Introducción a las descomposiciones de Heegaard y la homología de Heegaard-Floer.

ilustraciones

Autores:
Zapata Rendón, Sebastian
Tipo de recurso:
Fecha de publicación:
2021
Institución:
Universidad Nacional de Colombia
Repositorio:
Universidad Nacional de Colombia
Idioma:
spa
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oai:repositorio.unal.edu.co:unal/80340
Acceso en línea:
https://repositorio.unal.edu.co/handle/unal/80340
https://repositorio.unal.edu.co/
Palabra clave:
510 - Matemáticas
Variedades (Algebra universal)
Varieties (Universal algebra)
Heegaard-Floer
Descomposiciones de Heegaard
3 Variedades
Cuerpos de asas
Complejo de curvas
Espacios lenticulares
Heegaard-Floer
Heegaard splitting
3 Manifolds
Handlebody
Complex curve
Lens spaces
Rights
openAccess
License
Reconocimiento 4.0 Internacional
id UNACIONAL2_3162b7252d3bd1ccec9313113a8da85a
oai_identifier_str oai:repositorio.unal.edu.co:unal/80340
network_acronym_str UNACIONAL2
network_name_str Universidad Nacional de Colombia
repository_id_str
dc.title.spa.fl_str_mv Introducción a las descomposiciones de Heegaard y la homología de Heegaard-Floer.
dc.title.translated.eng.fl_str_mv Introduction to Heegaard splittings and Heegaard-Floer Homology.
title Introducción a las descomposiciones de Heegaard y la homología de Heegaard-Floer.
spellingShingle Introducción a las descomposiciones de Heegaard y la homología de Heegaard-Floer.
510 - Matemáticas
Variedades (Algebra universal)
Varieties (Universal algebra)
Heegaard-Floer
Descomposiciones de Heegaard
3 Variedades
Cuerpos de asas
Complejo de curvas
Espacios lenticulares
Heegaard-Floer
Heegaard splitting
3 Manifolds
Handlebody
Complex curve
Lens spaces
title_short Introducción a las descomposiciones de Heegaard y la homología de Heegaard-Floer.
title_full Introducción a las descomposiciones de Heegaard y la homología de Heegaard-Floer.
title_fullStr Introducción a las descomposiciones de Heegaard y la homología de Heegaard-Floer.
title_full_unstemmed Introducción a las descomposiciones de Heegaard y la homología de Heegaard-Floer.
title_sort Introducción a las descomposiciones de Heegaard y la homología de Heegaard-Floer.
dc.creator.fl_str_mv Zapata Rendón, Sebastian
dc.contributor.advisor.none.fl_str_mv Toro Villegas, Margarita María
dc.contributor.author.none.fl_str_mv Zapata Rendón, Sebastian
dc.subject.ddc.spa.fl_str_mv 510 - Matemáticas
topic 510 - Matemáticas
Variedades (Algebra universal)
Varieties (Universal algebra)
Heegaard-Floer
Descomposiciones de Heegaard
3 Variedades
Cuerpos de asas
Complejo de curvas
Espacios lenticulares
Heegaard-Floer
Heegaard splitting
3 Manifolds
Handlebody
Complex curve
Lens spaces
dc.subject.lemb.spa.fl_str_mv Variedades (Algebra universal)
dc.subject.lemb.eng.fl_str_mv Varieties (Universal algebra)
dc.subject.proposal.spa.fl_str_mv Heegaard-Floer
Descomposiciones de Heegaard
3 Variedades
Cuerpos de asas
Complejo de curvas
Espacios lenticulares
dc.subject.proposal.eng.fl_str_mv Heegaard-Floer
Heegaard splitting
3 Manifolds
Handlebody
Complex curve
Lens spaces
description ilustraciones
publishDate 2021
dc.date.accessioned.none.fl_str_mv 2021-09-29T20:44:45Z
dc.date.available.none.fl_str_mv 2021-09-29T20:44:45Z
dc.date.issued.none.fl_str_mv 2021
dc.type.spa.fl_str_mv Trabajo de grado - Maestría
dc.type.driver.spa.fl_str_mv info:eu-repo/semantics/masterThesis
dc.type.version.spa.fl_str_mv info:eu-repo/semantics/acceptedVersion
dc.type.content.spa.fl_str_mv Text
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status_str acceptedVersion
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dc.identifier.instname.spa.fl_str_mv Universidad Nacional de Colombia
dc.identifier.reponame.spa.fl_str_mv Repositorio Institucional Universidad Nacional de Colombia
dc.identifier.repourl.spa.fl_str_mv https://repositorio.unal.edu.co/
url https://repositorio.unal.edu.co/handle/unal/80340
https://repositorio.unal.edu.co/
identifier_str_mv Universidad Nacional de Colombia
Repositorio Institucional Universidad Nacional de Colombia
dc.language.iso.spa.fl_str_mv spa
language spa
dc.relation.references.spa.fl_str_mv P. Bandieri, P. Cristofori, and C. Gagliardi. A Census of Genus-Two Manifolds up to 42 Coloured Tetrahedra. Discrete Mathematics, 310:2469-2481, 2010.
R. Bhatia and K. Mukherjea. The Space of Unordered Tuples of Complex Numbers. Linear Algebra and its Applications, 52/53:765-768, 1983.
J. Birman, F. Gonz ales, and J. M. Montesinos. Heegaard Splittings of Prime 3-manifolds are not unique. Michigan Mathematical Journal, 23(2):97-103, 1976.
N. Bohm. Morse Theory and Handle Decompositions. University of Chicago Mathematics REU, 2019.
M. Boileau and J. Otal. Sur les Scindements de Heegaard du Tore T3. Journal of Diferential Geometry, 32(1):209-233, 1990.
F. Bonahon and J. Otal. Scindements de Heegaard des espaces lenticulaires . Comptes Rendus de l'Academie des Sciences- Series I - Mathematics, 294(17):585-587, 1982.
A. Casson and C. Gordon. Reducing Heegaard Splittings. Topology and its Applications, 27:275-283, 1987.
P. Cristofori, C. Gagliardi, and L. Grasselli. Heegaard and Regular Genus of 3- Manifold with Boundary. Revista Matemática de la Universidad Complutense de Madrid, 8(2):379-398, 1995.
B. Farb and D. Margalit. A Primer on Mapping Class Groups. Princeton University Press, 2012.
S. Fushida. Homology 3-Spheres. 381 Sloan Hall, Stanford University, CA, Course notes, 2020.
D. Gay and R. Kirby. Trisecting 4-Manifolds. Geometry and Topology, 20:3097-3132, 2016.
L. Goeritz. Die Heegaard-Diagramme des Torus. Abhandlungen aus dem Mathematischen Seminar der Universit at Hamburg, 9:187-188, 1932.
J. E. Greene. Heegaard Floer Homology. Notices of the American Mathematical Society, 68:19-33, 2021.
V. Gugenheim. Piecewise linear isotopy and embedding of elements and spheres. Proceedings of the London Mathematical Society, 3:29-53, 1953.
W. Haken. Some results on surfaces in 3-Manifolds. Studies in Modern Topology, Mathematical Association of America, Prentice-Hall, pages 34-98, 1968.
W. Harvey. Geometric Structure of Surface Mapping Class Groups. Homological group theory (Proc. Sympos., Durham,1977), London Mathematical Society Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, pages 255-269,1979.
A. Hatcher. Algebraic Topology. Cambridge University Press, 2002.
P. Heegaard. Forstudier til en topologisk Teori for de algebraiske Fladers Sammenhaeng. Doctoral Thesis, 1898.
J. Hempel. 3-Manifolds. American Mathematical Society Chelsea Publishing, 1976.
J. Hempel. 3-Manifolds as Viewed from the Curve Complex. Topology, 40(3):631-657, 2001.
A. Ido. An estimation of Hempel distance by using Reeb graph. Department of Mathematics, Nara Women's University, 2010.
J.Johnson. Notes on Heegaard Splittings. 2006.
J. Johnson. Stable Functions and Common Stabilizations of Heegaard Splittings. Transactions of the American Mathematical Society, 361, 2007.
J. Johnson. Stable Functions and Common Stabilizations of Heegaard Splittings. Transactions of the Ametican Mathematical Society, 361(7):3747-3765, 2009.
J. Karabas, P. Malicky, and R. Nedela. Three-Manifolds with Heegaard Genus atmost Two Represented by Crystallisations with at most 42 Vertices. Discrete Mathematics, 307(21):2569-2590, 2007.
S. Lins. Gems, Computers And Attractors for 3-Manifolds. World Scienti c. Series on Knots and Everything-Vol. 5, 1995.
J. Milnor. A Unique Factorization Theorem for 3-Manifolds. American Journal of Mathematics, 84:1-7, 1962.
J. Milnor. Morse Theory. Based on lecture notes by M.Spivak and R. Wells. Annals of Mathematics Studies. Princeton University Press, Princeton, 1963.
J. Milnor. Lectures on the h-Cobordism Theorem. Princeton University Press, 1965.
E. Moise. Affne structures in 3-manifolds.V. The Triangulation Theorem and Hauptvermutung. Annals of Mathematics, 56(2):96-114, 1952.
A. Nagasato. Estimations of distances for Heegaard splittings, and for bridge decomposition. Doctoral Thesis, Nara Women's University, 2013.
P. S. Ozvath and Z. Szabo. Holomorphic disks and topological invariants for closedthree-manifolds. Annals of Mathematics, 159(3):1027-1158, 2004.
P. S. Ozvath and Z. Szabo. An Introduction to Heegaard Floer Homology. American Mathematical Society, Floer Homology, Gauge Theory, and Low-Dimensional Topology:3-27, 2006.
H. Poincare. Analysis Situs. Journal de l'Ecole Polytechnique, 19(4):1-123, 1895.
E. Rivera. Enlaces de tres puentes. Tesis doctoral, Universidad Nacional de Colombia Sede Medellin, 2016.
H. Rubinstein and M. Scharlemann. Comparing Heegaard Splittings of non-Haken 3-Manifolds. Topology, 35(4):1005-1026, 1996.
C. Sadanand. A Two Dimensional Description of Heegaard Splittings. Doctoral thesis, Stony Brook University, 2017.
O. Saeki and T. Kobayashi. The Rubinstein-Scharlemann Graphic of a 3-Manifold as the Discriminant Set of a Stable Map. Paci c Journal of Mathematics, 195:101-156, 2000.
B. Sahamie. Introduction to the Basics of Heegaard Floer Homology. Annales de la Faculte des Sciences de Toulouse, 22(2):269-336, 2013.
N. Savaliev. Lectures on the Topology of 3-Manifolds. An Introduction to the Casson Invariant. De Gruyter, 2012.
M. Scharlemann. Heegaard Splittings of 3-Manifolds. Low dimensional topology. Lectures at the Morningside Center of Mathematics. Notes by Ruifeng Qiu, pages 25-39, 2003.
M. Scharlemann and A. Thompson. Thin position for 3-Manifolds. Contemporary Mathematics, 164:231-238, 1994.
J. Schultens. Heegaard splittings of Seifert bered spaces with boundary. Transactions of the American Mathematical Society, 347(7), 1995.
J. Schultens. Introduction to 3-Manifolds. American Mathematical Society, 2014.
J. Singer. Three-Dimensional Manifolds and Their Heegaard Diagrams. Transactions of the American Mathematical Society, 35:88-111, 1933.
L. Struth. Hakens Lemma. Hausarbeit zur Vorlesung Kirby-Kalk ul bei Dr. Marc Kegel. 2018.
F. Waldhausen. Heegaard-Zerlegungen der 3-Sph are. Topology, 7:195-203, 1986.
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dc.publisher.spa.fl_str_mv Universidad Nacional de Colombia
dc.publisher.program.spa.fl_str_mv Medellín - Ciencias - Maestría en Ciencias - Matemáticas
dc.publisher.department.spa.fl_str_mv Escuela de matemáticas
dc.publisher.faculty.spa.fl_str_mv Facultad de Ciencias
dc.publisher.place.spa.fl_str_mv Medellín, Colombia
dc.publisher.branch.spa.fl_str_mv Universidad Nacional de Colombia - Sede Medellín
institution Universidad Nacional de Colombia
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spelling Reconocimiento 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Toro Villegas, Margarita Maríaabaa6c07e2063dce2203518ab8a5075eZapata Rendón, Sebastianeea970f95f0a1c3ef96dc188e0f8dd182021-09-29T20:44:45Z2021-09-29T20:44:45Z2021https://repositorio.unal.edu.co/handle/unal/80340Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/ilustracionesSe estudia a fondo la construcción de las descomposiciones de Heegaard, haciendo énfasis en el por qué es una herramienta central en el estudio de las 3-variedades. Para esto, analizamos las 3-variedades desde las categorías topológica, suave y triángulable, en donde siempre podemos derivar el concepto de descomposición de Heegaard como algo arraigado a la 3-variedad misma. Luego de esto se explora uno de los invariantes de 3-variedades más recientes, la homología de Heegaard-Floer, centrándonos en la aparición de las descomposiciones de Heegaard en su construcción, sirviendo así como eje motivador para futuros proyectos. (Texto tomado de la fuente)The construction of Heegaard decompositions is studied in depth, emphasizing why it is a central tool in the study of 3-manifolds. For this, we analyze the 3-manifolds from the topological, smooth and triangular categories, where we can always derive the Heegaard decomposition concept as something rooted in the 3-manifold itself. After this, one of the most recent 3-manifold invariants is explored, the Heegaard-Floer homology, focusing on the appearance of Heegaard decompositions in its construction, and thus serving as a motivating axis for future projects.MaestríaMagíster en Ciencias - Matemáticas83 páginasapplication/pdfspaUniversidad Nacional de ColombiaMedellín - Ciencias - Maestría en Ciencias - MatemáticasEscuela de matemáticasFacultad de CienciasMedellín, ColombiaUniversidad Nacional de Colombia - Sede Medellín510 - MatemáticasVariedades (Algebra universal)Varieties (Universal algebra)Heegaard-FloerDescomposiciones de Heegaard3 VariedadesCuerpos de asasComplejo de curvasEspacios lenticularesHeegaard-FloerHeegaard splitting3 ManifoldsHandlebodyComplex curveLens spacesIntroducción a las descomposiciones de Heegaard y la homología de Heegaard-Floer.Introduction to Heegaard splittings and Heegaard-Floer Homology.Trabajo de grado - Maestríainfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/acceptedVersionTexthttp://purl.org/redcol/resource_type/TMP. Bandieri, P. Cristofori, and C. Gagliardi. A Census of Genus-Two Manifolds up to 42 Coloured Tetrahedra. Discrete Mathematics, 310:2469-2481, 2010.R. Bhatia and K. Mukherjea. The Space of Unordered Tuples of Complex Numbers. Linear Algebra and its Applications, 52/53:765-768, 1983.J. Birman, F. Gonz ales, and J. M. Montesinos. Heegaard Splittings of Prime 3-manifolds are not unique. Michigan Mathematical Journal, 23(2):97-103, 1976.N. Bohm. Morse Theory and Handle Decompositions. University of Chicago Mathematics REU, 2019.M. Boileau and J. Otal. Sur les Scindements de Heegaard du Tore T3. Journal of Diferential Geometry, 32(1):209-233, 1990.F. Bonahon and J. Otal. Scindements de Heegaard des espaces lenticulaires . Comptes Rendus de l'Academie des Sciences- Series I - Mathematics, 294(17):585-587, 1982.A. Casson and C. Gordon. Reducing Heegaard Splittings. Topology and its Applications, 27:275-283, 1987.P. Cristofori, C. Gagliardi, and L. Grasselli. Heegaard and Regular Genus of 3- Manifold with Boundary. Revista Matemática de la Universidad Complutense de Madrid, 8(2):379-398, 1995.B. Farb and D. Margalit. A Primer on Mapping Class Groups. Princeton University Press, 2012.S. Fushida. Homology 3-Spheres. 381 Sloan Hall, Stanford University, CA, Course notes, 2020.D. Gay and R. Kirby. Trisecting 4-Manifolds. Geometry and Topology, 20:3097-3132, 2016.L. Goeritz. Die Heegaard-Diagramme des Torus. Abhandlungen aus dem Mathematischen Seminar der Universit at Hamburg, 9:187-188, 1932.J. E. Greene. Heegaard Floer Homology. Notices of the American Mathematical Society, 68:19-33, 2021.V. Gugenheim. Piecewise linear isotopy and embedding of elements and spheres. Proceedings of the London Mathematical Society, 3:29-53, 1953.W. Haken. Some results on surfaces in 3-Manifolds. Studies in Modern Topology, Mathematical Association of America, Prentice-Hall, pages 34-98, 1968.W. Harvey. Geometric Structure of Surface Mapping Class Groups. Homological group theory (Proc. Sympos., Durham,1977), London Mathematical Society Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, pages 255-269,1979.A. Hatcher. Algebraic Topology. Cambridge University Press, 2002.P. Heegaard. Forstudier til en topologisk Teori for de algebraiske Fladers Sammenhaeng. Doctoral Thesis, 1898.J. Hempel. 3-Manifolds. American Mathematical Society Chelsea Publishing, 1976.J. Hempel. 3-Manifolds as Viewed from the Curve Complex. Topology, 40(3):631-657, 2001.A. Ido. An estimation of Hempel distance by using Reeb graph. Department of Mathematics, Nara Women's University, 2010.J.Johnson. Notes on Heegaard Splittings. 2006.J. Johnson. Stable Functions and Common Stabilizations of Heegaard Splittings. Transactions of the American Mathematical Society, 361, 2007.J. Johnson. Stable Functions and Common Stabilizations of Heegaard Splittings. Transactions of the Ametican Mathematical Society, 361(7):3747-3765, 2009.J. Karabas, P. Malicky, and R. Nedela. Three-Manifolds with Heegaard Genus atmost Two Represented by Crystallisations with at most 42 Vertices. Discrete Mathematics, 307(21):2569-2590, 2007.S. Lins. Gems, Computers And Attractors for 3-Manifolds. World Scienti c. Series on Knots and Everything-Vol. 5, 1995.J. Milnor. A Unique Factorization Theorem for 3-Manifolds. American Journal of Mathematics, 84:1-7, 1962.J. Milnor. Morse Theory. Based on lecture notes by M.Spivak and R. Wells. Annals of Mathematics Studies. Princeton University Press, Princeton, 1963.J. Milnor. Lectures on the h-Cobordism Theorem. Princeton University Press, 1965.E. Moise. Affne structures in 3-manifolds.V. The Triangulation Theorem and Hauptvermutung. Annals of Mathematics, 56(2):96-114, 1952.A. Nagasato. Estimations of distances for Heegaard splittings, and for bridge decomposition. Doctoral Thesis, Nara Women's University, 2013.P. S. Ozvath and Z. Szabo. Holomorphic disks and topological invariants for closedthree-manifolds. Annals of Mathematics, 159(3):1027-1158, 2004.P. S. Ozvath and Z. Szabo. An Introduction to Heegaard Floer Homology. American Mathematical Society, Floer Homology, Gauge Theory, and Low-Dimensional Topology:3-27, 2006.H. Poincare. Analysis Situs. Journal de l'Ecole Polytechnique, 19(4):1-123, 1895.E. Rivera. Enlaces de tres puentes. Tesis doctoral, Universidad Nacional de Colombia Sede Medellin, 2016.H. Rubinstein and M. Scharlemann. Comparing Heegaard Splittings of non-Haken 3-Manifolds. Topology, 35(4):1005-1026, 1996.C. Sadanand. A Two Dimensional Description of Heegaard Splittings. Doctoral thesis, Stony Brook University, 2017.O. Saeki and T. Kobayashi. The Rubinstein-Scharlemann Graphic of a 3-Manifold as the Discriminant Set of a Stable Map. Paci c Journal of Mathematics, 195:101-156, 2000.B. Sahamie. Introduction to the Basics of Heegaard Floer Homology. Annales de la Faculte des Sciences de Toulouse, 22(2):269-336, 2013.N. Savaliev. Lectures on the Topology of 3-Manifolds. An Introduction to the Casson Invariant. De Gruyter, 2012.M. Scharlemann. Heegaard Splittings of 3-Manifolds. Low dimensional topology. Lectures at the Morningside Center of Mathematics. Notes by Ruifeng Qiu, pages 25-39, 2003.M. Scharlemann and A. Thompson. Thin position for 3-Manifolds. Contemporary Mathematics, 164:231-238, 1994.J. Schultens. Heegaard splittings of Seifert bered spaces with boundary. Transactions of the American Mathematical Society, 347(7), 1995.J. Schultens. Introduction to 3-Manifolds. American Mathematical Society, 2014.J. Singer. Three-Dimensional Manifolds and Their Heegaard Diagrams. Transactions of the American Mathematical Society, 35:88-111, 1933.L. Struth. Hakens Lemma. Hausarbeit zur Vorlesung Kirby-Kalk ul bei Dr. Marc Kegel. 2018.F. Waldhausen. Heegaard-Zerlegungen der 3-Sph are. Topology, 7:195-203, 1986.InvestigadoresLICENSElicense.txtlicense.txttext/plain; charset=utf-83964https://repositorio.unal.edu.co/bitstream/unal/80340/1/license.txtcccfe52f796b7c63423298c2d3365fc6MD51ORIGINAL1020490802.2021.pdf1020490802.2021.pdfTesis Maestría en Ciencias - Matemáticasapplication/pdf5005742https://repositorio.unal.edu.co/bitstream/unal/80340/2/1020490802.2021.pdfd7f64de25011caef732012474948b146MD52THUMBNAIL1020490802.2021.pdf.jpg1020490802.2021.pdf.jpgGenerated Thumbnailimage/jpeg4929https://repositorio.unal.edu.co/bitstream/unal/80340/3/1020490802.2021.pdf.jpg0b8bce143e0ae9363185af73896b1408MD53unal/80340oai:repositorio.unal.edu.co:unal/803402023-07-29 23:03:25.892Repositorio Institucional Universidad Nacional de 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GVyZWNob3MgZGUgYXV0b3IgcXVlIGNvbmxsZXZlIGxhIGRpc3RyaWJ1Y2nDs24gZGUgZXN0b3MgYXJjaGl2b3MgeSBtZXRhZGF0b3MuCkFsIGhhY2VyIGNsaWMgZW4gZWwgc2lndWllbnRlIGJvdMOzbiwgdXN0ZWQgaW5kaWNhIHF1ZSBlc3TDoSBkZSBhY3VlcmRvIGNvbiBlc3RvcyB0w6lybWlub3MuCgpVTklWRVJTSURBRCBOQUNJT05BTCBERSBDT0xPTUJJQSAtIMOabHRpbWEgbW9kaWZpY2FjacOzbiAyNy8yMC8yMDIwCg==

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