Evgeniy V. Martyushev RESEARCH STATEMENT

Transcription

1 Evgeniy V. Martyushev RESEARCH STATEMENT My research interests lie in the fields of topology of manifolds, algebraic topology, representation theory, and geometry. Specifically, my work explores various interactions among these fields. Some of them arise naturally from the study of a new method of constructing invariants of 3-manifolds (and links in 3-sphere) introduced by I.G. Korepanov in [4, 6, 8, 9]. We call these invariants geometric because their construction is essentially based on the geometry of 3-dimensional Euclidean space. Let me describe this in more detail. It is well known that any two triangulations of a 3-manifold can be transformed into each other using a final sequence of four local rebuildings of the triangulation (Pachner moves). If we construct an algebraic expression depending on some values ascribed to a manifold triangulation and invariant under these moves, we get a value that is independent of the choice of triangulation. It is natural way to construct a manifold invariant. For example, Euler characteristic and Turaev-Viro type invariants can be proved to be invariant using exactly this method. Let M be a connected closed orientable 3-dimensional manifold, T its triangulation. Let us ascribe a real number λ ij > 0 to every edge e ij T, connecting vertices i and j, so that the following inequality holds for every tetrahedron of complex T with edges (e ij, e ik, e il, e jk, e jl, e kl ): λ 2 ij λ 2 ik λ 2 il 1 λ 2 ij 0 λ 2 jk λ 2 jl > 0. (1) 1 λ 2 ik λ 2 jk 0 λ 2 kl 1 λ 2 il λ 2 jl λ 2 kl 0 Recall that this condition guarantees the existence of a non-degenerate Euclidean tetrahedron with edge lengths (λ ij, λ ik, λ il, λ jk, λ jl, λ kl ) and this tetrahedron is unique up to isometries. Further, let us ascribe a sign + or to every tetrahedron τ T. For each its inner dihedral angle at this edge, taken with the sign ascribed to ith tetrahedron. This angle is a function of λ s. edge a, belonging to ith tetrahedron of complex T, we denote by ϕ (i) a Definition 1. Defect angle ω a at an edge a is a quantity ω a = i ϕ (i) a mod 2π, where the sum is taken over all tetrahedra sharing the edge a. Let us say that we are given an admissible coloring on T, if to every edge e ij T a number λ ij is ascribed so that the condition (1) holds, and to every tetrahedron from T a sign is ascribed so that the defect angle at every edge vanishes. The construction of geometric invariant is naturally divided into three main parts. 1

2 First, on a given admissible coloring of the triangulation T, we find a class [ρ] of equivalent representations ρ: π 1 (M) E(3), and also a covering of M corresponding to the kernel of ρ [ρ]. Then we map the triangulated covering space T into 3-dimensional Euclidean space according to the action of Im ρ as follows. The vertices from T of the same orbit are mapped into the points of R 3 and coordinates of these points are related by the elements of Im ρ. Each simplex of nonzero dimension is mapped into the convex linear shell of its vertex images in R 3. The following theorem states in fact that such a mapping is agreed with the initial admissible coloring. Theorem 1 ([13]). Let T be a simplicial complex for the universal covering space M. Let an admissible coloring be given on T. Then, there exists a continuous map Γ: T R 3 such that l ij = λ ij, i, j, where l ij is the length of Γ(e ij ). Any other map Γ : T R 3 for the same admissible coloring can be obtained from Γ by an orientation preserving isometry of R 3. Let T x M denote the tangent space to M at a point x, F be a fundamental family of simplices in T, i.e. such a family of simplices of T that over each simplex of T lies exactly one simplex of this family. Let N i denote the full number of i-dimensional simplices in F. Finally, we construct an algebraic complex of the following vector spaces with distinguished bases. The space e(3) ρ = {u e(3) Ad ρ(h) u = u, h π 1 (M)} = H 0 (M; Ad ρ ), where Ad ρ = Ad ρ: π 1 (M) Aut(e(3)). The space (dg) = T [ρ] R(M, E(3)) = H 1 (M; Ad ρ ), where R(M, E(3)) is the representation space of π 1 (M) in E(3). The space (dx) = T Γ R 3N0 x, where R 3N0 x vertices from F to R 3. is the space of all mappings of The space (dl) = T Γ R N1 l, where R N1 l is the space of all mappings of edges from F to R with the basis (l 1,..., l N1 ), l i is the Euclidean length of ith edge. The space (dω) = T Γ R N 1 ω, where R N 1 ω is the space of all mappings of edges from F to R with the basis (ω 1,..., ω N1 ), ω i is the defect angle at ith edge. The spaces e(3) ρ, (dx) and (dg) have the bases dual to e(3) ρ, (dx) and (dg) respectively. Theorem 2 ([9], [13]). The sequence of vector spaces and linear mappings f 1 f 2 f 3 =f 0 e(3) ρ (dx) (dg) (dl) T 3 (dω) f T 2 (dx) (dg) f T 1 e(3) ρ 0 (2) is an acyclic complex, i.e. the image of every mapping in this sequence is isomorphic to the kernel of next mapping. 2

3 Let τ denote the torsion of the acyclic complex (2) (definitions and properties of acyclic complexes and their torsions can be found in the remarkable book [15]). We define the geometric invariant as I ρ (M) = τ ( 6V ), (3) where V is the volume of tetrahedron, and the product is taken over all tetrahedra of F. The value I ρ (M) is independent of various details of its construction. Namely, it does not depend on the choice of the triangulation T, on the choice of the fundamental family F, and, finally, on the choice of Γ, i.e. it does not matter in which points of Euclidean space we place the vertices from the fundamental family (see [13] for details and proofs). Thus, the invariant (3) depends only on the manifold M and on the representation ρ or, more precisely, on the class [ρ]. For example, calculations show that for the trivial representation the invariant is expressed in terms of the first homology group of M. On the other hand, for abelian representations, our invariant seems to be related to the (abelian) Reidemeister torsion (see formula (6) and the paragraph below). The geometric invariant is multiplicative under the connected sum operation. Let M 1, M 2 be oriented closed 3-manifolds, ρ i : π 1 (M i ) E(3) a representation of the fundamental group of M i in E(3). Denote by ρ # : π 1 (M 1 ) π 1 (M 2 ) E(3) a representation of free product of π 1 (M 1 ) and π 1 (M 2 ) such that ρ = ρ # π1 (M i ) i. Theorem 3 ([13]). Let ρ 1 = θ be trivial. Then, the geometric invariant for M 1 #M 2 and ρ # is I ρ# (M 1 #M 2 ) = I θ (M 1 ) I ρ2 (M 2 ). (4) The sign minus in this formula appears because the invariant for 3-sphere equals 1. In the paper [9] we proposed a general formula of the geometric invariant for so-called lens spaces, which was then proved in [13]. Let p > q > 0 be two coprime integers. We identify S 3 with the subset {(z 1, z 2 ) C 2 z z 2 2 = 1} of C 2. The lens space L(p, q) is defined as the quotient manifold S 3 /, where denotes the action of the cyclic group Z p on S 3 given by: ζ (z 1, z 2 ) = (ζz 1, ζ q z 2 ), ζ = e 2πi/p. (5) Theorem 4 ([13]). The invariant I k (L(p, q)) for L(p, q) and a nontrivial representation ρ k of π 1 (L(p, q)) = Z p is equal to where k = 1,..., p 1. I k (L(p, q)) = 1 ( p 2 4 sin πk ) 4 πkq sin, (6) p p For coprime p and q, there are two integers a and b such that aq + bp = 1. Let ζ be a primitive root of unity of degree p. Then, the Reidemeister torsion of the lens space L(p, q) is defined by the formula ([15, theorem 10.6]): τ R (L(p, q)) = (1 ζ k ) 1 (1 ζ ka ) 1, 3

4 where k = 1,..., p 1. Comparing this with (6), we conclude that for a given p formula (6) yields τ R (L(p, q)) 4 up to a constant. It turns out that the construction of geometric invariants can be generalized for links in 3-sphere as follows. First, on a given oriented link L we construct an oriented triangulation of S 3 satisfying the following conditions: 1. the whole link L lies on some edges of the triangulation; 2. for any tetrahedron of the triangulation, not more than two of its vertices belong to L; 3. for any edge e of the triangulation, its vertices either are different or, if they coincide, e represents the corresponding meridian of some component of L. Here the edges with coinciding ends from the condition 3 are used to define the following simplicial move. Let an edge BD lie on a certain link component, and let there be a tetrahedron BDAA in the triangulation, with its edge AA representing a meridian of the corresponding link component. The move 1 2 is defined as follows: take a point C in the edge BD and replace the tetrahedron BDAA by two tetrahedra BCAA and CDAA. The move 2 1 is the inverse to that. Using the methods of the paper [10], one can prove the following analogue of the Pachner theorem (see [8] for details). Theorem 5 ([8]). A triangulation of sphere S 3 obeying the conditions 1 3 can be transformed into any other triangulation obeying the same conditions by a sequence of the following elementary moves: Pachner moves 2 3 and 1 4. Such moves are not affect the edges lying on the link L (however, the link may pass through edges and/or vertices lying in the boundary of the transformed cluster of tetrahedra); moves 1 2. Note that there exists a simple general algorithm of constructing the triangulation with properties 1 3 for arbitrary L. Further, on a given representation ρ: πl SO(3) of the link group in SO(3) we construct a covering of 3-sphere branched along the link. After that, the covering space is mapped into the 3-dimensional Euclidean space according to the action of Im ρ in the same manner as for the case of 3-manifolds. Then we construct an algebraic complex which is similar to (2). Finally, under the assumption of acyclicity of this complex, we find its torsion τ and the geometric invariant for a link: I ρ (L) = τ N (2 2 cos ϕ j ) n j j=1 l 2 ( 6V ). (7) Here N is the number of link components; n j is the number of vertices in F lying on jth component; ϕ j s are parameters of the representation ρ (rotation around 4

5 the overpass belonging to jth component through the angle ϕ j is a generator of πl in the Wirtinger presentation). In the denominator, the primed product is taken over whose edges from F which lie on the link, V is a tetrahedron volume and the second product is taken over all tetrahedra of F. Calculations of the geometric invariant for different knots and links let us propose the following Conjecture 1 ([13]). Let L S 3 be a link with N components, L (t 1,..., t N ) its Alexander polynomial, and ρ: πl SO(3) an abelian representation of the link group. Then, { L (e iϕ 1 ) 4 (2 2 cos ϕ 1 ) 2, N = 1 I ρ (L) = L (e iϕ 1,..., e iϕ N ) 4, N > 1. To conclude I would like to mention a few directions for the future research. Relative invariants and topological field theories. In the paper [3], a relative version of the geometric invariant was constructed for a 3-manifold and a framed knot therein. The next natural step in this direction is a developing of topological field theory on the base of relative geometric invariants in the spirit of Atiyah s axioms [1]. Quantum generalization. There is a very intriguing question about possible existence of some quantum values from which the geometric invariants can be obtained as a semiclassical limit. This question is motivated by the fact that some formulas in our construction can be obtained by a limiting procedure from similar relations for the quantum 6j-symbols (cf. [4, formula (5)]). One more promising idea in this direction, proposed recently by I.G. Korepanov, is a use (instead of the Euclidean geometry) of the geometry related to the action of group PSL(2, C) on a complex variable by fractional-linear transformations. Generalization to higher-dimensional manifolds and knots. In the papers [5, 6, 7], the geometric invariant was constructed for 4-dimensional manifolds. It would be interesting to apply this construction for knotted surfaces in 4-sphere (and, in general, for knotted (n 2)-dimensional surfaces in n-sphere). Higher-dimensional generalization will be especially interesting if combined with quantum generalization. My general mathematical interests include many questions in related areas. I can mention the volume conjecture, Chern-Simons and Seiberg-Witten invariants, Khovanov homologies. I would be glad to continue my research on geometric invariants and to explore the other branches of mathematics. References [1] M.F. Atiyah, Topological quantum field theory, Publications Mathématiques de l IHÉS, 68 (1988),

6 [2] J. Dubois, Torsion de Reidemeister non abelienne et forme volume sur l espace des représentations du groupe d un nœud: Ph.D. thesis, Université Blaise Pascal, 2003, 144 pages. [3] J. Dubois, I.G. Korepanov, E.V. Martyushev, Euclidean geometric invariant of framed knots in manifolds, preprint, arxiv:math/ , [4] I.G. Korepanov, Invariants of PL manifolds from metrized simplicial complexes, J. Nonlin. Math. Phys. 8 (2001), [5] I.G. Korepanov, Euclidean 4-simplices and invariants of four-dimensional manifolds: I. Moves 3 3, Theor. Math. Phys. 131 (2002), [6] I.G. Korepanov, Euclidean 4-simplices and invariants of four-dimensional manifolds: II. An algebraic complex and moves 2 4, Theor. Math. Phys. 133 (2002), [7] I.G. Korepanov, Euclidean 4-simplices and invariants of four-dimensional manifolds: III. Moves 1 5 and related structures, Theor. Math. Phys., 135 (2003), [8] I.G. Korepanov, Euclidean tetrahedra and knot invariants, Proceedings of the Chelyabinsk Scientific Center 24 (2004), 1 5. [9] I.G. Korepanov, E.V. Martyushev, Distinguishing three-dimensional lens spaces L(7, 1) and L(7, 2) by means of classical pentagon equation, J. Nonlin. Math. Phys. 9 (2002), [10] W.B.R. Lickorish, Simplicial moves on complexes and manifolds, Geometry and Topology Monographs 2 (1989), [11] E.V. Martuyshev, Euclidean simplices and invariants of three-manifolds: a modification of the invariant for lens spaces, Proceedings of the Chelyabinsk Scientific Center 19 (2003), No. 2, 1 5. [12] E.V. Martuyshev, Euclidean geometric invariants of links in 3-sphere, Proceedings of the Chelyabinsk Scientific Center 26 (2004), No. 4, 1 5. [13] E.V. Martuyshev, Geometric invariants of three-dimensional manifolds, knots and links: PhD thesis, South Ural State University, 2007, 103 pages. [14] U. Pachner, PL homeomorphic manifolds are equivalent by elementary shellings, Europ. J. Combinatorics 12 (1991), [15] V.G. Turaev, Introduction to combinatorial torsions, Boston: Birkhäuser, 2000, 144 pages. 6