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1. The Borsuk-Ulam theorem for closed $3$-manifolds having geometry $S^2\times \R$

Author

Bauval, A., Gonçalves, D. L. \, Hayat, C., and Zvengrowski, P.

Subjects
Mathematics - Algebraic Topology
Abstract

Let $M$ be a closed 3-manifold which admits the geometry $S^2\times \R$. In this work we determine all the free involutions $\tau$ on $M$, and the Borsuk-Ulam index of $(M,\tau)$.

Published
2020

3. The Borsuk-Ulam theorem for the Seifert manifolds having flat geometry

Author

Bauval, A., Gonçalves, D. L., and Hayat, C.

Subjects
Mathematics - Geometric Topology, 57M05 57M10 57M50
Abstract

Let M be a Seifert manifold which belongs to the geometry Flat. In this work we determine all the free involutions {\tau} on M, and the Borsuk-Ulam indice of (M,{\tau}).

Published
2018

4. Preuve simplifi\'ee du th\'eor\`eme de Serret sur les nombres \'equivalents

Author

Bauval, Anne

Subjects
Mathematics - Number Theory, 11A55
Abstract

If the continued fractions of two irrational numbers have a common complete quotient, then these two numbers are in the same orbit under the action of $\mathrm{PGL}(2,\mathbb{Z})$. The converse is Serret's well-known theorem, but we give a simpler and purely algebraic proof of it., Comment: in French

Published
2017

5. Th\'eor\`eme d'Eilenberg-Zilber en homologie cyclique enti\`ere

Author

Bauval, Anne

Subjects
Mathematics - K-Theory and Homology, 16E40, 19D55
Abstract

For simplicial modules, Eilenberg-Zilber's classical theorem states the existence of a product $sh : M\otimes N\to M\times N$ (the shuffle) and a coproduct $AW : M\times N\to M\otimes N$ (the Alexander-Whitney map), which are quasi-inverse of eachother. A cyclic version of this theorem was established in 1987 by Hood and Jones: they proved that $sh$ and $AW$ admit "coextensions" $sh_\infty$ and $AW_\infty$, using an acyclic-model method. Besides, an explicit formula for $sh_\infty$ has been discovered by several authors. But the question remained open of such an explicit formula for $AW_\infty$, and for the homotopies by which $sh_\infty$ and $AW_\infty$ are mutual quasi-inverses and are quasi-(co)-associative. We present a complete answer to this problem and show that all these -- now explicit -- maps extend continuously to entire cyclic complexes (associated to normed algebras)., Comment: in French, Preprint written and disseminated in 1998

Published
2016

6. Suites r\'ecurrentes lin\'eaires d'ordre 2 \`a divisibilit\'e forte

Author

Bauval, A.

Subjects
Mathematics - Number Theory, 11B39, 11A05
Abstract

We reprove twice, in a simpler but as elementary way, a result by Hor\'ak and Skula (1985) who determined, among all sequences of integers defined by $$u_1=1,\quad u_2=R,\quad u_{n+2}=Pu_{n+1}-Qu_n$$ for some integers $P,Q,R$, those which satisfy the strong divisibility condition $$\forall i,j\in\mathbb N^*\quad u_i\land u_j=\left|u_{i\land j}\right|,$$ where $\land$ denotes the greatest common divisor., Comment: in French

Published
2016

7. Reidemeister-Schreier's Algorithm For 2-Coverings of Seifert Manilfolds

Author

Bauval, A. and Hayat, C.

Subjects
Mathematics - Geometric Topology, Primary 57M10, Secondary 57M05, 57S25
Abstract

It is classical that given any Seifert structure on N, Reidemeister-Schreier's algorithm produces a presentation of all index 2 subgroups of the fundamental group of N, described as the fundamental group of some Seifert manifolds. The new result of this article is concise formulas that gather all possible cases., Comment: 12 pages

Published
2014

8. Littlewood's proof of the fundamental theorem of algebra: a simpler version

Author

Bauval, Anne

Subjects
Mathematics - History and Overview, Mathematics - Number Theory, 12D05, 30A10
Abstract

We present an elementary proof of the fundamental theorem of algebra, following Cauchy's version but avoiding his use of circular functions. It is written in the same spirit as Littlewood's proof of 1941, but reduces it to more elementary and constructive arguments.

Published
2014

9. An elementary proof of the halting property for Chakravala algorithm

Author

Bauval, Anne

Subjects
Mathematics - Number Theory, 11A: Elementary number theory
Abstract

In 1930, A.A.K. Ayyangar allegedly produced the missing proof that the ancient Indian Chakravala algorithm, designed to solve Pell's equation, always halts. Refining his own elementary arguments, we give a correct and shorter proof.

Published
2014

10. L'anneau de cohomologie de toutes les vari\'et\'es de Seifert

Author

Bauval, A. and Hayat, C.

Subjects
Mathematics - Algebraic Topology, Mathematics - Geometric Topology, 13D03, 57S25 (Primary) 55N45 (Secondary)
Abstract

The cohomology ring with coefficients in $\Z_p$, where $p$ is a prime integer, of a Seifert manifold $M$, orientable or not orientable is obtained from a simplicial decomposition of $M$. Many choices must be made before applying Alexander-Whitney formula to get the cup-products. The most difficult choices are those of liftings from the cellular complex to the simplicial complex when we add the condition to be cocycles., Comment: few typ corrections 20 figures

Published
2012

11. A classification of special 2-fold coverings

Author

Bauval, Anne, Goncalves, Daciberg L, Hayat, Claude, and Mello, Maria Herminia de Paula Leite

Subjects
Mathematics - Algebraic Topology, Mathematics - Geometric Topology, 57R15, 53C27
Abstract

Starting with an O(2)-principal fibration over a closed oriented surface F_g, g>=1, a 2-fold covering of the total space is said to be special when the monodromy sends the fiber SO(2) = S^1 to the nontrivial element of Z_2. Adapting D Jonhson's method [Spin structures and quadratic forms on surfaces, J London Math Soc, 22 (1980) 365-373] we define an action of Sp(Z_2,2g), the group of symplectic isomorphisms of (H_1(F_g;Z_2),.), on the set of special 2-fold coverings which has two orbits, one with 2^{g-1}(2^g+1) elements and one with 2^{g-1}(2^g-1) elements. These two orbits are obtained by considering Arf-invariants and some congruence of the derived matrices coming from Fox Calculus. Sp(Z_2,2g) is described as the union of conjugacy classes of two subgroups, each of them fixing a special 2-fold covering. Generators of these two subgroups are made explicit., Comment: This is the version published by Geometry & Topology Monographs on 29 April 2008

Published
2009
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12. Classifications of special double-coverings associated to a non-orientable surface

Author

Bauval, Anne and Hayat, Claude

Subjects
Mathematics - Geometric Topology, Mathematics - Algebraic Topology, 57M10, 57R15, 57S05, 57M60
Abstract

This paper investigates some actions "\`a la Johnson" on the set, denoted by ${\cal E}$, of Spin-structures which are interpreted as special double-coverings of a trivial $S^1-$fibration over a non-orientable surface $N_{g+1}$. The group acting is first a group of orthogonal isomorphisms assoiciated to $N_{g+1}$. A second approach is to consider the subspace of ${\cal E}$ (with $2^{g}$ elements) coming from special double-coverings of $S^1\times F_g$, where $F_g$ is the orientation covering of $N_{g+1}$. The group acting now is a subgroup of the group of symplectic isomorphisms associated to $F_{g}$. In both situations, we obtain results on the number of orbits and the number of elements in each orbit. Except in one case, these results do not depend on any necessary choices. We compare both previous classifications to a third one: weak-equivalence of coverings, Comment: 22 pages, 4 figures

Published
2008

15. Cosmic Womb: The Seeding of Planet Earth

Author

Wickramasinghe, Chandra and Bauval, Robert

Subjects
Cosmic Womb: The Seeding of Planet Earth (Nonfiction work) -- Wickramasinghe, Chandra -- Bauval, Robert -- Book reviews, Literature/writing
Abstract

Cosmic Womb: The Seeding of Planet Earth Chandra Wickramasinghe, PhD and Robert Bauval Bear & Company/Inner Traditions 9781591433071, $20.00 www.BearandCompanyBooks.com Cosmic Womb: The Seeding of Planet Earth belongs in any [...]

Published
2018

16. A New Interpretation of a Rare Old Kingdom Dual Title: The King’s Chief Librarian and Guardian of the Royal Archives of Mehit

Author

Manu Seyfzadeh, Robert M. Schoch, and Robert Bauval

Subjects
Civilization, Sphinx, Interpretation (philosophy), media_common.quotation_subject, General Medicine, Art, Ancient history, Genealogy, Royal Archives, Kingdom, Philology, Guardian, The Symbolic, media_common
Abstract

To date, no unequivocal textual reference to the Great Sphinx has been identified prior to Egypt’s New Kingdom. Here, we present evidence that the monument we now know as the Great Sphinx was called Mehit and that this name was part of an exclusive title held only by the highest officials of the royal Egyptian court going back to at least early dynastic times, i.e. prior to the time of the Great Sphinx’s generally presumed construction during the 4th Dynasty. Furthermore, the symbolic origins of this title precede the 4th Dynasty by at least five centuries, going back to the very cradle of writing during the earliest dynastic era of the early Nile civilization. Based on this philological evidence corroborating geological and archeo-astronomical evidence previously published, we conclude that a lion-like stone monument existed on the Giza Plateau long before the Great Sphinx is generally believed to have been made and that early dynastic Egyptians referred to it in writing.

Published
2017
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18. Preuve simplifiée du théorème de Serret sur les nombres équivalents

Author

Bauval, Anne, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics - Number Theory, [MATH]Mathematics [math], 11A55
Abstract

If the continued fractions of two irrational numbers have a common complete quotient, then these two numbers are in the same orbit under the action of $\mathrm{PGL}(2,\mathbb{Z})$. The converse is Serret's well-known theorem, but we give a simpler and purely algebraic proof of it., Comment: in French

Published
2019

19. The Borsuk-Ulam theorem for the Seifert manifolds having flat geometry

Author

Bauval, Anne, Gonçalves, Daciberg L., Hayat, Claude, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Instituto de Matemática e Estatística (IME), Universidade de São Paulo (USP), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), and Universidade de São Paulo = University of São Paulo (USP)

Subjects
Mathematics - Geometric Topology, 57M05 57M10 57M50, [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], FOS: Mathematics, Mathematics::General Topology, Mathematics::Metric Geometry, Geometric Topology (math.GT), [MATH]Mathematics [math], Mathematics::Algebraic Topology, Mathematics::Symplectic Geometry, Mathematics::Geometric Topology
Abstract

Let M be a Seifert manifold which belongs to the geometry Flat. In this work we determine all the free involutions {\tau} on M, and the Borsuk-Ulam indice of (M,{\tau}).

Published
2018

21. Preuve simplifi��e du th��or��me de Serret sur les nombres ��quivalents

Author

Bauval, Anne

Subjects
FOS: Mathematics, Number Theory (math.NT), 11A55
Abstract

If the continued fractions of two irrational numbers have a common complete quotient, then these two numbers are in the same orbit under the action of $\mathrm{PGL}(2,\mathbb{Z})$. The converse is Serret's well-known theorem, but we give a simpler and purely algebraic proof of it., in French

Published
2017
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23. Suites r\'ecurrentes lin\'eaires d'ordre 2 \'a divisibilit\'e forte

Author

Bauval, Anne, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics - Number Theory, 11B39, 11A05, [MATH]Mathematics [math], [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]
Abstract

We reprove twice, in a simpler but as elementary way, a result by Hor\'ak and Skula (1985) who determined, among all sequences of integers defined by $$u_1=1,\quad u_2=R,\quad u_{n+2}=Pu_{n+1}-Qu_n$$ for some integers $P,Q,R$, those which satisfy the strong divisibility condition $$\forall i,j\in\mathbb N^*\quad u_i\land u_j=\left|u_{i\land j}\right|,$$ where $\land$ denotes the greatest common divisor., Comment: in French

Published
2016

24. Suites r��currentes lin��aires d'ordre 2 �� divisibilit�� forte

Author

Bauval, A.

Subjects
FOS: Mathematics, 11B39, 11A05, Number Theory (math.NT)
Abstract

We reprove twice, in a simpler but as elementary way, a result by Hor��k and Skula (1985) who determined, among all sequences of integers defined by $$u_1=1,\quad u_2=R,\quad u_{n+2}=Pu_{n+1}-Qu_n$$ for some integers $P,Q,R$, those which satisfy the strong divisibility condition $$\forall i,j\in\mathbb N^*\quad u_i\land u_j=\left|u_{i\land j}\right|,$$ where $\land$ denotes the greatest common divisor., in French

Published
2016
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27. Reidemeister-Schreier's Algorithm For 2-Coverings of Seifert Manilfolds

Author

Bauval, Anne, Hayat, Claude, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics::Group Theory, Mathematics - Geometric Topology, [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], FOS: Mathematics, Primary 57M10, Secondary 57M05, 57S25, Geometric Topology (math.GT), [MATH]Mathematics [math], Mathematics::Symplectic Geometry, Mathematics::Geometric Topology
Abstract

It is classical that given any Seifert structure on N, Reidemeister-Schreier's algorithm produces a presentation of all index 2 subgroups of the fundamental group of N, described as the fundamental group of some Seifert manifolds. The new result of this article is concise formulas that gather all possible cases., 12 pages

Published
2014

29. L'anneau de cohomologie des variétés de Seifert

Author

Anne Bauval, Claude Hayat, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
010102 general mathematics, 0103 physical sciences, 57N10 (57M05), 010307 mathematical physics, General Medicine, 0101 mathematics, 01 natural sciences, Humanities, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

Resume En subdivisant une decomposition cellulaire dʼune variete de Seifert, on obtient une decomposition Δ-simpliciale et un quasi-isomorphisme. On exhibe des cocycles Δ-simpliciaux qui relevent les generateurs cellulaires usuels de la cohomologie de la variete. Il ne reste plus quʼa appliquer la formule dʼAlexander–Whitney pour decrire explicitement la structure dʼanneau de cohomologie a coefficients dans Z / p Z , avec p premier, de cette variete de Seifert.

Published
2013
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30. The Borsuk-Ulam theorem for double coverings of Seifert manifolds

Author

Bauval, Anne, L. Gonçalves, Daciberg, Hayat, Claude, Zvengrowski, Peter, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2012

31. L'anneau de cohomologie de toutes les vari\'et\'es de Seifert

Author

Bauval, Anne, Hayat, Claude, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
13D03, 57S25 (Primary) 55N45 (Secondary), Mathematics - Geometric Topology, [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], Mathematics - Algebraic Topology, Mathematics::Geometric Topology, Mathematics::Algebraic Topology
Abstract

The cohomology ring with coefficients in $\Z_p$, where $p$ is a prime integer, of a Seifert manifold $M$, orientable or not orientable is obtained from a simplicial decomposition of $M$. Many choices must be made before applying Alexander-Whitney formula to get the cup-products. The most difficult choices are those of liftings from the cellular complex to the simplicial complex when we add the condition to be cocycles., Comment: few typ corrections 20 figures

Published
2012

32. A classification of special 2-fold coverings

Author

Claude Hayat, Daciberg Lima Gonçalves, Maria Hermínia de Paula Leite Mello, Anne Bauval, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discrete mathematics, 57R15, 53C27, Group (mathematics), 010102 general mathematics, Fibration, Geometric Topology (math.GT), Surface (topology), 01 natural sciences, 57R15 (57M12), Combinatorics, Mathematics - Geometric Topology, Conjugacy class, Monodromy, 0103 physical sciences, FOS: Mathematics, Congruence (manifolds), Algebraic Topology (math.AT), Mathematics - Algebraic Topology, 010307 mathematical physics, 0101 mathematics, Element (category theory), Mathematics, Symplectic geometry
Abstract

Starting with an O(2)-principal fibration over a closed oriented surface F_g, g>=1, a 2-fold covering of the total space is said to be special when the monodromy sends the fiber SO(2) = S^1 to the nontrivial element of Z_2. Adapting D Jonhson's method [Spin structures and quadratic forms on surfaces, J London Math Soc, 22 (1980) 365-373] we define an action of Sp(Z_2,2g), the group of symplectic isomorphisms of (H_1(F_g;Z_2),.), on the set of special 2-fold coverings which has two orbits, one with 2^{g-1}(2^g+1) elements and one with 2^{g-1}(2^g-1) elements. These two orbits are obtained by considering Arf-invariants and some congruence of the derived matrices coming from Fox Calculus. Sp(Z_2,2g) is described as the union of conjugacy classes of two subgroups, each of them fixing a special 2-fold covering. Generators of these two subgroups are made explicit., This is the version published by Geometry & Topology Monographs on 29 April 2008

Published
2008
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33. Théorème d'Eilenberg-Zilber en homologie cyclique entière

Author

Bauval, Anne, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
16E40, 19D55, [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], Mathematics - K-Theory and Homology, [MATH]Mathematics [math]
Abstract

For simplicial modules, Eilenberg-Zilber's classical theorem states the existence of a product $sh : M\otimes N\to M\times N$ (the shuffle) and a coproduct $AW : M\times N\to M\otimes N$ (the Alexander-Whitney map), which are quasi-inverse of eachother. A cyclic version of this theorem was established in 1987 by Hood and Jones: they proved that $sh$ and $AW$ admit "coextensions" $sh_\infty$ and $AW_\infty$, using an acyclic-model method. Besides, an explicit formula for $sh_\infty$ has been discovered by several authors. But the question remained open of such an explicit formula for $AW_\infty$, and for the homotopies by which $sh_\infty$ and $AW_\infty$ are mutual quasi-inverses and are quasi-(co)-associative. We present a complete answer to this problem and show that all these -- now explicit -- maps extend continuously to entire cyclic complexes (associated to normed algebras)., Comment: in French, Preprint written and disseminated in 1998

Published
1998

41. THE VATICAN HERESY.

Author

Bauval, Robert and Hohenzollern, Chiara

Subjects
RELIGIOUS orthodoxy, NONFICTION
Published
2014

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