ICM 2006 Short Communications Abstracts Section 06 · ICM 2006 – Short Communications. Abstracts. Section 06 The ring C(X) modulo its socle and principal ideals F. Azarpanah*, O

ICM 2006

Short Communications

Abstracts

Section 06Topology

ICM 2006 – Short Communications. Abstracts. Section 06

Uniform type conoids

Teresa Amaral Abreu*, Eusebio Corbacho, Vaja Tarieladze

D. de Matematica, Instituto Politecnico do Cavado e do Ave, Urbanizacao Quintadas Formigas, 4750-Barcelos, Portugal; D. de Matematica Aplicada 1, E. T. S.E. de Telecomunicacion, Universidad de Vigo, Vigo, Spain; MuskhelishviliInstitute of Computational Mathematics, Georgian Academy of Sciences,Tbilisi–93, [email protected]; [email protected]; [email protected]

2000 Mathematics Subject Classification. 54E15

A conoid is a (not necessarily cancellative) Abelian semigroup (X, +) suchthat for each x ∈ X and for every non-negative real number α the productx ·α ∈ X is defined in such a way that the following properties are satisfied:

[A.1] (x1 + x2) · α = x1 · α + x2 · α ∀x1, x2 ∈ X, ∀α > 0[A.2] (x · α1) · α2 = x · (α1 · α2) ∀x ∈ X, ∀α1, α2 > 0[A.3] x · (α1 + α2) = x · α1 + x · α2 ∀x ∈ X, ∀α1, α2 ∈> 0[A.4] x · 1 = x ∀x ∈ X

A similar notion with the name ”a semivector space” had been alreadyconsidered by G. Birkhoff [3] and it was also also implicitly used by H.Radstrom (1952). Later this concept was rediscovered and studied by G.Godini (1962; with the name a semilinear space), R.E. Worth (1970; withthe same name), by R. Urbanski(1976; with the name an abstract convexcone), by E. Pap (1980; with the name a semi-vector space), by K. Keimaland W. Roth ([5]; with the name a cone), etc. Every real vector spacecan be viewed as a conoid. The inspiring examples of conoids for us werethe hyperspace of all convex subsets of given real vector space and its sub-conoids. We shall discuss the concept of a local quasi-uniform conoid, whichwe define as a conoid X equipped with a local quasi-uniformity Q with re-spect to which + is uniformly continuous. The notion of a quasi-uniformsemigroups has been studied earlier by V.S. Krishnan (1957), R. Kopper-man (1982) (cf. also [4]). The basic facts of the theory of quasi-uniformconoids are collected in [1], where they are used for developing an Integra-tion Theory for functions with values in such structures. Our talk is basedon [2].

References

[1] Abreu, T. ; Integration on Quasi-uniform Conoids, Thesis Doctoral, Vigo, 2005.

[2] Abreu, T., Corbacho, E. and V. Tarieladze, Uniform type conoids, Preprint(2006).

ICM 2006 – Madrid, 22-30 August 2006 1


[3] Birkhof, G., Integration of functions with values in a Banach space, Trans.Amer. Math. Soc., 1935, 38, 357–378.

[4] Corbacho,E.; Dikranjan D.; Tarieladze, V., Absorption adjunctable semigroups,Research and Exposition in Mathematics, 2000, 24, 77–103.

[5] Keimal K.; Roth, W., Ordered cones and approximation, Lecture Notes inMath., 1992, 1517.

2 ICM 2006 – Madrid, 22-30 August 2006


The ring C(X) modulo its socle and principal ideals

F. Azarpanah*, O. A. S. Karamzadeh and S. Rahmati

Department of Mathematics, Chamran University, Ahvaz, [email protected]

2000 Mathematics Subject Classification. 54C40

The essentiality of prime ideals and z-ideals of C(X) modulo the socleand principal ideals of C(X) are investigated. We show that the Goldiedimension of C(X) modulo its socle is greater than or equal to that of C(X)and the equality may not always hold. Topological spaces X for which theJacobson radical of C(X) modulo its socle is zero are characterized. Usingthis characterization, it turns out that for a compact space X, the Jacobsonradical of C(X) modulo its socle is zero if and only if the set of isolatedpoints of C(X) is finite.

References

[1] F. Azarpanah, Intersection of essential ideals in C(X), Proc. Amer. Math. Soc.125(7) (1997), 2149–2154.

[2] O.A.S. Karamzadeh and M. Rostami, On the intrinsic topology and some re-lated ideals of C(X), Proc. Amer. Math. Soc., 93(1) (1985), 179–184.



Co-H-spaces and almost localization

Cristina Costoya*, Norio Iwase

Departamento de Alxebra, USC, 15782 Santiago de Compostela, Spain; GraduateSchool of Mathematics, Kyushu University, Fukuoka 810-0044, [email protected]; [email protected]

2000 Mathematics Subject Classification. 55P45

When X is a simply-connected co-H-space it is well known that the p-localization X(p) is also a co-H-space for any prime p. The immediate prob-lem is: Does the converse statement hold or not? In [2], the first authorsettles the answer in positive when X is a finite complex.

In this work, we extend the above result to non-simply connected, finitespaces X in terms of a fibrewise p-localization by Bendersky [1] and May[5] (or almost p-localization by the second author [4]) rather than a usualp-localization, since X is non-nilpotent unless X ' S1 by Hilton-Mislin-Roitberg [3].

Simply-connected spaces and connected co-H-spaces are typical exam-ples of a space X with a co-action of Bπ1(X) along rX : X → Bπ1(X) theclassifying map of the universal covering cX : X → X. We paraphrase thefibrewise property as ‘almost’ property for rX , following earlier authors. Ifsuch a space X is actually a co-H-space, then the almost p-localization ofX is an almost co-H-space, for any prime p. We show the converse state-ment holds. That is, let X be a connected, finite complex with a co-actionof Bπ1(X) along rX such that the almost p-localization of X is an almostco-H-space for every prime p, then X is a co-H-space.

References

[1] M. Bendersky, A functor which localizes the higher homotopy groups of anarbitrary CW-complex, Lecture Notes in Math. 418, Springer Verlag, Berlin(1975), 13-21.

[2] C. Costoya-Ramos, Co-H-spaces and localization, Submitted.

[3] P. Hilton, G. Mislin and J. Roitberg, On co-H-spaces, Comment. Math. Helv.53(1978), 1–14.

[4] N. Iwase, Co-H-spaces and the Ganea conjecture, Topology 40(2001), 223-234.

[5] J.P. May, Fibrewise localization and completion, Trans. Amer. Math. Soc,258(1980), 127-146.



Topology in robot motion planning

Michael Farber

Department of Mathematical Sciences, South Road, Durham DH1 3LE, UK

2000 Mathematics Subject Classification. [email protected]

I will discuss some topological problems inspired by robotics. The motionplanning problem of robotics deals with algorithms which take as inputpairs A,B of states of a given mechanical system (robot) and produce asoutput a continuous motion of the system starting at A and ending at B.The robot motion planning problem translates into the problem of findinga section of a specific fibration over the product X×X; here X denotes theconfiguration space of the system.

With any path-connected topological space X one associated a numer-ical invariant TC(X) measuring the complexity of the problem of naviga-tion in X, viewed as the configuration spaces of a mechanical system [1],[2]. The main properties of TC(X) are (1) homotopy invariance and (2) alower bound in terms of the cohomology algebra of X, see [1]. The numberTC(X) determines the structure of motion planning algorithms in X, bothdeterministic and random [2].

The invariant TC(X) can be explicitly computed in many examples [2].In the case of the real projective space X = RPn (where n 6= 1, 3, 7) thenumber TC(RPn)−1 equals the minimal dimension of the Euclidean spaceinto which RPn can be immersed [4]. The number TC(X) plays a role inthe study of collision free control of many particles moving in the Euclideanspace [5] or on a graph [3].

In the talk I will discuss some further results.

References

[1] Farber M., Topological Complexity of Motion Planning, Discrete and Compu-tational Geometry, 29 (2003), 211–221.

[2] Farber M., Topology of robot motion planning., In Morse Theoretic Methodsin Nonlinear Analysis and in Symplectic Topology (ed. by Paul Biran, OctavCornea, Francois Lalonde), Springer, 2006, 185 – 230.

[3] Farber M., Collision free motion planning on graphs. In Algorithmic Founda-tions of Robotics IV (ed. by M. Erdmann, D. Hsu, M. Overmars, A. Frank vander Stappen), Springer, 2005, 123 – 138.



[4] Farber M., Tabachnikov S., Yuzvinsky S., Topological Robotics: Motion Plan-ning in Projective Spaces, International Math. Research Notices, 34(2003),1853–1870.

[5] Farber M., Yuzvinsky S., Topological Robotics: Subspace Arrangements andCollision Free Motion Planning, Transl. of AMS, 212(2004), 145-156.



Relative Seiberg-Witten and Ozsvath-Szabo invariants forsurfaces in four-manifolds

Sergey Finashin

Department of Mathematics, Middle East Technical University, Ankara 06531,[email protected]

2000 Mathematics Subject Classification. 57M

I present my work [1], where the relative Seiberg-Witten (SW) and Ozsvath-Szabo (OS) invariants, for surfaces in 4-manifolds, were introduced andstudied. C. Taubes [2] introduced such relative invariants for tori, and weconsider the case of higher genus. Refining the classical, “absolute” SW andOS invariants, the relative invariants have a similar package of properties,which may look more natural in the relative case. The product formula looksas a usual product of polynomials (similar to the genus 1 case of Taubes),and the adjunction inequality that estimates genus of membranes on a givensurface, has the most simple, “classical” form, without positivity assump-tion on the self-intersection of a membrane. As a consequence, we obtainminimality of symplectic and Lagrangian membranes (say, on Lagrangianand respectively symplectic surfaces), which seems to be a new observation.

References

[1] Finashin, S., Relative Seiberg-Witten and Ozsvath-Szabo 4-dimensional invari-ants with respect to embedded surfaces, math.GT/0401345

[2] Taubes, C., The Seiberg-Witten invariants and 4-manifolds with essential tori,Geometry and Topology, 5 (2001) 441–519



About the BZ/p-homotopy of classifying spaces

Ramon J. Flores

Departamento de Estadıstica, Universidad Carlos III; Edificio Miguel deUnamuno, Campus de Colmenarejo, 28270, Colmenarejo, Madrid, [email protected]

2000 Mathematics Subject Classification. 55P20; 55P80

The nullification and cellularization functors were developed in the firstnineties by E. Dror-Farjoun [3] and W. Chacholski [2] , in the context ofA-homotopy theory. Roughly speaking, the A-nullification functor kills thestructure of a certain space X that can be detected through the mappingspace map∗(A,X), while the A-cellularization preserves only this structure.These functors are dual in a quite subtle way.

In the cases we are interested, A will always be BZ/p, the classifyingspace of the group of p elements, for p prime. The study of the mod phomotopy theory of the spaces through the mapping space map∗(BZ/p, X)has deserved a prominent place in Homotopy Theory since the resolutionby Miller of the Sullivan conjecture [5], the discovering of the T -functor byLannes [4] and the development of localization techniques by Bousfield [1]and others.

In our talk, we will give explicit characterizations of the BZ/p-nullificationsand cellularizations of classifying spaces of different families of groups andtheir homotopic analogues. More concretely, we will deal with finite groups,infinite discrete groups with finite proper dimension, compact Lie groupsand realizations of linking systems of p-local finite groups. We will also showin addition some applications of our results, computations concerning con-crete groups, and some interesting conjectures that arise in this framework.

References

[1] A.K. Bousfield, D.M. Kan, Homotopy limits, completions and localizations. Lec-ture Notes in Maths. 304, Springer, 1972.

[2] Dror-Farjoun, E., Cellular spaces, null spaces and homotopy localization. Lec-ture Notes in Maths. 1622, Springer, 1995.

[3] W. Chacholski, On the functors CWA and PA, Duke J. Math. 84 (1996), no.3, 599–631.

[4] J. Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’ungroupe abelien elementaire, Publ. Math. Inst. Hautes Etudes Sci. 75 (1992),135–244.



[5] H. Miller, The Sullivan conjecture on maps from classifying spaces, Ann. ofMath. (2) 120 (1984), 39–87.



On the reciprocity law of the Fourier-Dedekind sums

Yoshihiro FukumotoDepartment of Information Systems, Tottori University of EnvironmentalStudies, 1-1-1 Wakabadai-Kita Tottori, [email protected]

2000 Mathematics Subject Classification. 11F20, 14K25, 57M27, 57N13,57R20, 57N70

In this talk, we give a geometric interpretation of the reciprocity law of theFourier-Dedekind sums σm(a1, . . . , an; p) given by M. Beck and S. Robins[1]. In fact, we show that the V -index of the spinc Dirac operator over theweighted projective space CP(a0, a1, . . . , an) is equal to the dimension ofthe space of all weighted homogeneous polynomials of weight (a0, a1, . . . , an)and degree m, and this equality gives precisely the Beck-Robins reciprocitylaw. In this equality, σm(a1, . . . , an; p) appears as a localization term ofthe V -index and is essencially the eta invariant of (2n − 1)-dimensionallens spaces L(p; a1, . . . , an) associated to the spinc Dirac operator withspinc structure m ∈ Z/p [3]. In particular, when

∑nk=0 ak is even, then

CP(a0, a1, . . . , an) admits a V -spin structure, and we can consider the Diracoperator and obtain a reciprocity law of the cosecant sums δ(p; a1, . . . , an).This result can be seen as a spin version of the reciprocity law of the higherdimensional Dedekind sums def(p; a1, . . . , an) of D. Zagier [5]. Note in par-ticular that in the case n = 2, the combination def(p; q, r) + 8δ(p; q, r) givesa homology spin cobordism invariant of 3-dimensional lens spaces L(p; q, r)[4] and satisfies a reciprocity law of the theta multiplier given by B. Berndt[2].

References

[1] Beck, M., Robins, S., Dedekind sums: a combinatorial-geometric viewpoint,DIMACS Ser. Discrete Math. Theoret. Comput, Sci. 64 (2004), 25–35.

[2] Berndt, B., Analytic Eisenstein series, theta-functions, and series relations inthe spirit of Ramanujan, J. Reine Angew. Math. 303/304 (1978), 332–365.

[3] Fukumoto, Y., The index of the spinc Dirac operators on the weighted projec-tive space and the reciprocity law of the Fourier-Dedekind sum, J. Math. Anal.Appl. 309 (2005), 674–685.

[4] Fukumoto, Y., Furuta, M., Ue, M., W-invariants and Neumann-Siebenmanninvariants for Seifert homology 3-spheres, Topology Appl. 116 (2001), 333–369.

[5] Zagier, D., Higher dimensional Dedekind sums, Math. Ann. 202 (1973), 149–172.



On a conjecture of Whittaker on the uniformization ofhyperelliptic curves

Ernesto Girondo*, Gabino Gonzalez-Diez

Departamento de Matematicas, Universidad Autonoma de Madrid, Campus deCantoblanco, 28049 Madrid, [email protected]; [email protected]

2000 Mathematics Subject Classification. 30F10, 14H15

This work concerns an old conjecture due to E. T. Whittaker [5], aimingto describe the group that uniformizs an arbitrary hyperelliptic Riemannsurface y2 =

∏2g+2i=1 (x − ai) as an index two subgroup of the monodromy

group of an explicit second order linear differential equation with singu-larities at the values ai. Whittaker and collaborators in the thirties, andR. Rankin [4] some twenty years later, were able to prove the conjecturefor several families of hyperelliptic surfaces, characterized by the fact thatthey admit a large group of symmetries. However, general results of theanalytic theory of moduli of Riemann surfaces, developed later, imply thatWhittaker’s conjecture cannot be true in its full generality. More recently,numerical computations showed that Whittaker’s prediction is incorrect forrandom surfaces, and in fact it was conjectured that it only holds for theknown cases of surfaces with a large group of automorphisms [1].

We prove the following results:

• Having many automorphisms is not a necessary condition for a surfaceto satisfy Whittaker’s conjecture [2].

• Whittaker sublocus of moduli space of Riemann surfaces, this is thesubset of points representing hyperelliptic curves which satisfy Whit-taker’s conjecture, is compact [3].

• R.A. Rankin stated more than forty years ago that the conjecturehad not been proved for any algebraic equation containing irremovablearbitrary constants. We combine our compactness result with otherfacts coming from Teichmuller theory to show that in the most nat-ural interpretations of this sentence we can think of, this is, in fact,impossible [3].

References

[1] Chudnovsky, D.V., Chudnovsky, G.V., Computer Algebra in the service ofmathematical physics and number theory. Computers in mathematics. Lecture



Notes in Pure and Applied Mathematics, 125, Dekker, New York, 1990, 109–232.

[2] Girondo, E., Gonzalez-Diez, G., On a conjecture of Whittaker concerning uni-formization of hyperelliptic curves, Trans. American Math. Soc. 356, no. 2(2003), 691–702.

[3] Girondo, E., Gonzalez-Diez, G., On the structure of Whittaker sublocus ofmoduli space of algebraic curves, Proc. Royal Soc. Edinburgh, to appear.

[4] Rankin, R.A., The differential equations associated with the uniformization ofcertain algebraic curves, Proc. Roy. Soc. Edinburgh sect. A 65, (1958), 35–62.

[5] Whittaker, E.T., On hyperlemniscate functions. A family of automorphic func-tions, J. London Math. Soc., 3 (1929), 274–278.



Dynamical types of isometries of the hyperbolic 5-space

Ravi S. Kulkarni, Krishnendu Gongopadhyay*

School of Mathematics, Harish-Chandra Research Institute, Chhatnag Road,Jhunsi, Allahabad 211019, [email protected]; [email protected]

2000 Mathematics Subject Classification. 51M10; 37C85, 15A33, 58D15

In this short communication we aim to give an algebraic characterizationof the dynamical types of the orientation-preserving isometries of the hy-perbolic 5-space and to find the parameter spaces of the isometries with afixed dynamical type.

Classically, based on their fixed points the dynamical types of the isome-tries of the hyperbolic n-space Hn were classified as elliptic, parabolic orhyperbolic. In dimension 3, the isometries of H3 were identified with the2×2 invertible matrices over C and the classical dynamical types were char-acterized algebraically in terms of the traces of the matrices. An account ofthis classification can be found, for eg. in [2]. In dimension 4, the isometriescan be identified with a suitable subgroup of GL(2, H), the group of all 2×2matrices over the skew-field of quaternions H. Using this representationParker et al [2] have given an algebraic characterization of the dynamicaltypes of isometries of H4. Parker et al have done it with respect to a finerclassification of the dynamical types of isometries of H4.

In this talk, we wish to give a finer classification of the dynamical typesof isometries of Hn. This classification is obtained by attaching “rotationangles” to an isometry and is valid for all dimensions. We will focus atdimension 5. With respect to the finer classification we have the followingdynamical types of the orientation-preserving isometries of H5:

2-rotatory elliptic, 1-rotatory elliptic, 1-rotatory translation, pure trans-lation, 2-rotatory stretch, 1-rotatory stretch and pure stretch.

We will give an algebraic characterization of these dynamical types byidentifying the orientation-preserving isometries of H5 with PGL(2, H). Thegroup GL(2, H) can be embedded in GL(4, C). Using this embedding wehave considered the co-efficients ci’s of the characteristic polynomials ofthe corresponding elements in GL(4, C) as the conjugacy invariants. Thedesired algebraic characterization is obtained in terms of suitable fractionsof the ci’s.

Finally we will parameterize the orientation-preserving isometries witha fixed dynamical type by topological manifolds. For example, the set of alltranslations can be identified with S4×R4, the set of all 2-rotatory stretches



can be identified with a manifold of dimension 15 etc. A version of this workis available on the arxiv: math.GT/0511444.

References

[1] A. F. Beardon, The geometry of discrete groups, Graduate Texts in Mathemat-ics 91, Springer- Verlag, Berlin, 1983.

[2] John R. Parker, W. Cao, X. Wang, On the classification of quaternionic Mobiustransformations, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 2, 349–361.



Some topological properties of dynamically defined wild knots

Gabriela Hinojosa*, Alberto Verjovsky

Facultad de Ciencias, UAEM, Ave. Universidad 1001 Col. Lomas de Chamilpa,Cuernavaca, [email protected]

2000 Mathematics Subject Classification. Primary: 57M30. Secondary: 57M45,57Q45, 30F14

In this paper we study kleinian groups of Schottky type whose limit setis a wild knot in the sense of Artin and Fox [1]. We show that, if the“original knot” fibers over the circle then the wild knot Λ also fibers overthe circle. As a consequence, the universal covering of S3 − ∗ is R3. Weprove that the complement of a dynamically-defined fibered wild knot cannot be a complete hyperbolic 3-manifold[2]. We also prove that this typeof wild knots are homogeneous: given two points p, q ∈ K there exists ahomeomorphism f of the sphere such that f(K) = K and f(p) = q. Wealso show that if the wild knot is a fibered knot then we can choose an fwhich preserves the fibers[3].

References

[1] E. Artin, Zur Isotopie zweidimensionalen Flachen im R4. Abh. Math. Sem.Univ. Hamburg (1926), 174-177.

[2] G. Hinojosa, Wild knots as limit set of Kleinian Groups Contemporary Math-ematics. 389 (2005), 125-139.

[3] G. Hinojosa, A. Verjovsky Homogeneity of dynamically defined wild knot. Rev.Mat. Complut. 19 (2006) no.1 101-111



Bounds on boundary slopes for Montesinos knots

Kazuhiro Ichihara*, Shigeru Mizushima

College of General Education, Osaka Sangyo University, Nakagaito 3-1-1, Daito,Osaka 574-8530, Japan; Department of Mathematical and Computing Sciences,Tokyo Institute of Technology, 12–1 Ohokayama, Meguro, Tokyo 152–8552, [email protected]; [email protected]

2000 Mathematics Subject Classification. 57M25

This talk is based on our joint works appearing as [2], [3] and [4]. In thetalk, we will give a number of bounds about boundary slopes of essentialsurfaces embedded in the exteriors of Montesinos knots in the 3-sphere. Wewill actually consider the denominators of boundary slopes, the differencesbetween pairs of boundary slopes, and the diameters of the boundary slopesets for Montesinos knots. Our main tool to establish the bounds is thealgorithm developed by A. Hatcher and U. Oertel in [1] which enumeratesall boundary slopes for a given Montesinos knot.

References

[1] Hatcher, A., Oertel, U., Boundary slopes for Montesinos knots, Topology 28(1989), 453–480.

[2] Ichihara, K., Mizushima, S., Bounds on numerical boundary slopes for Mon-tesinos knots, preprint available at arXiv:math.GT/0503190.

[3] Ichihara, K., Mizushima, S., Crossing number and diameter of boundary slopeset of Montesinos knot, preprint available at arXiv:math.GT/0510370.

[4] Ichihara, K., Mizushima, S., A lower bound of the boundary slope diameter forMontesinos knot, in preparation.



Generalization of Thurston’s double limit theorem-solution tohis conjecture

Inkang Kim

Department of Mathematics, Seoul National University, Seoul 151-742, [email protected]

2000 Mathematics Subject Classification. 30F40; 57M50

For a given hyperbolic 3-manifold M with a compressible boundary S, ifconformal structures at infinity on S corresponding to geometrically finitehyperbolic metrics, converge to a doubly incompressible projective lami-nation, then after passing to a subsequence, the sequence converges alge-braically [1]. This settle downs a Thurston’s conjecture regarding Masurdomain [2].

References

[1] Inkang Kim, Cyril Lecuire and Ken’ichi Ohshika, Convergence of freely decom-posable Kleinian groups, preprint.

[2] H. Masur, Measured foliations and handlebodies, Ergodic Theory and Dynam.Systems 6 (1986), 99-116.



Homotopy dominations of polyhedra

Danuta Ko lodziejczyk

Faculty of Mathematics and Informational Sciences, Warsaw University ofTechnology, pl. Politechniki 1, 00-661 Warsaw, [email protected]

2000 Mathematics Subject Classification. 55P15, 55P55

In this talk we will discuss the problem:

Does every polyhedron dominate only finitely many different homotopytypes?

This question was stated by K. Borsuk in 1968, at the Topological Con-ference in Herceg-Novi (in an equivalent, shape-theoretical formulation) andpublished in the 1970’s in his papers, for example in [1]. By a polyhedronwe mean here a finite one.

We showed that the answer to the Borsuk’s question is negative: thereexists a polyhedron (even of dimension 2), which homotopy dominates in-finitely many polyhedra of different homotopy types. (From the classicalresults of homotopy theory, each 1-dimensional polyhedron dominates onlyfinitely many different homotopy types, see [5].)

Furthermore, such examples are even frequent: we proved that, for in-stance, for every non-abelian poly-Z-group G and integer n ≥ 3, there existsa polyhedron P with this property with fundamental group G and of di-mension n [2].

Therefore there exist polyhedra with nilpotent fundamental group domi-nating infinitely many polyhedra of different homotopy types. On the otherhand, we proved that polyhedra with finite fundamental group and nilpotentpolyhedra dominate only finitely many different homotopy types [3].

We will present these and further related results. We will also answersome other questions concerning homotopy dominations of polyhedra pub-lished by K. Borsuk in his papers and book [4].

References

[1] Borsuk, K., Some problems in the theory of shape of compacta, Russian Math.Surveys 34:6 (1979), 24–26.

[2] Borsuk, K., Theory of Shape. Polish Scientific Publishers 59, Warsaw, 1975.

[1] Ko lodziejczyk, D., Homotopy dominations within polyhedra, Fund. Math. 178(2003), 189-202.



[2] Ko lodziejczyk, D., Polyhedra with finite fundamental group dominate finitelymany different homotopy types, Fund. Math. 180 (2003), 1-9.

[5] Wall, C. T. C., Finiteness Conditions for CW-complexes, Ann. of Math. 81(1965), 56–69.



The universal sl3-link homology

Marco Mackaay*, Pedro VazDepartamento de Matematica, Campus de Gambelas, Universidade do Algarve,8005-139 Faro, [email protected]

2000 Mathematics Subject Classification. 18G60, 57M27, 81R50

Based on Khovanov’s work on sl2 and sl3-link homologies [4, 5] we definethe universal sl3-link homology Ua,b,c(L) over the polynomial ring Z[a, b, c].We follow a diagrammatic approach similar to the one Bar-Natan [1] usedfor sl2 and prove the invariance of Ua,b,c(L) under the Reidemeister moves.We also show that Ua,b,c(L) is functorial up to sign with respect to linkcobordisms.

In the second part of the talk we show that, when a, b, c are com-plex numbers, there are three isomorphism classes to which Ua,b,c(L, C)can belong depending on the number of roots of the polynomial f(X) =X3−aX2− bX− c. If f has one root of multiplicity three, then Ua,b,c(L, C)is isomorphic to Khovanov’s original sl3-link homology with a = b = c = 0.If f has three distinct roots, then Ua,b,c(L, C) is isomorphic to Gornik’s sl3-link homology [3] with (a, b, c) = (0, 0, 1). The last case, when f has twodistinct roots one of which with multiplicity two, is new and we show that

U∗a,b,c(L, C) ∼=⊕

L′⊆L

KH∗−j(L′)(L′, C),

where the direct sum is taken over all sublinks of L and KH∗−j(L′)(L′, C)is Khovanov’s original sl2-link homology with a shift in the homologicaldegree depending on L′. This result proves part of the conjectures in [2] forsl3.

References

[1] D. Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol.9 (2005), 1443-1499.

[2] N. M. Dunfield, S. Gukov, J. Rasmussen, The superpolynomial for knot ho-mologies, preprint 2005, math.GT/0505662.

[3] B. Gornik, Note on Khovanov link cohomology, preprint 2004,math.QA/0402266.

[4] M. Khovanov, sl(3) link homology, Alg. Geom. Top. 4 (2004), 1045-1081.

[5] M. Khovanov, Link homology and Frobenius extensions, preprint 2004,math.QA/0411447



Stabilizers and orbits of smooth functions

Sergiy Maksymenko

Topology dept., Institute of Mathematics, NAS of Ukraine, Tereshchenkivs’kastr., 3, Kyiv, 01601, [email protected]

2000 Mathematics Subject Classification. 32S20, 57R70, 58B05

Let M be a smooth connected compact manifold, f : M → R1 a smoothfunction, DM the group of diffeomorphisms of M , and DR1(f) the group ofpreserving orientation diffeomorphisms of M that leave invariant the imagef(M) of f . There are two natural actions of the groups DR1(f)×DM andDM on C∞(M,R1):

DR1(f)×DM × C∞(M,R1) → C∞(M,R1) (φ, h) · f = φ ◦ f ◦ h−1,DM × C∞(M,R1) → C∞(M,R1) h · f = f ◦ h−1,

where φ ∈ DR1(f) and h ∈ DM . Let SMR1 and SM be correspondingstabilizers and OMR1 and OM the corresponding orbits of f with respectto these actions. We endow all these spaces with the corresponding C∞

Whitney topologies.For z ∈ M let C∞z (M) be the algebra of germs of smooth functions at z

and let ∆(f, z) be the Jacobi ideal in C∞z (M) generated by germs of partialderivatives of f at z. We put on f the following three conditions:

(1) f is constant at every connected component of ∂M ;(2) there are only finitely many critical values of f ;(3) for every critical point z of f the germ of the function f(x) − f(z)

belongs to ∆(f, z), i.e. there is a vector field F near z such that f(x)−f(z) =df(F )(x).

Condition (3) holds for a very large class of singularities. In particularnon-degenerate and simple singularities and even formal series satisfy (3).It is also preserved by stable equivalence of singularities.

Let DcR1(f) be the subgroup of DR1(f) consisting of diffeomorphisms

that also fix every boundary or critical value of f .Definition. Say that a critical point z of f is essential if for every neigh-borhood Uz of z there exists a neighborhood Vf of f in C∞(M) with C∞-topology such that every g ∈ Vf has a critical point in Uz. E.g. if f(x) = x2

and g(x) = x3, then 0 ∈ R1 is essential for f but not for g.

Theorem [1] (A) Suppose that f satisfies (1)-(3). Then we have an exactsequence 1 → SM → SMR1 → Dc

R1(f) → 1, i.e. the stabilizer SMR1 isan extension of SM by Dc

R1(f). This sequence split by a homomorphismθ : Dc

R1(f) → SMR1, whence the embedding SM × idM ⊂ SMR1 extends to a



homeomorphism between SM ×DcR1(f) and SMR1. In particular, SM × idM

is a strong deformation retract of SMR1.(B) In addition, suppose that every critical level-set of f includes either

an essential critical point or a connected component of ∂M . Then the em-bedding OM ⊂ OMR1 extends to a homeomorphism OM × Rn−2 ≈ OMR1,where n is the total number of critical and boundary values of f .

References

[1] S. Maksymenko, Stabilizers and orbits of smooth functions, to appear in Bul-letin des Sciences Mathematiques, http://xxx.lanl.gov/math.FA/0411612



Khovanov homology for virtual knots and minimality problems

Vassily Olegovich Manturov

Chair of Geometry, Moscow State Regional University, Moscow, 129344,Eniseyskaya, 16/21 -34, [email protected]


The Khovanov homology [except Z2-case, [2]] does not have a straightfor-ward generalization for the case of virtual knots, because of possible 1-1bifurcations. Such a generalization is possible for the case of “alternatibleknots” or “knots with orientable atoms”, where an atom is a checkerboard2-surface corresponding to a (virtual) knot. One can construct such homol-ogy theory by using orienting covering over atoms and cablings, which leadto orientable atoms. Another approach is to use twisted coefficients in theHopf algebra representing the homology of the unknot, see [1, 3].

The theories decribed above, also generalize Bar-Natan’s topologicaltheory of Khovanov homology for tangles and cobordisms [4] and Kho-vanov’s theory using Frobenius extension [3].

For these theories, tree spanning expansion is generalized and somecrossing number estimates are obtained [genus of the atom allows to es-timate the width of the Khovanov homology, which allows to generalize theKauffman-Murasugi theorem in many directions], [1].

References

[1] Manturov, V.O., Teoriya Uzlov (Knot Theory), RCD, Moscow-Izhevsk, 2005(in Russian).

[2] Manturov, V.O., The Khovanov polynomial for virtual knots, Doklady Math.,69, N. 2., pp. 164-167.

[3] Manturov V.O. (2005), Kompleks Hovanova dlya virtual’nykh uzlov (The Kho-vanov complex for virtual links), Fundamental and Applied Mathematics, vol.11, N.4. pp. 127-152 (in Russian).

[4] Turaev, V.G., Turner, P.(2005), Link homology and unoriented topologicalquantum field theory, arXiv: math. GT/0506229; v1.



A categorical quaternary box bracket operation

Howard J. Marcum

Department of Mathematics, The Ohio State Universtiy at Newark, 1179University Drive, Newark, Ohio 43055 [email protected]

2000 Mathematics Subject Classification. 55Q35

A classical approach to computing homotopy groups of spheres (and otherspaces) utilizes composition methods as initiated by Toda in his seminal1962 book [5]. Central to the method are various secondary homotopy op-erations with the principal one being the Toda bracket. But at times thequaternary Toda bracket plays an essential role.

A more recent trend has been the development of Toda bracket typeoperations in settings other than the topological category, as given in theseries of articles [1]–[4]. Indeed new operations of Toda bracket type haveemerged, of which the box bracket [4] and the 2-sided matrix Toda bracket[3] seem quite useful, and new formulae have been found. Significantly lack-ing however has been a categorical definition of the quaternary Toda bracket(whose definition even in the topological case is complicated). Here we de-fine and study as an alternative a related but different operation called thequaternary box bracket that does allow a categorical definition. This newoperation satisfies a triviality theorem which yields a mod zero congruencyrelation for Toda brackets in analogy with one obtained by Toda. As a com-putational example a new relation in the 23-stem of the homotopy groupsof spheres is observed.

References

[1] Hardie, K. A., Kamps, K. H. and Marcum, H. J., A categorical approach tomatrix Toda brackets, Trans. Amer. Math. Soc. 347 (1995), 4625–4649.

[2] Hardie, K. A., Kamps, K. H. and Marcum, H. J., The Toda bracket in thehomotopy category of a track bicategory, J. Pure Appl. Algebra 175 (2002),109–133.

[3] Hardie, K. A., Kamps, K. H., Marcum, H. J. and Oda, N., Triple brackets andlax morphism categories, Appl. Categ. Structures 12 (2004), 3–27.

[4] Hardie, K. A., Marcum, H. J. and Oda, N., Bracket operations in the homotopytheory of a 2-category, Rend. Istit. Mat. Trieste 33 (2001), 19–70.

[5] Toda, H., Composition Methods in Homotopy Groups of Spheres. Annals ofMathematics Studies 49, Princeton University



Uniformly continuous maps between ends of infinite trees

Alvaro Martınez Perez

Department of Geometry and Topology, Universidad Complutense de Madrid,Plza de las Ciencias 3, 28040 Madrid, [email protected]

2000 Mathematics Subject Classification. 57M99, 54E35, 55P55

There is a well-known correspondence between infinite trees and ultrametricspaces which can be interpreted as an equivalence of categories and comesfrom considering the end space of the tree.

In this equivalence, uniformly continuous maps between the end spacesare translated to some classes of coarse maps (or even classes of metricallyproper lipschitz maps) between the trees.

This construction, set out for shape theory, also allows us to give a newgeometric description of the category of inverse sequences of sets and someproperties related with the Mittag-Leffler condition.

References

[1] Hughes, B. R-trees and ultrametric spaces: a categorial equivalence. Advancesin Mathematics. 189,(2004) 148–191.

[2] Mardesic, S., Segal, J. Shape theory. North-Holland (1982).

[3] Moron, M.A., Ruiz del Portal, F.R. Shape as a Cantor completion process.Mathematische Zeitschrift. 225, (1997) 67–86.

[4] Roe, J. Lectures on coarse geometry. University Lecture Series, vol.31 Ameri-can Mathematical Society (2003).



3-Manifold Recognizer

Sergei Matveev

Department of Mathematics, Chelyabinsk State University, Kashirin Brothersstreet, 129, Chelyabinsk 454021, [email protected]

2000 Mathematics Subject Classification. 57-04, 57N10

Low-dimensional topology remains one of the most intriguing and inter-esting branches of modern mathematics. The 3-Manifold Recognizer is acomputer program which is designed to determine the structure and topo-logical type of a given 3-manifold and calculate several of its invariants.It was written by V. Tarkaev according to an efficient partial recognitionalgorithm elaborated by S. Matveev and other members of the ChelyabinskTopology Group.

The program works as follows (see [1, 2]): if we input a presentation ofa 3-manifold M , the computer constructs a special spine of M and appliesto it different moves which yield a simpler spine of the same manifold.Surgery moves consisting in cutting M along tori and annuli are also used.They can change M , but only in controlled manner. Performing the movesfor as long as possible, we decompose M into a so-called labelled moleculewhose atoms and edges correspond to some simpler 3-manifolds and gluinghomeomorphisms between them. If all the atoms are known (for example, ifall of them are Seifert), the reverse assembling process determines the exacttype of M . If not, then nevertheless we get an important information onM , such as information about its JSJ-structure.

The program can work with manifolds presented by their triangulations,singular triangulations, special spines, framed links, genus two Heegaarddiagrams, or crystallizations. Depending on the manifold, the result canbe given in the form of a standard descriptions of Seifert, Stallings, orgeneralized Waldhausen manifold.

The program is easy to operate and has passed several extensive tests.It successfully recognized all closed irreducible orientable 3-manifold up tocomplexity 12 (altogether more than 30 000 manifolds), as well as severalmonster-sized examples prepared specifically for testing.

The zip-compressed executable file of the program (written in C++) isabout 700Kb large and can run under Windows 98 and its later versions.The 3-Manifold Recognizer is available for download from http://www.topology.kb.csu.ru/∼mr



References

[1] Matveev , S. V., Computer recognition of three-manifolds, Experimental Math-ematics, 7(1998), No. 2, 153-161.

[2] Matveev, S. V. Algorithmic topology and classification of 3-manifolds. SpringerACM-monographs, V. 9, Springer, 2003.



Continuous homology

Leonard Mdzinarishvili

Department of higher mathematics, Georgian Technical University, 77, M.Kostava St., 0175 Tbilisi, [email protected]

2000 Mathematics Subject Classification. 55N05

Denote by F (Z,X) the set of all continuous functions f : Z → X, given thecompact-open topology. Let F (Y ) = {F (∆q, X)} be a topological simplicialcomplex of Y , and G be a topological abelian group. Denote by Mn(Y, G)the group of all continuous maps ϕ : F (∆n, Y ) → G, by Mn

c (Y, G) the sub-group of constant maps, and by M n(Y, G) = Mn(Y, G)/Mn

c (Y, G). Considerthe functional spectrum {Fm(X), τm} of X, Fm(X) = F (X, Sm), which in-duces an inverse system of the cochain complexes {M m−∗(Fm(X), G)}. Theinverse limit M∗(X, G) = lim

←{M m−∗(Fm(X), G)} is a chain complex and

the homology h∗(X, G) of it will be called continuous homology.

Theorem 1. If X is a compact space, G = R, then for q ≥ 0 there is anisomorphism

hq(X, R) ≈ HMq (X, R),

where HM∗ is the homology of Milnor [1].

Theorem 2. If X is a compact space and dim X ≤ n, G = S1, then forq ≥ 0 there is an isomorphism

hq(X, S1) ≈ HMq (X, R).

Corollary (the Steenrod duality theorem). Let X be a compact subset ofSn+1.

a) If G = R, then for 0 < q < n there is an isomorphism

hn−q(X, R) ≈ h q(Sn+1 \X, R),

b) If G = S1, then for 1 < q < n there is an isomorphism

hn−q(X, S1) ≈ h q(Sn+1 \X, S1),

and for q = 1 there is an exact sequence

0 −→ H 1(Sn+1 \X, Z) −→ hn−1(X, S1) −→ h 1(Sn+1 \X, S1) −→ 0,

where H ∗ is the Alexander-Spanier cohomology, h q is the partially contin-uous Alexander-Spanier cohomology [2].



References

[1] Milnor, J., On the Steenrod homology theory. Mimeographed Notes, Princeton,1960.

[2] Mdzinarishvili, L., Partially continuous Alexander–Spanier cohomology theory,Topologic and nichtkommutative geometric. Universitat Heidelberg Mathema-tisches Institut, Heft No. 130, 1996.



Linking pairing and foliated cohomology

Yoshihiko Mitsumatsu

Department of Mathematics, Chuo University, 1-13-27, Kasuga, Bunkyo-ku,Tokyo, 112-8551, [email protected]

2000 Mathematics Subject Classification. 57R30, 57R17

For a closed oriented 3-manifold M with a preferred volume form dvol, letXh denote the kernel of the natural surjective homomorphism from Xd toH1(M ; R), where Xd is the Lie algebra of all smooth divergence free vectorfields on M .

We can define so called asymptotic linking pairing between X and Y ∈Xh ([1], [2]), which is non-degenerate and symmetric. Consider the problemthat whether if we can define the signature of this pairing. It is clear thatthe dimensions of maximal positive and maximal negative subspaces areinfinite, so that definig the signature is to give a meaning to this ∞−∞.

The results presented in this talk are as follows. First, in the presence ofa foliation of codimension 1, we can ‘renormalize’ ∞−∞ and in some caseswe get finite number. Second, it is shown that this computation reduces tothat of 1st foliated(leafwise) cohomology. Thirdly, using our previous resultson the foliated cohomology [3], in the case of algebraic Anosov foliations,we get 0 for suspension type Anosov and 1 for geodesic flow type Anosov asthe signature. Of course this process may depend on the choice of foliationsand it does not imply that it gives rise to an invariant of the manifold itself.

A possible application of this framework to contact topology, namely ananalytic definition of torsion invariant is also proposed and some analyticaldifficulties that have been confronted are discussed.

References

[1] V. I. Arnol’d, The asymptotic Hopf invariant and its applications, Selecta Math.Soviet. 5 (1986), 327-345.

[2] J.-M. Gambaudo & E. Ghys, Enlacements asymptotiques, Topology, 36 (1997),1355-1379.

[3] Sh. Matsumoto & Y. Mitsumatsu, Leafwise cohomology and rigidity of certainLie group actions, Ergodic Theory and Dynamical Systems, 23-6 (2003), 1839-1866.



Classification of manifolds – recent results

Himadri Kumar Mukerjee

Department of Mathematics, North-Eastern Hill University, NEHU Campus,Shillong-793022, [email protected], [email protected]

2000 Mathematics Subject Classification. 55P10; 55P25; 55S99; 57R19;57R67

Surgery on manifolds of dimension ≥ 5 and its application to clssification ofmanifolds within a homotopy type had been a central problem in topologyin 1960s and 1970s (Milnor, Kervaire, Browder, Novikov, Wall, Sullivan,Kirby, Siebenmann [2] and many others contributed in this period). Af-ter that there has been intermitant developments, algebraic reformulation,computation of surgery, applications to Novikov’s conjecture etc. (Madsen,Milgram, Hambleton, Taylor, Williams, Ranicki, Farrel, Jones and manyothers contributed in this period). There has also been some isolated devel-opment leading to classification in the homotopy type of newer manifolds(Kharshiladze [1], Rudyak and others contributed in this). The author plansto present his recent results on the classification of manifolds homotopyequivalent to Dold manifolds [3], real Milnor manifolds [4] and Wall’s mani-folds [5]. These manifolds form generators of the cobordism groups of man-ifolds. The results were obtained by determining the sets, groups and themaps in the Sullivan–Wall surgery exact sequences involving the structuresets of these manifolds.

References

[1] Kharshiladze, A. F., Surgery on manifolds with finite fundamental group, Us-pekhi Mat. Nauk. 42:4 (1987), 55–85.

[2] Kirby, R. C., Siebenmann, L. C., Foundational Essays On Topological Mani-folds, Smoothings, And Triangulations. Annals of Math. Studies 88, PrincetonUniversity Press, Princeton, NJ, 1977

[3] Mukerjee, H. K., Classification of homotopy Dold manifolds, New York J. Math.9 (2003), 1-23.

[4] Mukerjee, H. K., Classification of homotopy real Milnor manifolds, Topologyand its Appl. 139 (2004), 151-184.

[5] Mukerjee, H. K., Classification of homotopy Wall’s manifolds, Topology and itsAppl., to appear.



Characterization and topological rigidity of Nobeling manifolds

Andrzej Nagorko

Department of Topology, Institute of Mathematics of the Polish Academy ofSciences, ul. Sniadeckich 8 P. O. Box 21 00-956 Warszawa, [email protected]

2000 Mathematics Subject Classification. 55M10, 54F45

We develop a theory of Nobeling manifolds similar to the theory of Hilbertspace manifolds and to the theory of Menger manifolds developed by M. Bestv-ina [1]. In particular we prove the Nobeling manifold characterization con-jecture.

We define the n-dimensional universal Nobeling space νn to be the setof points of R2n+1 with at most n rational coordinates. To enable compar-ison with the infinite dimensional case we let ν∞ denote the Hilbert space.We define an n-dimensional Nobeling manifold to be a Polish space locallyhomeomorphic to νn. The following theorem for n = ∞ is the characteriza-tion theorem of H. Torunczyk [4]. We establish it for n < ∞, where it wasknown as the Nobeling manifold characterization conjecture.Characterization theorem. An n-dimensional Polish ANE(n)-space isa Nobeling manifold if and only if it is strongly universal in dimension n.

The following theorem was proved by D. W. Henderson and R. Schori [2]for n = ∞. We establish it in the finite dimensional case.Topological rigidity theorem. Two n-dimensional Nobeling manifoldsare homeomorphic if and only if they are n-homotopy equivalent.

We also establish the open embedding theorem, the Z-set unknottingtheorem, the local Z-set unknotting theorem and the sum theorem forNobeling manifolds.

References

[1] Bestvina, M., Characterizing k-dimensional universal Menger compacta. Mem.Amer. Math. Soc. 71 (1988), no. 380.

[2] Henderson, D. W. and Schori, R., Topological classification of infinite dimen-sional manifolds by homotopy type. Bull. Amer. Math. Soc. 76 (1970), 121–124.

[3] Nagorko, A., Characterization and topological rigidity of Nobeling manifolds.Available online from http://arxiv.org/abs/math/0602574. PhD Thesis,Warsaw University, 2006.

[4] Torunczyk, H., Characterizing Hilbert space topology. Fund. Math. 111 (1981),247–262.



C∞(X) and related ideals

F. Azarpanah, M. Namdari*

Department of Mathematics, Chamran University, Ahvaz, [email protected]

2000 Mathematics Subject Classification. 54C40

We have characterized the spaces X for which the smallest z-ideal contain-ing C∞(X) is prime. The intersection of all free maximal ideals in C∗(X)is denoted by C∞(X) which precisely consists of all continuous functionsf in C(X) vanish at infinity, i.e., {x ∈ X : |f(x)| ≥ 1

n} is compact, forall integer n. C∞(X) is investigated as a ring in [1] and as an ideal ofC(X) in [2].Some interesting ideals related to C∞(X) are introduced andcorresponding to the relations between these ideals and C∞(X), topologi-cal spaces X are characterized. Some compactness concepts are explicitlystated in terms of ideals related to C∞(X). Finally we have shown that aσ-compact space X is Baire if and only if every ideal containing C∞(X) isessential.

References

[1] A.R. Aliabad, F. Azarpanah, M. Namdari, Rings of continuous functions van-ishing at infinity, Comment. Mat. Univ. Carolinae, 45,3(2004), 519-533

[2] F. Azarpanah, T. Soundararajan, When the family of functions vanishing atinfinity is an ideal of C(X), Rocky Mountain J. Math., 31.4 (2001), 1-8.



Algebraic topology in constrained variational equations

Agostino Prastaro

Department of Methods and Mathematical Models For Applied Sciences,University of Roma “La Sapienza”, Via A. Sacrpa, 16 - 00161 Roma, [email protected]

2000 Mathematics Subject Classification. 55N22, 58B32, 58E15, 58K65,81T75

Following our previous results on the algebraic topological characterizationsof (quantum, super) PDE’s [1–5], we will consider, in this talk, variationalproblems, in the category QS of quantum supermanifolds, constrained bypartial differential equations. We get theorems of existence for local andglobal solutions. The characterization of global solutions is made by meansof integral bordism groups.

References

[1] A. Prastaro, Quantum manifolds and integral (co)bordism groups in quantumpartial differential equations, Nonlinear Analysis, 47/4:2609–2620, 2001, MR2004c:35343.

[2] A. Prastaro, Quantized Partial Differential Equations, World Scientific Pub-lishing, River Edge, NJ, 2004, MR 2086084; ZM 02113703.

[3] A. Prastaro, (Co)bordism groups in quantum super PDE’s. I: Quan-tum supermanifolds, Nonlinear Analysis: Real World Appls., DOI:101016/j.nonrwa.2005.12.008, in press.

[4] A. Prastaro, (Co)bordism groups in quantum super PDE’s. II: Quan-tum super PDE’s, Nonlinear Analysis: Real World Appls., DOI: 101016/j.nonrwa.2005.12.007, in press.

[5] A. Prastaro, (Co)bordism groups in quantum super PDE’s. III: Quantumsuper Yang-Mills equations, Nonlinear Analysis: Real World Appls., DOI:101016/j.nonrwa.2005.12.005, in press.



An invariant for Stallings manifolds from a TQFT

P. Semiao

Departamento de Matematica, Universidade do Algarve, Faculdade de Ciencias eTecnologia 8000-062 Faro, [email protected]

2000 Mathematics Subject Classification. 57R56

We obtain an invariant of Stallings [1] manifolds using a Topological Quan-tum Field Theory (TQFT) [2] approach. The TQFT’s are specified by ma-trix equations, which are solved using unitary matrices. In our approach [3]a TQFT is a monoidal functor [4] from a “topological” monoidal category toan “algebraic” monoidal category. The objects of the topological categoryare triples (X, A,m), where X is an orientable connected 3-dimensionalmanifold with boundary, A is an orientable connected 2-dimensional man-ifold without boundary, and m : A → X is an “inclusion” morphism. Themorphisms of this category are isomorphisms and gluing morphisms. Onthe algebraic side, the objects of the category are pairs (V, x), where V isa vector space over a field with an involution, and x is an element of V .The morphisms are linear maps which preserve the elements of the respec-tive spaces. The TQFT functor preserves an additional structure given bytwo monoidal endofunctors, corresponding to change of orientation on thetopological side and changing the scalar multiplication using the involutionon the algebraic side.

A fundamental feature of TQFT is the gluing of two spaces, and thepresent approach describes this operation in terms of morphisms in thetopological category, but describes equally well the self-gluing of a singlespace. E.g. if S is a 2-dimensional manifold without boundary endowedwith a fixed orientation and ϕ : S → S is an orientation-preserving auto-morphism, then the self-gluing of the cylinder S × I, where I is the stan-dard closed unit interval, is a 3-dimensional manifold Sϕ := S×I

∼ϕknown

as a Stallings manifold, where ∼ϕ is the relation generated by the relation(x, 0) ∼ (ϕ(x), 1).

References

[1] Stallings, J., On Fibering Certain 3-Manifolds, from: “Topology of 3-Manifoldsand related topics, Proceedings of the University of Georgia Institute, 1961”,M. K. Fort, Jr. Ed., Prentice-Hall, New Jersey, 1962, 95–100.

[2] Atiyah, M., Topological Quantum Field Theories, Publ. Math. Inst. HautesEtudes Sci., 68 (1989), 175–186.



[3] Picken, R., Semiao, P., A new framework for TQFT’s, submitted to AGT.

[4] Mac Lane, S., Categories for the Working Mathematician, 2nd edition, SpringerVerlag, 1998.



Stability and thickness of the homology of torus knots

Marko Stosic

Departamento de Matematica, Instituto Superior Tecnico, Av. Rovisco Pais 1,1049-001 Lisboa, [email protected]


In this talk we present the properties of Khovanov homology and of Khovanov-Rozansky homology for torus knots.

First of all, we relate the homology groups of T (p, q) and T (p, q + 1),and we show that up to certain homological degree, their homology groupscoincide. As corollaries, we calculate the homology of torus knots for lowhomological degrees, and we show that non-alternating torus knots are ho-mologically thick. Finally we prove that there exists stable Khovanov ho-mology for torus knots, conjectured by Dunfield, Gukov and Rasmussen in[1].

Furthermore, we show analogous properties for the Khovanov-Rozansky(sl(n)) homology of torus knots. Even though the particular definition ofKhovanov-Rozansky homology makes it practically incalculable, we man-aged to apply the similar methods as in Khovanov homology case (cube ofresolutions, long exact sequences in homology), and thus to obtain the anal-ogous results. Namely, we show that the sl(n)-homology groups of T (p, q)and T (p, q + 1) coincide up to certain homological degree, and that thereexists stable sl(n)-homology for torus knots.

References

[1] N. Dunfield, S. Gukov and J. Rasmussen: The Superpolynomial for knot ho-mologies, arXiv:math.GT/ 0505662.

[2] M. Khovanov: A categorification of the Jones polynomial, Duke Math. J.101:359-426 (2000)

[3] M. Khovanov, L. Rozansky: Matrix Factorizations and link homology,arXiv:math.QA/ 0401268.

[4] M. Stosic: Homological thickness of torus knots, arXiv:math.GT/0511532



On the perfectness of the group of real analytic diffeomorphisms

Takashi Tsuboi

Graduate School of Mathematical Sciences, University of Tokyo, Komaba Meguro,Tokyo 153-8914, [email protected]

2000 Mathematics Subject Classification. Primary 57R30, 58F18; Secondary58F15, 53C30, 53C15, 53C12

The group of real analytic diffeomorphisms Diffω(M) of a real analyticmanifold M is a rich group. It is dense in the group Diff∞(M) of smoothdiffeomorphisms. Herman [2] showed that for the n-dimensional torus Tn,the identity component Diffω(Tn)0 of Diffω(Tn) is a simple group. He con-jectured that the identity component Diffω(M)0 of Diffω(M) is simple fora real analytic closed manifold M .

We consider a little weaker problem and we would like to show thatDiffω(M)0 is a perfect group for a wider class of manifolds. We show that fora real analytically multi U(1) fibered manifold M , the identity componentDiffω(M)0 is a perfect group. Here, a closed manifold M of dimension n ismulti U(1) fibered if there exist n distinct principal U(1) bundle structureswith the tangent spaces of fibers spanning the tangent space TxM of M ateach point x ∈ M . For example, compact Lie groups and the 7 dimensionalsphere S7 are multi U(1) fibered. We also show that Diffω(M)0 is a perfectgroup for a product M of spheres.

The following result is an important step to show our main results and itis shown by using a result of Arnold [1]. Let p : M −→ B be a real analyticprincipal U(1) bundle over a closed manifold B. Let Diffω(F) denotes thegroup of real analytic diffeomorphisms mapping each fiber of the projectionp : M −→ B to itself. The identity component Diffω(F)0 of Diffω(F) is aperfect group.

References

[1] V.I. Arnol’d Small denominators. I, Mappings of the circumference onto it-self, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961) 21–86 = Amer. Math. Soc.Translations ser 2. 46 (1965), 213–284.

[2] M. Herman, Sur le groupe des diffeomorphismes R-analytiques du tore, (Proc.Colloq., Dijon, 1974), pp. 36–42. Lecture Notes in Math., Vol. 484, Springer,Berlin, 1975.



Homeomorphism groups of noncompact 2-manifolds

Tatsuhiko Yagasaki

Division of Mathematics, Kyoto Institute of Technology, Matsugasaki, Sakyoku,Kyoto 606-8585, [email protected]

2000 Mathematics Subject Classification. 57S05, 58D05, 57N05, 57N20

Topological properties of homeomorphism groups of compact 2-manifoldsand their subgroups have been studied by various authors. In this expositionwe report some extensions of these results to the case of non-compact 2-manifolds.

Suppose M is a non-compact connected 2-manifold with ∂M = ∅. LetH(M) denote the group of homeomorphisms of M with the compact-opentopology. For a subgroup G of H(M), let G0 denote the connected compo-nent of idM in G. The scripts c, PL and QC denote “compact support”,“piecewise linear”, “quasiconformal” respectively. Our results are summa-rized as follows.

Theorem (1) ([2], [3]) (i) H(M)0 is an `2-manifold.

(ii) Homotopy Type:(a) H(M)0 ' S1 if M = R2, S1 × R or Open Mobius Band(b) H(M)0 ' ∗ in all other cases.

(2) ([4]) Hc(M)0 is HD (homotopy dense) in H(M)0.

(3) ([4]) If M is a non-compact connected PL 2-manifold with ∂M = ∅,then(i) HPL,c(M)0 is an `f

2 -manifold, (ii) HPL,c(M)0 is HD in H(M)0.

(4) ([1]) If M is a connected Riemann surface, then(i) HQC(M)0 is a Σ-manifold, (ii) HQC,c(M)0 is HD in H(M)0.

In [5] we have classified the homotopy type and the topological typeof the connected components of the space of embeddings of a compactconnected polyhedron into M . Recently we have obtained some results ongroups of measure-preserving homeomorphisms of noncompact 2-manifolds(math.GT/0507328, 0512231).



References

[1] Yagasaki, T., The groups of quasiconformal homeomorphisms on Riemann sur-faces, Proc. Amer. Math. Soc. 127 (1999), 2727–2734.

[2] Yagasaki, T., Spaces of embeddings of compact polyhedra into 2-manifolds,Topology Appl. 108 (2000), 107–122.

[3] Yagasaki, T., Homotopy types of homeomorphism groups of noncompact 2-manifolds, Topology Appl. 108 (2000), 123–136.

[4] Yagasaki, T., The groups of PL and Lipschitz homeomorphisms of noncompact2-manifolds, Bulletin of the Polish Academy of Sciences, Mathematics 51(4)(2003), 445–466.

[5] Yagasaki, T., Homotopy types of the components of spaces of embeddings ofcompact polyhedra into 2-manifolds, Topology Appl. 153 (2005), 174–207.



Automorphisms of braid groups on surfaces

Ping Zhang

Department of Mathematics, Eastern Mediterranean University, Gazimagusa,North [email protected]

2000 Mathematics Subject Classification. 57N05, 20F28, 20F36

Consider the surface braid group Bn(M) of n strings as a normal subgroupof the isotopy group G(M, n) of homeomorphisms of the surface M permut-ing n fixed distinguished points. A classical theorem of Nielsen says thatOut(π1(M)) = Out(B1(M)) is isomorphic to G(M) if M is a closed sur-face which is not a sphere ([2]). Exploiting presentations of Bn(M) and thepure braid group SBn(M) (see, say, [3]), we investigate a generalization ofNielsen’s theorem, that is, whether an automorphism of Bn(M) or SBn(M)is geometric in the sense that it is induced by a homeomorphism of M inthe form of conjugate action. Here are some of our results.

1. If M is a closed surface of negative Euler characteristic, Aut(Bn(M)),as well asAut(SBn(M)), is isomorphic to G(M, n), and Out(Bn(M)) is isomorphicto G(M), where G(M) is the isotopy group of all homeomorphisms of M([4]). This resembles the conclusions of [1] when M is orientable.

2. Let n ≥ 4. Then Aut(Bn(S2)) ∼= Z2 × Inn(Bn(S2)) × Z2, andOut(Bn(S2)) ∼= Z2 ⊕ Z2, where the first Z2 is generated by the reflec-tion of S2 in an equatorial plane and the second by an involution inducedby the multiplication of the full twist, which is the generator of the centreof Bn(S2).

3. Let n ≥ 3. Then Aut(Bn(RP 2)) ∼= Z2×Inn(Bn(RP 2)), and Out(Bn(RP 2))∼= Z2, where the generator of Z2 is the involution induced by the multipli-cation of the natural image in Bn(RP 2) of the full twist in Bn(S2).

References

[1] Irmak, E., Ivanov, N. V., McCarthy, J. D., Automorphisms of surface braidgroups, arXiv:math.GT/ 0306069 v1, 3 Jun 2003.

[2] Nielsen, J., Die Struktur periodischer Transformationen von Flachen, Danskevid. Selsk. Mat.-Fis. (1937), 1–37.

[3] Scott, G. P., Braid groups and the group of homeomorphisms of a surface,Proc. Camb. Phil. Soc. 68(1970), 605–617.



[4] Zhang, P., Automorphisms of braid groups on closed surfaces which are notS2, T 2, P 2 or the Klein bottle, J. Knot Theory Ramifications, to appear.


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