel@

30 Typical foliations Paulo Gusmão 1 Abstract

In this talk we will present the classifying space associated to a foliation and some results and questions about typical foliations in 3-manifolds.

1 Federal Fluminense University. [email protected]

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A generalization of Novikov’s Theorem on the existence of Reeb components in codimension one foliations Fernando Alcalde Cuesta, 1 Gilbert Hector, 2 and Paul A. Schweitzer, S.J. 3

Abstract

We study the structure and existence of generalized Reeb components in codimension one foliations. We show that under a certain condition a connected homological (m − 2) -dimensional vanishing cycle in a C2 transversely oriented codimension one foliation F of a closed m-manifold M lies on the boundary of a homological Reeb component. This extends Novikov’s famous theorem [1] to higher dimensions. The homological vanishing cycle is given as an immersion φ : B×[0 , 1] → M, where B is a connected oriented (m − 2) -manifold, such that Bt = φ(B × { t}) lies on a leaf Lt for every t, 0 6= [ B0] ∈ Hm−2(L0), and [Bt] = 0 ∈ Hm−2(Lt) for every t > 0. The additional hypothesis is that the self-intersections of Bt for t > 0 do not change as t varies. A generalized Reeb component with connected boundary is a compact foliated manifold whose interior fibers over the circle with the leaves as fibers and whose boundary is a single compact leaf. This is a partial result of twenty years of research efforts. We hope in the future to complete the proof in the general case, when the vanishing cycle is not connected and consequently L0 may be a finite union of leaves. References

[1] S.P. Novikov, Topology of foliations. Trans. Moscow Math. Soc. , 14 (1965), 268-304.

1 University of Santiago de Compostela. [email protected] 2 University of Lyon. [email protected] 3 PUC-Rio de Janeiro. [email protected]

31 Geometric invariants for group extensions with applications to twisted conjugacy classes Nic Koban 1 and Pe ter Wong 2

Abstract

The Bieri–Neumann–Strebel (BNS) invariant Σ1(G) of a finitely generated group G has many connections with other branches of mathematics. For example, the sigma invariant of the fundamental group of a 3-manifold is precisely the projection of the interior faces of the Thurston-norm ball. The invariant Σ1(G) also gives information on the finite generation of the commutator subgroup [G, G ]. In this talk, we will discuss the computation of Σ1 and of a related invariant Ω1 for certain group extensions. We use the results to construct hyperbolic 3-manifold groups G for which [G, G ] is finitely generated. Finally, we construct examples of group extensions that have property R∞, that is, every automorphism ϕ has an infinite number of ϕ-twisted conjugacy classes.

1 University of Maine - Farmington. [email protected] 2 Bates College. [email protected]

32 Fixed point sets of equivariant fiber-preserving maps Rafael Mor eira de Souza 1 and Peter Ngai-Sing Wong 2

Abstract

In 1990, H. Schirmer made use of the relative Nielsen fixed point theory [1] 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 [2]. Subsequently, generalizations of Schirmer’s result were given. In particular, Christina L. Soderlund improved the result of Schirmer by assuming that A is a locally contractible subset of X [3], and Robert F. Brown and Soderlund generalized the result to the setting of fiber preserving maps [4]. 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 respects 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 [5] and subsequently in [6,7,8]. Combining the setting of [4] together with the equivariant setting of [8], we determine necessary and sufficient conditions for the realization of a locally contractible G-subset A of X as the fixed point set of a fiber-preserving map h : X → X in a given G-fiber-preserving homotopy class. Here, G is a finite group and X is a compact smooth G-manifold as well as the total space of a G-fiber bundle. References

[1] H. Schirmer , A relative Nielsen number , Pacific J. Math. 122 (1986), 459–473. [2] H. Schirmer , Fixed pointsets in a prescribed homotopy class , Topology Appl. 37 (1990), 153–162. [3] C. L. Soderlund , Fixed point set of maps homotopic to a given map , Fixed Point Theory Appl. (2006), Article ID 46052, 20pp. [4] R. F. Brown and C. Soderlund , Fixed point sets of fiber-preserving maps , J. Fixed Point Theory Appl. 2 (2007), 41–53. [5] E. Fadell and P. Wong , On deforming G-maps to be fixed point free , Pacific J. Math. 132 (1988), 277–281. [6] P. Wong , Equivariant Nielsen fixed point theory for G-maps , Pacific J. Math. 150 (1991), 179–200. [7] P. Wong , On the location of fixed points of G-deformations , Topology Appl. 39 (1991), 159–165. [8] P. Wong , Equivariant Nielsen numbers , Pacific J. Math. 159 (1993), 153–175.

1 Federal University of Uberlândia. [email protected] 2 Bates College. [email protected]

33 Deformations of the discrete Heisenberg group Severin Barmeier 1 Abstract

We will talk about deformations of the discrete Heisenberg group in the context of deformations of discrete groups acting properly discontinuously on a non-Riemannian homogeneous space. In particular, I will determine when the discrete Heisenberg group acts properly discontinuously from the left and right on the (continuous) Heisenberg group H, written as the homogeneous space H × H/D (H), where D : H → H × H is the diagonal embedding, and give a complete description of the deformation space. This stands in contrast to the Riemannian setting, where deformations of discontinuous groups for a Riemannian symmetric space G/K are only admitted when G is locally isomorphic to SL 2 R. In this case the prime example of a deformation space of discontinuous groups is the Teichmüller space of a Riemann surface. References

[1] S. Barmeier , Deformations of the discrete Heisenberg group , Proc. Japan Acad. 89 , Ser. A (2013), 55–59.

1 State University of Campinas (UNICAMP). [email protected]

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New examples of Neuwirth–Stallings pairs and non-trivial real Milnor fibrations Taciana O. Souza, 1 R. Araújo dos Santos, 2 M. A. B. Hohlenwerger, 3 and O. Saeki 4

Abstract

We use the topology of configuration spaces to give a characterization of Neuwirth–Stallings pairs (S5, K ) with dim K = 2 . As a consequence, we construct polynomial map germs (R6, 0) → (R3, 0) with an isolated singularity at the origin such that their Milnor fibers are not diffeomorphic to a disk, thus putting an end to Milnor’s non-triviality question. Furthermore, for a polynomial map germ (R2n, 0) → (Rn, 0) or (R2n+1 , 0) → (Rn, 0) , n ≥ 3, with an isolated singularity at the origin, we study the conditions under which the associated Milnor fiber has the homotopy type of a bouquet of spheres. We then construct, for every pair (n, p ), a new example of a polynomial map germ (Rn, 0) → (Rp, 0) with an isolated singularity at the origin such that its Milnor fiber has the homotopy type of a bouquet of a positive number of spheres.

1 FAMAT - UFU. [email protected] 2 ICMC - USP. [email protected] 3 ICMC - USP. [email protected] 4 Kyushu University. [email protected]

34 Lefschetz coincidence class for several maps Thaís F. M. Mo nis 1 and Stanisław Spiez 2

Abstract

Let f1,...,f k : X → Y be functions, k ≥ 2. A coincidence point of them is a point x ∈ X such that f1(x) = · · · = fk(x). The set of all coincidence points is denoted by Coin (f1,...,f k). There is a vast literature with information about Coin (f1, f 2), the set of coincidence of two maps. In [1] the authors start a study about the set of coincidence of more than two maps, defining a n(k−1) coincidence class L(f1,...,f k) ∈ H (X; Q) when Y is a closed connected oriented n-manifold. In this work we revisit the definition given in [1] and we present a more natural way to define a class of coincidence for several maps. Also, using twisted coefficients, we present an extension of the definition of L(f1,...,f k) given in [1] to the case where Y is non-orientable. References

[1] C. Biasi, A. K. M. Libardi and T. F. M. Monis , The Lefschetz coincidence class of p-maps , Forum Math., DOI 10.1515/forum-2013-0038 (to appear). [2] E. Spanier , Duality in topological manifolds , in: Colloque de Topologie Tenu à Bruxelles, Centre Belge de Recherches Mathématiques (1966), 91–111. [3] D. L. Gonçalves and J. Jezierski , Lefschetz coincidence formula on non-orientable manifolds , Fundamenta Mathematicae 153 (1997), 1–23. [4] D. L. Gonçalves, J. Jezierski and P. Wong , Obstruction Theory and Coincidences in Positive Codimen- sion , Acta Mathematica Sinica, English Series 22 (2006), No. 5, 1591–1602. [5] E. Spanier, E. , Algebraic Topology , McGraw-Hill, New York, 1966. [6] J. W. Vick , Homology Theory. An introduction to Algebraic Topology , 2nd Edition. Springer-Verlag, New York, 1994. [7] G. Whitehead , Elements of Homotopy Theory. Springer-Verlag, New York, 1978.

1 Instituto de Geociências e Ciências Exatas, UNESP - Univ Estadual Paulista. [email protected] 2 Institute of Mathematics, Polish Academy of Sciences. [email protected]

35 The Borsuk–Ulam property for homotopy classes of functions between surfaces Vinicius Casteluber Laass 1 Abstract

Let τ : M → N be a fixed point free involution. The triple (M, τ, N ) is said to satisfy the Borsuk–Ulam property (PBU) if for every continuous map f : M → N, there exists a point x ∈ M such that f(τ(x)) = f(x). This definition arose from the classical Borsuk–Ulam theorem: for every continuous map f : Sn → Rn, there exists a point x ∈ Sn such that f(−x) = f(x). In [1], there exists a complete classification of the triples (M, τ, N ) has or has not PBU, in the case where M and N are closed surfaces. When (M, τ, N ) does not have PBU, it means there exists a continuous map f : M → N such that f(τ(x)) 6= f(x) for every x ∈ M. This is not a good measure for the quantity of functions that collapse an orbit of the action τ. From this work, a natural question appears: given a triple (X, τ, S ) which does not have the PBU, classify the homotopy classes of [X, S ] with the property that there exists a representative f : M → N such that f(τ(x)) 6= f(x) for every x ∈ M. I will say something about the cases where N is the 2-sphere, the projective plane and the Klein bottle. 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.

1 IME–Universidade de São Paulo. [email protected]

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Functions and vector fields on C(CP n)-singular manifolds Alice Kimie Miwa Libardi 1 and Vladimir V. Sharko 2

Abstract

In this paper we study functions and vector fields with isolated singularities on a C(CP n)- singular manifold. In general, a C(CP n)-singular manifold is obtained from a smooth (2 n + 1) - manifold with boundary which is a disjoint union of complex projective spaces CP n t CP n t · · · t CP n and subsequent capture of the cone over each component CP n of the boundary. We calculate the Euler characteristic of a compact C(CP n)-singular manifold M 2n+1 with finite isolated singular points. We also prove the Poincaré–Hopf Index Theorem for an almost smooth vector field with a finite number of zeros on a C(CP n)-singular manifold.

1 IGCE Unesp. [email protected] 2 Institute of Mathematics, National Academy of Sciences of Ukraine, Ukraine. [email protected]

36 Bourgin-Yang version of the Borsuk-Ulam theorem and related topics Wacław Marzan tow icz, 1 Denise de Mattos, 2 and Edivaldo dos Santos 3

Abstract

In 1933 S. Ulam posed and K. Borsuk & showed that if n > m then it is impossible to map f : Sn → Sm preserving symmetry: f(−x) = −f(x) .

Next in 1954-55, C. T. Yang, and D. Bourgin, showed that if f : Sn → Rm+1 preserves this symmetry then dim f −1(0) ≥ n − m − 1.

We will present versions of the latter for some other groups of symmetries and also discuss the case n = ∞. To emphasize the importance of this kind of topological theorem we present a short survey of their applications to combinatorics and non-linear analysis.

1 UAM, Poznań. [email protected] 2 ICMC–USP. [email protected] 3 DM-UFSCar. [email protected]

37 PAINÉIS (POSTERS)

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k Z2-actions fixing KP (2 m + 1) t KP (2 n + 1) Al lan Ed ley Ramos de An drade, 1 Pedro Pergher, 2 and Sergio Tsuyoshi Ura 3

Abstract

In [1], D. C. Royster classified, up to equivariant cobordism, involutions fixing disjoint unions of two real projective spaces. Motivated by [1], [2] and [3], we obtained a classification, up to k equivariant cobordism, of Z2-actions fixing a disjoint union of two real, complex or quaternionic projective spaces of odd dimensions, KP (2 m + 1) t KP (2 n + 1) . Specifically, we prove that such k Z2-action bounds equivariantly. This project was supported by FAPESP and CNPQ. References

[1] D. C. Royster , Involutions ﬁxing the disjoint union of two projective spaces , Indiana University Mathematics Journal 29 (1980), n 2, 267–276. [2] P. L. Q. Pergher , Bordism of two commuting involutions , Proc. Amer. Math. Soc. 126 (1998), 191–195. k s [3] P.L.Q. Pergher and A. Ramos , Z2 -Actions ﬁxing KdP (2 ) ∪ KdP (even ), Topology and its Applications (2009).

1 UFMS. [email protected] 2 UFSCar. [email protected] 3 UFSCar. [email protected]

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Some properties of the invariant E∗(G, S) Amanda Fer reira de Lima 1 and Maria Gorete Carreira Andrade 2

Abstract

Let G be a group, S = {Si, i ∈ I} a family of subgroups of G with infinite index and M a Z2G-module. The algebraic invariant E(G, S, M ) was defined by Andrade and Fanti ([1]) by using the theory of cohomology of groups. Based on this invariant, Andrade and Gazon ([2]) defined another algebraic invariant, dual to E(G, S, M ), denoted by E∗(G, S, M ), by using the theory of homology of groups instead of cohomology, and they obtained some results for M = Z2. In G this work we present some properties of the particular invariant E∗(G, S, Coind {1}Z2) denoted by E∗(G, S), obtaining necessary conditions for a pair (G, S) to be a Poincaré duality pair. 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; A. B. Gazon, A dual homological invariant and some properties , Int. Journal of Applied Math., 27 (1), (2014), 13-20.

1 Universidade Federal de São Carlos. mandinha −[email protected] 2 IBILCE - UNESP - S.J.Rio Preto. [email protected]

38 The Borsuk-Ulam property for closed surfaces Ana Maria Math ias Morita 1 and Maria Gorete Carreira Andrade 2

Abstract

The classical Borsuk-Ulam theorem states that if f : Sn −→ Rn is a continuous map, then there exists a point x in the sphere such that f(x) = f(−x). Since the publication, many generalizations of that result have been studied. Some generalizations consist in replacing either the domain (Sn, A ), where A is the antipodal involution, by another free involution pair (X, T ), or the target space Rn by more general topological spaces Y . In that case, we say that (( X, T ); Y ) satisfies the Borsuk-Ulam property if given any continuous map f : X −→ Y , there exists a point x in X such that f(x) = f(T (x)) . In this work, which is part of our master’s dissertation, we detail the proof of a classification result presented by Gonçalves in [1], that provides necessary and sufficient conditions for a closed surface to satisfy the Borsuk-Ulam property. References

[1] D. L. Gonçalves , The Borsuk-Ulam theorem for surfaces , Quaestiones Mathematicae 29 (2006), p. 117-123. [2] D. L. Gonçalves, J. Guaschi , The Borsuk-Ulam theorem for maps into a surface , Topology and its Appli- cations 157 (2010), p. 1742-1759. [3] W. S. Massey Algebraic Topology: an Introduction , Springer-Verlag, 1967. [4] J. R. Munkres Elements of Algebraic Topology , Addison-Wesley, 1984.

1 USP–ICMC. [email protected] 2 Orientadora. UNESP–IBILCE. [email protected]

39 The homology groups of tetrahedral spherical space forms Ligia Laís Fêmina, 1Ana Paula Tremura Galves, 2 and Oziride Manzoli Neto 3

Abstract

Given a free isometric action of the binary tetrahedral group 2 2 2 −1 −1 −1 3 4 P24 = hx, y, z |x = ( xy ) = y , zxz = xy, xyx = y , z = x = 1 i on an odd dimensional sphere, we obtain an explicit ﬁnite cellular decomposition of the tetrahedral spherical space forms, using the concept of fundamental domain . The cellular structure gives an explicit description of the associated cellular chain complex over the group P24 . The main purpose of this presentation is to use the chain complex to compute the homology groups of the tetrahedral spherical space forms with various coeﬃcients. 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] Fêmina, L.L.; Galves, A.P.T.; Neto, O.M.; Spreafico, M.F. Fundamental domain and cellular decomposition of tetrahedral spherical space forms. (Submitted paper) . [4] Swan, R.G. Periodic resolutions for finite groups. Annals of Mathematics , v.72, p.267-291, 1960. [5] Wolf, J.A. Spaces of constant curvature . McGraw-Hill Inc., 1967. [6] Zimmermann, B. On the classification of finite groups acting on homology 3-spheres. Pacific Journal of Math- ematics , v.217, n.2, p.387-395, 2004.

1 FAMAT-UFU. [email protected] 2 FAMAT-UFU, [email protected] 3 ICMC-USP. [email protected]

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A connected compact hyperreal line José Carlos Cifuentes Vasquez 1 and An der son Luis Gama 2

Abstract

In general a Hyperreal Line is a saturated model that extends the real numbers preserving first order logic; such models proved to be useful in Non-Standard Analysis. However while first order, for instance algebraic, properties are preserved, second order, for instance topological, ones may change significantly: in fact a Hyperreal Line is always Dedekind incomplete and the corresponding topology totally disconnected. In this work we show how a second extension may be used to recover a connected and compact topology that resembles the compact real line R∪{∞, −∞} but preserves saturation. We also show how this creates a connection between internal continuity and external (i.e. topological) continuity, opening up the possibility of external analysis. References

[1] R. Goldblat , Lectures on the Hyperreals: An Introduction to Nonstandard Analysis , Springer (1998), 1st edition.

1 Federal University of Paraná. [email protected] 2 Federal University of Paraná. [email protected]

40 Some results about the Borsuk–Ulam theorem for double coverings of surface bundles Anderson Paião dos Santos 1 Abstract

Let (X, τ ) denote a space X with a free involution τ on X. Given a space Y , we say that the Borsuk-Ulam theorem holds for the triple (X, τ ; Y ) if for any continuous map f : X → Y there exists a point x ∈ X such that f(x) = f(τ(x)) . Now, let M be homotopy equivalent to the total space of an F -bundle over B, where the base B and the ﬁber F are closed surfaces, 1 and consider a non-trivial class [ϕ] ∈ H (M; Z2). In [7] we studied the Borsuk-Ulam theorem for n triples (Mϕ, τ ϕ; R ), where Mϕ is the double covering of M and τϕ is the free involution on Mϕ such that Mϕ/τ ϕ = M. In this work we give some results about this problem in case the ﬁber F is K(π, 1) and the base is S2. References

[1] K. Borsuk , Drei Sätze über die n-dimensionale Euklidische Sphäre , Fund. Math. 20 (1933), 177–190. [2] D. L. Gonçalves , The Borsuk-Ulam theorem for surfaces , Quaestiones Mathematicae 29 (2006), 117–123. [3] D. L. Gonçalves, J. Guaschi , The Borsuk-Ulam theorem for maps into a surface , Topology and its applications 157 (2010), 1742–1759. [4] 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. [5] J. A. Hillman , The Algebraic Characterization of Geometric 4-Manifolds , London Mathematical Society, Lecture Note Series 198, Cambridge University Press, (1994). [6] J. A. Hillman , Four-manifolds, geometries and knots, Geometry and Topology Monographs , Volume 5, (2002). [7] A. P. Santos , Involuções e o teorema de Borsuk-Ulam para algumas variedades de dimensão 4 , PhD thesis, IME-USP, (2012).

1 Universidade Tecnológica Federal do Paraná. [email protected]

41 Conley index theory and spectral sequences in the Morse–Bott setting Dahisy Lima, 1 Ketty de Rezende, 2 and Mariana Silveira 3

Abstract

Algebraic-topological tools have been widely used in dynamical systems in order to determine structural properties which remain invariant under small perturbations, as in Conley index theory [1]. The Conley index provides a topological description of the local dynamics around the Morse sets associated to a Morse decomposition of a given isolated invariant set. The connection matrices introduced by Franzosa [3] are algebraic-topological tools which enable us to study the connections between Morse sets. Roughly speaking, a connection matrix for a Morse decomposition is a matrix which has as entries homomorphisms between the homology Conley indices of Morse sets. In the Morse setting, given a Morse function on a smooth closed n-manifold, one can construct a Morse chain complex (C∗, ∂ ∗) generated by the critical points of f and graded by their Morse indices such that the boundary operator counts (with signs) flow lines between consecutive critical points. Salamon proves in [5] that the Morse boundary operator ∂∗ is a special case of the connection matrix for the gradient flow generated by f. Endowing a Morse chain complex with an increasing filtration, one can associate to it a spectral sequence (Er, d r). In [2], a sweeping algorithm is introduced which generates a collection of connection and transition matrices from which one can recover the modules Er and differentials dr of the spectral sequence. These results led to the question of how closely the dynamics follows the spectral sequence. As one “turns the pages” of the spectral sequence, i.e. considers progres- sively modules Er, one observes algebraic cancellation occurring within the Er’s. These algebraic cancellations are dynamically interpreted as the history of birth and death of connecting orbits of the flow caused by the cancellation of consecutive critical points. A connection matrix theory approach is presented for flows associated to Morse-Bott functions on smooth closed n-manifolds by characterizing the set of connection matrices in terms of Morse- Smale perturbations, see [4]. Our goal is to obtain dynamical results from the spectral sequences associated to Morse-Bott complexes. References

[1] C. Conley , Isolated Invariant Sets and the Morse Index. CBMS Regional Conference Series in Math. AMS, Providence, RI 38 (1978). [2] O. Cornea, K.A. de Rezende and M. R. da Silveira , Spectral sequences in Conley’s theory. Ergodic Theory and Dynamical Systems 30 (4) (2010) 1009–1054 . [3] R. Franzosa , The Connection Matrix Theory for Morse Decompositions, Transactions of the American Math- ematical Society 311 (1989) 561-592. [4] D.V.S. Lima. and K.A. de Rezende. Connection Matrices for Morse-Bott ﬂows . To appear in Topological Methods in Nonlinear Analysis. [5] D. A. Salamon , The Morse theory, the Conley index and the Floer homology, Bull. London Math. Soc. 22 (1990) 113–240.

1 Universidade Estadual de Campinas. [email protected] 2 Universidade Estadual de Campinas. [email protected] 3 Universidade Federal do ABC. [email protected]

42 A notion of asymptotic dimension and finite decomposition complexity Évelin Me negesso Bar baresco 1 and Flávia Souza Machado da Silva 2

Abstract

The asymptotic dimension of a metric space X, defined by Gromov [1], is the smallest integer n so that for every R there is a uniformly bounded cover of X so that no R-ball in X meets more than n + 1 elements of the cover. The concept of finite decomposition complexity for metric spaces was introduced by Guentner, Tessera, and Yu [2, 3], and is a generalization of the notion of asymptotic dimension of Gromov. Roughly speaking, finite decomposition complexity measures the difficulty of decomposing a metric space into uniformly bounded pieces that are well-separated from one another. The aim of this work is to present some concepts and initial results of asymptotic dimension and finite decomposition complexity, and also examples of metric spaces with finite decomposition complexity. References

[1] M. Gromov , Asymptotic invariants of inﬁnite groups , in: Geometric Group Theory 2, Sussex, 1991, in: London Math. Soc. Lecture Note Ser. 182, Cambridge University Press, Cambridge, 1993, pp. 1–295. [2] E. Guentner, R. Tessera, and G. Yu , Discrete groups with ﬁnite decomposition complexity . Groups Geom. Dyn. 7 no. 2 (2013), 377–402. [3] E. Guentner, R. Tessera, and G. Yu , A notion of geometric complexity and its application to topological rigidity , Invent. Math. 189 no. 2 (2012), 315–357. [4] P. W. Nowak, G. Yu , Large Scale Geometry , EMS Textbooks in Mathematics, Zürich: European Mathe- matical Society (EMS), (2012).

1 IBILCE, Unesp São José do Rio Preto. [email protected] 2 IBILCE, Unesp São José do Rio Preto. [email protected]

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On covering maps and deck transformations Gi vanildo Donizeti de Melo 1 and Thiago de Melo 2

Abstract

This work was developed during an undergraduate course. We will present preliminary notions on covering spaces such as path and function lifting properties which are necessary in the sequel. Then we will deal with the action of the fundamental group of the base space of a covering on the ﬁbre. Finally we will classify the covering maps of a given space (up to some kind of equivalence) making use of some subgroups of the fundamental group of that space and the deck transformations. References

[1] Bredon, G. E. Topology and Geometry , Graduate Texts in Mathematics 139, Springer (2010).

1 IGCE–Unesp Rio Claro. [email protected] 2 Orientador. IGCE–Unesp Rio Claro. [email protected]

43 H-spaces and cyclic maps Guil herme Vi turi 1 and Thiago de Melo 2

Abstract

In this work, we study a subset of the based CW-complexes known as H-spaces as well as its dual (in some sense), the co-H-spaces . We also deal with cyclic maps, and link these two objects making use of a result from [1, Proposition 3.3], which states that X is an H-space if and only if the identity map 1X : X → X is a cyclic map. Moreover, in this case, if G(A, X ) denotes the Gottlieb group (of homotopy classes of cyclic maps f : A → X) and [A, X ] denotes the set of homotopy classes of maps from A to X, then we have G(A, X ) = [ A, X ], for every based CW-complex A. References

[1] K. D. Lim , On Cyclic Maps , J. Austral. Math. Soc. (1980), 349–357. [2] K. Varadarajan , Generalised Gottlieb Groups , Journal of Indian Math. Soc. (1969), 141–164. [3] G. W. Whitehead , Elements of Homotopy Theory , Springer-Verlag (1978)

1 IGCE–Unesp Rio Claro. [email protected] 2 Orientador. IGCE–Unesp Rio Claro. [email protected]

44 The nonabelian tensor product of groups Gustavo C. I. Figueiredo 1

Abstract

The nonabelian tensor square G ⊗ G of a group G was introduced by R. K. Dennis [8] “in a search for new homology functors having a close relationship to K-theory” and it is based on the work of C. Miller [9]. Subsequently R. Brown and J.-L. Loday [2] discovered a topological significance for the tensor square, namely, that the third homotopy group of the suspension of an Eilenberg MacLane space K(G, 1) satisfies π3 SK (G, 1) =∼ ker( κ), where κ : G⊗G → G is the “commutator homomorphism”: κ(g ⊗ h) = [ g, h ] = ghg −1h−1, for all g, h ∈ G. They also defined the tensor product G ⊗ H of two distinct groups acting “compatibly” on each other and showed that it arose in a certain “universal crossed square.” The main purpose of this work is to present the first properties of the nonabelian tensor product of groups and some of its applications. References

[1] Brown, R., Loday, J.-L., Excision homotopique en basse dimension, C. R. Acad. Sci. Paris Sér. I Math. 298, No. 15 (1984), 353-356. [2] Brown, R., Loday, J.-L., Van Kampen theorems for diagrams of spaces, Topology 26, No. 3 (1987), 311-335. [3] Brown, R., A nonabelian tensor product of groups, Algebra-Tagung Halle 1986, 59-72, Wissensch. Beitr., 33, Martin-Luther Univ., Halle Wittenberg, Halle (Saale) (1987). [4] Brown, R., Johnson, D. L., Robertson, E. F., Some computations of nonabelian tensor product of groups, J. Algebra 111, No. 1 (1987), 177-202. [5] McDermott, A., The nonabelian tensor product of groups: Computations and structural results, PhD thesis, National Univ. of Ireland, Galway (1998). [6] Nakaoka, I. N., Sobre o produto tensorial não abeliano de grupos, dissertação de mestrado, Universidade Estadual de Campinas (UNICAMP), Campinas (1994). [7] Whitehead, J. H. C., A certain exact sequence, Ann. of Math. 52 (1950), 60-71. [8] Dennis, R. K., In search of new homology functors having a close relationship to K-theory, Cornell University preprint (1976). [9] Miller, C., The second homology group of a group, Proc. Amer. Math. Soc. (1952), 588-595.

1 UFSCar - Master Student - [email protected]

45 Separation axioms in approach spaces Soraya Rosana Torres Kudri 1 and Izael do Nasci mento 2

Abstract

Approach spaces form a supercategory for the category of topological spaces ( TOP ). Those spaces were introduced by the mathematician Robert Lowen in order to solve some algebraic “gaps” found in TOP , but ended up becoming an interesting object of study and with applications in several areas. Following Lowen and Sioen ([10]) we characterize, in a more detailed fashion, some separation axioms in approach spaces without using categorical concepts. References

[1] M.L. Colosante, C. Uzcátegui and J. Vielma. Low separation axioms via the diagonal , Applied General Topology, Universidad Politécnica de Valencia, Volume 9, No. 1, 39-50. [2] A. Csaszar. Separation axioms for generalized topologies , Acta Math. Hungar. 104, 2004, 63-69. [3] P. Das and M.A. Rashid. Certain separation axioms in a space , Korean J. Math. Sciences, Vol. 7, 2000, 81-93. [4] I. do Nascimento. Axiomas de separação em espaços de aproximação , Dissertação de Mestrado, Programa de Pós-graduação em Matemática Aplicada, UFPR, Curitiba, 2013. [5] H.H. Domingues. Espaços métricos e introdução à topologia , Atual, São Paulo, 1982. [6] J. Dontchev and M. Ganster. On δ-generalized closed sets ans T 3 spaces , Mem. Fac. Sci., Kochi Univ., Ser. A, 4 Math 17, 1996. [7] W. Dunham. T 1 -Spaces , Kyungpook Math. J. Volume 17, Num. 2 - December, 1977. 2 [8] N. Levine. Generalized closed sets in topology , Rend. Circ. Mat. Palermo 19(2), 1970, 89-96. [9] R. Lowen. Approach spaces: the missing link in the topology-uniformity-metric triad , Oxford Mathematical Monographs, Oxford University Press, 1997. [10] R. Lowen and M. Sioen. A note on separation in AP , preprint. [11] M.N. Mukherjee and B. Roy. A unified theory for R 0, R 1 and certain other separation properties and their variant forms , Bol. Soc. Paran. Mat. (3s) v. 28(2), 2010, 15-24. [12] J.R. Munkres. Topology: a first course , Prentice-Hall, Englewood Cliffs, New Jersey, 1999. [13] B.M. Munshi. Separation axioms , Acta Science, Indica 12, No. 2, 140-144, 1986.

1 Universidade Federal do Paraná. [email protected] 2 Universidade Federal do Paraná. [email protected]

46 Spaces with operators and cohomology of groups under a topological viewpoint Jes sica Cristina Rossi nati Ro drigues da Costa 1 and Maria Gorete Carreira Andrade 2

Abstract

The theory of (co)homology of groups arose from studies in algebra and topology. The starting point for the topological aspect of the theory was the work by Hurewicz in 1936 on aspherical spaces, that is a space X with fundamental group π1(X) = G and higher homotopy groups πn(X) = 0 n > 1. Hurewicz showed, among other things, that the homotopy type of an aspherical space X, having the homotopy type of a CW -complex, depends only on its fundamental group G and, in particular, the cohomology groups of X depend only on the group G. For this reason, the cohomology of a group G, denoted by H∗(G), can be seen as the cohomology of any aspherical space X, having the homotopy type of a CW -complex, with fundamental group G. There is a purely algebraic definition of cohomology groups of G. In that definition the concept of projective resolutions of Z over ZG is used, where ZG denotes the group ring of G. Here we present the algebraic definition of H∗(G) and, based in [1] and [2], we detail the proof of the result which says that if X is an acyclic space and G operates properly on X, the quotient space X/G is an aspherical space with fundamental group G and the singular cohomology of X/G is the cohomology of the group G. References

[1] Brown, K. S. , Cohomology of Groups, G.T.M. 87 Spring-Verlag, New York (1982). [2] MacLane, S. , Homology, Spring-Verlag, Berlin (1967). [3] Vick, J. W. Homology Theory, Academic Press, 1973.

1 UNESP–IBILCE. [email protected] - Aluna de Mestrado - FAPESP - Proc. 2013/23980-0 2 Orientadora - UNESP–IBILCE. [email protected]

47 A note on topological classification of singularities João Carlos Ferreira Costa 1 Abstract

In this work we present some advances in the study of topological classification of singularities. We consider the topological classification in the set of all smooth map germs (Rn, 0) → (Rp, 0) with respect to two equivalence relations: the classical topological equivalence (which is given by change of coordinates in the source and target of the germ) and the topological contact equivalence (which is the topological version of classical contact equivalence introduced by John Mather). In addition, will be shown some relationships between these two classifications. References

[1] S. Alvarez, L. Birbrair, J. C. F. Costa and A. Fernandes , Topological K-equivalence of analytic function germs , Cent. Eur. J. Math. 8, no. 2 (2010) 338–345. [2] V. I. Arnold , Topological classification of Morse functions and generalisations of Hilbert’s 16th problem , Math. Phys. Anal.˙ Geom. 10 (2007), 227–236. [3] L. Birbrair, J. C. F. Costa and A. Fernandes , Finiteness theorem for topological contact equivalence of map germs , Hokkaido Math. J. 38 (2009) 511-517. [4] J. C. F. Costa , A note on topological contact equivalence , Real and Complex Singularities. Edited by M. Manoel, M. C. Romero Fuster, C. T. C. Wall (Org.) London Math. Soc. Lecture Notes Series 380 - Real and Complex Singularities. 1 ed. New York: Cambridge University Press, 380 (2010) 114–124. [5] J. C. F. Costa and J. J. Nuno-Ballesteros , Topological K-classification of finitely determined map germs , Geom. Dedicatae 166 (2013), 147–162. [6] T. Fukuda , Local topological properties of differentiable mappings I , Invent. Math. 65 no. 2 (1981), 227–250. [7] T. Fukuda , Local topological properties of differentiable mappings II , Tokyo J. Math. 8 no. 2 (1985), 501–520. [8] H. C. King , Topological type in families of germs , Invent. Math. 62 (1980) 1–13. [9] T. Nishimura , Topological K-equivalence of smooth map-germs , Stratifications, Singularities and Differential Equations, I (Marseille 1990, Honolulu, HI 1990), Travaux en Cours, 54 , Hermann, Paris (1997) 82–93. [10] T. Nishimura , Criteria for right-left equivalence of smooth map germs , Topology, 40 (2001), 433–462.

1 IBILCE/UNESP. [email protected]

48 Fibre maps and fixed points on certain surface bundles D. L. Gonçalves, 1 A. K. M. Libardi, 2 D. Penteado, 3 and J. P. Vieira 4

Abstract

The main purpose of this work is to study fixed points of fibre- preserving maps over S1 on the trivial surface bundles S1 × S where S is a closed surface of negative Euler characteristic. The case where S is equal to S2, i.e., the closed orientable surface of genus 2, is already known. We classify all such maps that can be deformed fibrewise to a fixed point free map. References

[1] E. Fadell and S. Husseini , A fixed point theory for fiber-preserving maps , Lecture Notes in Mathematics 886, Springer Verlag, (1981), 49–72. [2] E. Fadell and S. Husseini , The Nielsen number on surfaces ,in: Topological methods in non linear functional analysis, Toronto, Ont., (1982), in : Contemporary Mathematics 21 , Amer. Math. Soc., Providence, RI, (1983), 59–98. [3] D. L. Gonçalves, A. K. M. Libardi, D. Penteado and J. P. Vieira , Fixed points on trivial surface bundles , Proceedings of the Institute of Mathematics of the National Academy of Sciences of Ukraine, Brazilian- Polish Topology Workshop, 6(6)(2013), 67–85. [4] D. L. Gonçalves, D. Penteado and J. P. Vieira , Fixed Points on Torus Fiber Bundles over the Circle , Fundamenta Mathematicae 183(1)(2004), 1–38. [5] D. L. Gonçalves, D. Penteado and J. P. Vieira , Fixed points on Klein bottle fiber bundles over the circle , Fundamenta Mathematicae 203(3)(2009), 263–292. [6] W. Magnus, A. Karrass and D. Solitar , Combinatorial Group Theory - Presentations of Groups in Terms of Generators and Relations , Dover Publications,Inc., New York (1976). [7] G. P. Scott , Braid groups and the group of homeomorphisms of a surface , Proc. Camb. Phil. Soc. 68 (1969), 605-617. [8] G. W. Whitehead , Elements of Homotopy Theory , Graduate texts in Mathematics 61 , Springer Verlag, (1978).

1 IME-USP. [email protected] 2 I.G.C.E - Unesp - Univ Estadual Paulista. [email protected] 3 Universidade Federal de São Carlos. [email protected] 4 I.G.C.E - Unesp - Univ Estadual Paulista. [email protected]

49 Some results about a certain cohomological invariant Letí cia Sanches Silva 1 and Ermínia de Lourdes Campello Fanti 2

Abstract

Let G be group, W a G-set (such that [G : Gw] = ∞ for all w ∈ E, a set of orbit representa- Z G G G tives in W ), M a 2G-module and E(G, W, M ) := 1 + dimker res W , where res W = res Gw w∈E : 1 1 1 G 1 1 H (G; M) → H (W; M) := Qw∈E H (G w; M) , where res Gw : H (G; M) → H (G w; M) is the restric- tion map, for all w ∈ E (as deﬁned in [2]). The number E(G, W, M ) is independent of the set of G-orbit representatives in W and has some interesting properties. In this work we present another way to see that the deﬁnition of E(G, W, M ) is independent of the set of G-orbit representatives in W, and we present some particular results about this invariant when the G-action is free, or M is the Z2G-module FT G, or M is the Z2G-module Z2(G/T ), where T is a subgroup of G. 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 , The cohomological invariant E’(G,W) and some properties , International Journal of Applied MathematicsManuscripta 25 (2012), 183–190. [3] E. L. C. Fanti; L. S. Silva , Some properties of the cohomological invariant E(G, W, M ) (in preparation). [4] L. S. Silva , O invariante E(G, W, M ): algumas propriedades e aplicações na teoria de decomposição de grupos . Dissertação (Mestrado), IBILCE-UNESP, São José do Rio Preto (2013).

1 UNESP/IBILCE - S.J.Rio Preto. [email protected] 2 UNESP/IBILCE - S.J.Rio Preto. [email protected]

50 Contracting homotopy and diagonal maps for binary tetrahedral groups Ligia Laís Fêmina, 1Ana Paula Tremura Galves, 2 and Oziride Manzoli Neto 3

Abstract

In this work, using the 4-periodic resolution for the binary tetrahedral group, we define a contracting homotopy and we construct a diagonal map. From the diagonal map, we compute the ring structure of the cohomology groups of a Tetrahedral Spherical Space Forms with various coefficients. The 4-periodic resolution was defined in [3]. We studied the action of the binary tetrahedral group 2 2 2 −1 −1 −1 3 4 P24 = hx, y, z |x = ( xy ) = y , zxz = xy, xyx = y , z = x = 1 i on the spheres in this paper. We found a fundamental domain for the free action on the spheres which gave the quotient spaces, called Tetrahedral Spherical Space Forms. Through these regions we built a convenient chain complex of these spaces and now we use it to calculate their cohomology ring, by the contracting homotopy and diagonal map. 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] Fêmina, L.L.; Galves, A.P.T.; Neto, O.M.; Spreafico, M.F. Fundamental domain and cellular decomposition of tetrahedral spherical space forms. (Submitted paper) . [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.

1 FAMAT-UFU. [email protected] 2 FAMAT-UFU, [email protected] 3 ICMC-USP. [email protected]

51 On singular foliations on the solid torus José L. Arraut, 1 Lu ciana F. Mar tins, 2 and Dirk Schütz 3

Abstract

We study smooth foliations on the solid torus S1 × D2 having S1 × { 0} and S1 × ∂D 2 as the only compact leaves and S1 × { 0} as singular set. We denote the family of such foliations by A. We show that all other leaves can only be cylinders or planes. This is the case for foliations in A studied in [3] which are deﬁned by orbits of an action of R2 and it was the main motivation of this work. If we allow compact regular leaves other than S1 × ∂D 2 to exist, we show that these necessarily have to be tori. This is then used to prove our main result.

Theorem ([1]) Let F ∈ A and L be a non-compact leaf of F. Then the inclusion L ⊂ S1 × (D2 − { 0}) induces an injection on the fundamental groups and, consequently, L is diﬀeo- morphic to R2 or S1 × R.

We also give necessary conditions for the foliation to be a suspension of a diﬀeomorphism of the disc. To quote L. Conlon from the MathSciNet of [4]: “Foliations with singularities are a real zoo, but they do arise in nature (e.g., the orbit foliation of a Lie group action). In order to get any reasonable structure theory, it is necessary to severely restrict the types of singularities.” We hope that our study can contribute for this theory. References

[1] J.L. Arraut, L.F. Martins, D. Schuetz, On singular foliations on the solid torus , Topology and its Applications, 160 (2013), 1659-1674. [2] C. Camacho, A. Lins Neto, Geometric Theory of Foliations . Birkhäuser, Boston, Massachusetts, 1985. [3] C. Maquera, L.F. Martins, Orbit Structure of certain R2-actions on solid torus , Ann. Fac. Sci. Toulouse Math. (6) 17 (2008), 613-633. [4] B. Scárdua, J. Seade, Codimension one foliations with Bott-Morse singularities. II. , J. Topology. 4 (2011), No. 2, 343-382.

1 Universidade de São Paulo, Câmpus de São Carlos. [email protected] 2 IBILCE-UNESP, Câmpus de São José do Rio Preto. [email protected] 3 University of Durham, UK. [email protected]

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Minimal Whitney stratification and Euler obstruction of discriminants Marcelo José Saia 1 Abstract

In this work we show how to obtain the minimal Whitney stratification of the discriminant of finitely determined map germs from (Cm, 0) to (Cp, 0) when p = 2 m − 1 or m = n + p with n ≥ 0. We apply the theory developed by Gaffney which shows how to compute a Whitney stratification of discriminants of any finitely determined holomorphic map germ in the nice dimensions of Mather, or in its boundary. For the pairs cited above we show that both stratifications coincide. We also compute the local Euler obstruction at 0 in a class of discriminants of co-rank one finitely determined map germs from Cn+p to Cp with n ≥ 0.

This is joint work with G. F. Barbosa and N. G. Grulha Junior.

1 Universidade de São Paulo, Instituto de Ciências Matemáticas e de Computação-USP, São Carlos-SP. Brazil. [email protected]

52 Cohomology of the fixed point sets of semifree actions on a space X of type (a, b ) Marcio de Jesus Soares 1 Abstract

Toda, in [1], studied the cohomology ring of a space X such that Hj(X; Z) = Z, para j = 0 , n, 2n and 3n, for some positive integer n, and the others are trivial. Let ui be the generator of Hin (X; Z), for i = 0 , 1, 2 and 3. The definition then requires that 2 u1 = au 2 and u1u2 = bu 3, for some pair of integers (a, b ). The ring structure of H∗(X; Z) is completely determined by this pair. This space X is said to be of type (a, b ). In this work the cohomology of the fixed points of Zp semifree action on a space X of type (a, b ) is studied. For this study the multiplicative structure of one of two spectral sequences associated to the double complex is used. This tool was used in [2] for determining the cohomology of fixed points sets for p-group actions on spaces that have the same cohomology ring as products two or three spheres modulo p. References

[1] H. Toda , Note on cohomology ring of certain spaces , Proc. Amer. Math. Soc., 14 (1963), 89–95. [2] M. J. Soares , Ações de p-grupos sobre produtos de esferas, co-homologia dos grupos virtualmente cíclicos (Za o Zb) o Z e [Za o (Zb × Q2i )] o Z e Co-homologia de Tate , Tese de doutoradamento, USP/São Carlos, 2008. [3] G. E. Bredon , Introduction to compact transformation groups , Academic Press, 1972.

1 Universidade Federal de São Carlos [email protected]

53 Introduction to the theory of knots and polynomial invariants

Neemias Mar tins1 and Simone M. Moraes 2

Abstract

We begin this work with a history of the origin and evolution of Knot Theory. Then we introduce the basic concepts of the theory and we present the first tools studied within this theory in the search for classification of knots, namely, number of crossings, three-colorability, and Reidemeister Movements. We finish the work with the study of polynomial invariants of knots, we construct the Alexander polynomial and introduce relations that define the polynomials of Conway and Jones. Furthermore, we present examples of knots of which are not equivalent but are distinguished by these polynomials. References

[1] C. C. Adams , Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. W. H. Freeman and Company}, New York, (2001). [2] E. Colli , Introdução à Teoria dos Nós , IME-USP, Disponível em < http://www.ime.usp.br/ matemateca/textos/nos.pdf>. Acesso em: 15 jan. 2014. [3] R. H. Crowell , e R. H. Fox Introduction to Knot Theory , Dover Publications (2008). [4] S. Dias , Introdução à Teoria de Nós . Dissertação de Mestrado em Matemática Universidade do Minho, Minho. (2004). [5] D. Hacon , Introdução à Teoria de Nós em R3 . IMPA – 15 º Colóquio Brasileiro de Matemática, (1985). [6] O. Manzoli Neto , Mini-curso: Nós (e Enlaçamentos) . XVII Encontro Brasileiro de Topologia, PUC-Rio, (2010). Disponível em www.mat.puc-rio.br/ebt2010/notes/Oziride.pdf . Acesso em 15. jan. 2014 [7] D. Rolfsen , Knots and Links . Publish or Perish, Inc., 1976.

1 Universidade Federal de Viçosa [email protected] 2 Universidade Federal de Viçosa - Orientadora [email protected]

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Graphical calculus for the Hecke algebra

Pablo Gon za lez Pagotto 1 and Alistair Savage 2

Abstract

We develop a graphical calculus for the Hecke algebra of type A, when its parameter is nonzero and not a nontrivial root of unity. Then we proceed to generalize the concepts of symmetrizer and antisymmetrizer to this algebra. We also explore some of their properties, recursive relations and their interaction with each other. References

[1] Cvitanović, Predrag , Group theory , Princeton University Press, Princeton, NJ, (2008). [2] Licata, Anthony and Savage, Alistair , Hecke algebras, ﬁnite general linear groups, and Heisenberg cate- goriﬁcation , Quantum Topol., 4, (2013), 125–185. [3] Mathas, Andrew , Iwahori-Hecke algebras and Schur algebras of the symmetric group , 15 , American Mathe- matical Society, Providence, RI, (1999). [4] Biggs, N. L. , The roots of combinatorics , Historia Mathematica, 6, (1979). [5] Kleiner, Israel , The evolution of group theory: a brief survey , Mathematics Magazine, 59 , (1986), 195–215.

1 IGCE–Unesp. [email protected] 2 University of Ottawa. [email protected]

54 Nonabelian algebraic topology

Renato Vieira 1 Abstract

∗ It’s a well known fact that the fundamental group functor π1 : Top → Grp and the classification space functor B : Grp → Top ∗ induce an equivalence between the homotopy category of the homotopy 1-types and the category of groups. In this sense we can say that groups model homotopy 1-types. In Whitehead’s work [6, 7, 8] he observed that given a fibration F → U → X we have a u −1 homomorphism ∂ : π1(F ) → π1(U) and a π1(U) action on π1(X) such that ∂( x) = u∂ (x)u and xx 0x−1 = ∂(x)x0. He calls any homomorphism of groups with an action of the codomain on the domain satisfying the above equations as a crossed module. He also defined the concept of quasi-isomorphisms between crossed modules and, with MacLane [5], proved that the localization of the crossed module category obtained by inverting the quasi-isomorphisms is equivalent to the homotopy category of the homotopy 2-types. The key to generalizing Whitehead’s work to arbitrary dimensions is to observe that the crossed module category is equivalent to the category of the categories internal to groups, that is the category of cat 1-groups. From this Loday [4] showed, using a generalization of fibrations denoted n-cubes of fibrations, that the homotopy category of the homotopy (n + 1) -types is equivalent to a certain localization of the category of n-categories internal to groups, that is the category of cat n-grupos. An important result by Brown and Loday [1] is that the fundamental cat n-group functor Π that gives us the aforementioned equivalence satisfies a certain generalization of the Seifert-van Kampen theorem, that is it preserves certain colimits. This suggests that if might be possible to compute the fundamental cat n-group of spaces using methods analogous to computations of fundamental groups. Given this theorem it is useful to compute colimits in the category of cat n- groups. To this end it is convenient to work in the category of crossed n-cubes of groups, that Ellis and Steiner proved is equivalent to the category of cat n-groups [3], where certain colimits have a natural presentation. The poster is a quick survey of these results, including applications. We shall define the crossed n-cubes of groups and the n-cubes of fibrations, and also state the generalized Seifert-van Kampen theorem. We then sketch the proof by Elis and Mikhailov [2] of a combinatorial formula for the homotopy groups of the sphere S2. We also exhibit a new proof of the fact that the wedge sum of Eilenberg-MacLane spaces K(G, 1) ∨ K(H, 1) is homotopic to K(G ∗ H, 1) . References

[1] R. Brown, J.L. Loday, van Kampen theorem for diagram of spaces , Topology, 26 (1987), 311-334. [2] G. Ellis, R. Mikhailov, A colimit of classifying spaces , Advances in Math., 223 (2010), 2097-2113. [3] G. Ellis, R. Steiner, Higher-dimensional crossed modules and the homotopy groups of (n+1)-ads , J. Pure Applied Algebra, 46 (1987), 117-136. [4] J.L. Loday, Spaces with ﬁnitely many non-trivial homotopy groups , J. Pure Applied Algebra, 24 (1982), 179-202. [5] S. Mac Lane, J.H.C. Whitehead, On the 3-type of a complex , Proc. Nat. Acad. Sci., 36 (1950), 41-58. [6] J.H.C. Whitehead, On adding relations to homotopy groups , Annals of Math., 42 (1941), 409-428. [7] J.H.C. Whitehead, Note on a previous paper , Annals of Math., 47 (1946), 806-810. [8] J.H.C. Whitehead, Combinatorial homotopy II , Bull. Amer. Math. Soc, 55 (1949), 453-496.

1 IME–USP. [email protected]

55 A periodic resolution for the binary icosahedral group

S. To moda, 1 O. Manzoli Neto, 2 M. Spreaﬁco, 3 L.L. Fêmina, 4 and A.P.T. Galves 5

Abstract

The study of the cohomology ring of Seifert manifolds first appeared in the work of Bryden, Hayat-Legrand, Zieschang, and Zvengrowski[1]. They considered the Seifert manifolds with infinite fundamental group. In [5], the cohomology ring of some Seifert manifolds with finite fundamental group were investigated.

Currently, we are working on the Poincaré homology sphere. Its fundamental group, P120 , is referred to as the binary icosahedral group, and has the symmetric presentation 2 2 P120 = hA, B : AB A = BAB, BA B = ABA i, given in Coxeter and Moser[2]. The resolution was given in [5]; however, the verification was done via computer program. We have constructed a contracting homotopy for this resolution. The associated diagonal map using the result of Handel[4] was also defined, which enables one to compute the ring structure of the group cohomology with arbitrary coefficients. References

[1] J. Bryden, C. Hayat-Legrand, H. Zieschang, and P. Zvengrowski, The cohomology ring of a class of Seifert manifolds , Topology Appl. 105 (2000), no. 2, 123–156. [2] H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups , fourth ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 14, Springer-Verlag, Berlin, 1980. [3] L.L. Fêmina, A.P.T. Galves, O. Manzoli Neto, and M. Spreaﬁco, Cellular decomposition and free resolution for split metacyclic spherical space forms , Homology, Homotopy and Applications 15 (2013), no. 1, 253-278. [4] D. Handel, On products in the cohomology of the dihedral groups , Tohoku Math. J. (2) 45 (1993), no. 1, 13–42. [5] S. Tomoda and P. Zvengrowski, Remarks on the cohomology of ﬁnite fundamental groups of 3-manifolds , Geometry and Topology Monographs 14 (2008), 519–556.

1 Okangan College, Kelowna, Canada. [email protected] 2 Universidade de São Paulo, São Carlos, Brazil. [email protected] 3 Universitá del Salento, Lecce, Italy. [email protected] 4 Universidade Federal de Uberlândia, Brazil. [email protected] 5 Universidade Federal de Uberlândia, Brazil. [email protected]

56 Zk-actions fixing KP m KP n 2 (2 + 1) t (2 + 1) Allan Edley Ramos de Andrade, 1 Pedro Pergher, 2 and Ser gio Tsuyoshi Ura 3

Abstract

In [1], D. C. Royster classified, up to equivariant cobordism, involutions fixing disjoint unions of two real projective spaces. Motivated by [1], [2] and [3], we obtained a classification, up to Zk equivariant cobordism, of 2-actions fixing a disjoint union of two real, complex or quaternionic projective spaces of odd dimensions, KP (2 m + 1) t KP (2 n + 1) . Specifically, we prove that such Zk 2-action bounds equivariantly. References

1 UFMS. [email protected] 2 UFSCar. [email protected] 3 UFSCar. [email protected]

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Dual higher order Whitehead products

Thi ago de Melo 1 and Marek Golasiński 2

Abstract

In this work we will deﬁne the higher order Whitehead product for co-H-spaces G1, . . . , G n by means of the product ‘ ◦’ introduced by Gray ([2]) and also obtain the same properties as Porter. To do this we make use of ψ θ G ◦ H −→ Σ(Ω G ∧ ΩH) −→ G ◦ H where θ is projection and θψ ' 1. Let G1, . . . , G n be co-H-spaces. We deﬁne 0 φn : ( G1 ◦ G2) ∧ G3 ∧ · · · ∧ Gn → F W (G).

The next step is to follow Porter’s paper to show, for example, that given fi : Gi → Y for i = 1 ,...,n , if Y is an H-space then 0 ∈ [f1,...,f n]. Also, we plan to characterize all spaces Y with 0 ∈ [f1,...,f n]. Finally, Hardie ([3]) and then Porter ([4]) have deﬁned higher Whitehead products for maps fi : ΣAi → X. But Arkowitz ([1]) has deﬁned a dual Whitehead product for two maps fi : X → ΩAi. So it is natural to ask about the existence of a dual higher order Whitehead product. This is the purpose of this work. References

[1] M. Arkowitz, The generalized Whitehead product , Paciﬁc J. Math. 12 (1962), 7–23. [2] B. Gray, On generalized Whitehead products , Trans. Amer. Math. Soc. 11 (2011), 6143–6158. [3] K. A. Hardie, On a construction of E. C. Zeeman , [4] G. J. Porter, Higher order Whitehead products , Topology 3 (1965), 123–135.

1 IGCE–Unesp Rio Claro, Brazil. [email protected] 2 Institute of Mathematics, Casimir the Great University, pl. Weyssenhoﬀa 11 85–072, Bydgoszcz, Poland. [email protected]

57 Study of certain singularity sets associated to a polynomial map

Thuy Nguyen Thi Bich 1

Abstract

We present a method, called “la méthode des façons,” to study and classify the asymptotic set n n SF of a dominant polynomial map F : C → C . In 2010 Anna and Guillaume Valette constructed n n a real pseudomanifold VF associated to a polynomial map F : C → C . The singular part of VF is contained in SF ∪ K0(F ), where K0(F ) is the set of critical values of F . In case n = 2 they proved that if F is a polynomial map with K0(F ) = ∅, then it is not proper if and only if the intersection homology of VF is non-trivial in dimension 2. We give a generalization of this result, n n to the case of a polynomial map F = ( F1, .. . ,F n) : C → C . This result has relationships to the Jacobian conjecture. In order to calculate the intersection homology of VF , we take a stratiﬁcation of the asymptotic set SF . This is the motivation of this work. Through “la méthode des façons,” we can also characterize the asymptotic set SF .

1 IML–Institut de Mathématiques de Luminy.

A Typeset by L TEX (July 27, 2014) – Total of abstracts: 68 = 41 talks + 27 posters

58 1 Alexandre Paiva Barreto UFSCar – São Carlos - SP 2 Alice Kimie Miwa Libardi UNESP – Rio Claro - SP 3 Allan Edley Ramos de Andrade UFMS – Três Lagoas - MS 4 Amanda Ferreira de Lima UFSCar – São Carlos - SP 5 Ana Maria Mathias Morita USP – São Carlos - SP 6 Ana Paula Tremura Galves UFU – Uberlândia - MG 7 Anderson Luis Gama UFPR – Curitiba - PR 8 Anderson Paião dos Santos UTFPR – Cornélio Procópio - PR 9 António Manuel Freitas Gomes Cunha Salgueiro Universidade de Coimbra 10 Brian Callander UNICAMP – Campinas - SP 11 Bruno Caldeira Carlotti de Souza UNESP – São José do Rio Preto - SP 12 Carolina de Miranda e Pereiro UFSCar – São Carlos - SP 13 Catarina Mendes de Jesus UFV – Viçosa – MG 14 Claude Hayat Institut Mathématiques de Toulouse - France 15 Claudemir Aniz UFMS – Campo Grande - MS 16 Clotilzio Moreira dos Santos UNESP – São José do Rio Preto - SP 17 Daciberg Lima Goncalves USP – São Paulo – SP 18 Dahisy Valadão de Souza Lima UNICAMP – Campinas - SP 19 Daniel Vendrúscolo UFSCar – São Carlos - SP 20 Darlan Girao Universidade Federal do Ceará - Fortaleza - CE Benemérita Universidad Autónoma de Puebla - 21 David Herrera Carrasco Mexico 22 Denise de Mattos USP – São Carlos - SP 23 Dirceu Penteado UFSCar – São Carlos - SP 24 Edivaldo Lopes dos Santos UFSCar – São Carlos - SP 25 Eduardo Hoefel UFPR – Curitiba - PR 26 Elizabeth Terezinha Gasparim UNICAMP – Campinas - SP 27 Erica Boizan Batista UNESP - São José do Rio Preto - SP 28 Ermínia de Lourdes Campello Fanti UNESP - São José do Rio Preto - SP 29 Évelin Meneguesso Barbaresco UNESP - São José do Rio Preto - SP 30 Fabiana Santos Cotrim UFSCar – São Carlos - SP Benemérita Universidad Autónoma de Puebla - 31 Fernando Macías Romero Mexico 32 Flávia Souza Machado da Silva UNESP - São José do Rio Preto - SP 33 Gabriel Calsamiglia UFF – Niterói - RJ 34 Givanildo Donizeti de Melo UNESP - Rio Claro - SP 35 Guido Gerson Espiritu Ledesma UNICAMP – Campinas - SP 36 Guilherme Vituri Fernandes Pinto UNESP - Rio Claro - SP 37 Gustavo Cazzeri Innocencio Figueiredo UFSCar – São Carlos - SP 38 Gustavo de Lima Prado USP - São Paulo – SP 39 Izael do Nascimento UFPR – Curitiba - PR 40 Jean-Paul Brasselet CNRS - France 41 Jéfferson Luiz Rocha Bastos UNESP – São José do Rio Preto - SP 42 Jérôme Los CNRS - France 43 Jessica Cristina Rossinati Rodrigues da Costa UNESP – São José do Rio Preto - SP 44 João Carlos Ferreira Costa UNESP – São José do Rio Preto - SP 45 João Miguel Nogueira Universidade de Coimbra - Portugal 46 João Peres Vieira UNESP - Rio Claro - SP 47 John Mc Cleary Vassar College, USA 48 José Gregorio Rodríguez Nieto National University of Colombia - Colombia 49 Juan Carlos Rocha Barriga UFSCar – São Carlos - SP 50 Juan Valentín Mendoza Mogollón UFV – Viçosa - MG 51 Kisnney Emiliano de Almeida UEFS – Feira de Santana - BA 52 Leticia Sanches Silva UNESP – São José do Rio Preto - SP 53 Ligia Laís Fêmina UFU – Uberlândia - MG 54 Luciana de Fátima Martins UNESP – São José do Rio Preto - SP 55 Luiz Roberto Hartmann Junior UFSCar – São Carlos - SP 56 Marcelo Jose Saia USP - São Carlos - SP 57 Marcio de Jesus Soares UFSCar – São Carlos - SP 58 Marek Golasinski Casimir the Great University - Poland 59 Maria Gorete Carreira Andrade UNESP – São José do Rio Preto - SP 60 Mariana Rodrigues da Silveira UFABC – Santo André - SP 61 Michael J. Hopkins Harvard University - USA 62 Michael Richard Kelly Loyola University New Orleans - USA 63 Michelle Ferreira Zanchetta Morgado UNESP – São José do Rio Preto - SP 64 Neemias Silva Martins UFV – Viçosa - MG 65 Nelson Antonio Silva USP - São Carlos - SP 66 Nguyen Thi Bich Thuy USP - São Carlos - SP 67 Northon Canevari Leme Penteado USP - São Carlos - SP 68 Osamu Saeki Kyushu University – Japan 69 Oscar Ocampo UFBA – Salvador - BA 70 Oziride Manzoli Neto USP - São Carlos - SP 71 Pablo Gonzalez Pagotto UNESP - Rio Claro - SP 72 Pantaleón David Romero Sánchez Universidad CEU Cardenal Herrera - Spain 73 Parameswaran Sankaran IMSc - Chennai - India 74 Paul A. Schweitzer, S.J. PUC - Rio de Janeiro - RJ 75 Paulo Henrique Cabido Gusmão UFF – Niterói - RJ 76 Pedro Luiz Queiroz Pergher USP - São Carlos - SP 77 Peter Wong Bates College - USA 78 Rafael Moreira de Souza UFU – Ituiutaba - MG 79 Renato Vasconcellos Vieira USP – São Paulo - SP 80 Rosa Elvira Quispe Ccoyllo UFES – Vitória - ES 81 Satoshi Tomoda Okanagan College - Canada 82 Sergio Tsuyoshi Ura UFSCar – São Carlos - SP 83 Severin Barmeier UNICAMP – Campinas - SP 84 SIMONE MORAES UFV – Viçosa - MG 85 Stephanie Akemi Raminelli UNICAMP – Campinas - SP 86 Taciana Oliveira Souza UFU – Uberlândia - MG 87 Thaís Fernanda Mendes Monis UNESP - Rio Claro - SP 88 Thiago de Melo UNESP - Rio Claro - SP 89 Tomas Edson Barros UFSCar – São Carlos - SP 90 Vinicius Casteluber Laass USP – São Paulo - SP 91 Vladimir V. Sharko National Academy of Sciences of Ukraine 92 Wacław Bolesław Marzantowicz UAM – Poznań - Poland