Link Theory in Manifolds

Kaiser, Uwe

doi:10.1007/bfb0092686

Cited by 19 publications

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“…For oriented closed rational homology 3-spheres the linking numbers in Q are classical and defined e.g. in [12]. The linking numbers in Q are defined by mapping the universal linking numbers (defined in the commutative ring R(M ) := Z[…”

Section: Theorem 6 the Inclusion Zlmentioning

confidence: 99%

Deformation of string topology into homotopy skein modules

Kaiser

2003

Algebr. Geom. Topol.

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Relations between the string topology of Chas and Sullivan and the homotopy skein modules of Hoste and Przytycki are studied. This provides new insight into the structure of homotopy skein modules and their meaning in the framework of quantum topology. Our results can be considered as weak extensions to all orientable 3-manifolds of classical results by Turaev and Goldman concerning intersection and skein theory on oriented surfaces. AMS Classification 57M25; 57M35, 57R42Keywords 3-manifold, string topology, deformation, skein module, torsion, link homotopy, free loop space, Lie algebra IntroductionIn 1999 Moira Chas and Dennis Sullivan discovered the structure of a graded Lie algebra on the equivariant homology of the free loop space of an oriented smooth or combinatorial d-manifold [2]. Later Cattaneo, Fröhlich and Pedrini developed ideas about the quantization of string topology in the framework of topological field theory [1]. It is the goal of this paper to study VassilievKontsevitch and skein theory of links in 3-manifolds in relation with string topology. Our approach is intrinsically 3-dimensional. It hints towards a general deformation theory for a category of modules over the Chas-Sullivan Lie bialgebra (see also [3]) of an oriented 3-manifold. But the line of thought will not follow the ideas of classical quantization as in [1].Here is the main idea of the paper: For M a 3-dimensional manifold, the Chas-Sullivan structure measures the oriented intersections between the loops in a 1-dimensional family with those in a 0-dimensional familiy (which is just a formal linear combination of free homotopy classes of loops). The resulting pairing takes values in formal linear combinations of free homotopy classes. It is easy to see that the construction extends to collections of loops. It is our Our main method is transversality of families, which is at the heart of Vassiliev theory [21]. We will show that the process of replacing homotopy classes of maps by equivalence classes of transverse objects (oriented links in the case of 0-dimensional homology of the mapping space) naturally deforms the ChasSullivan type intersection theory into well-known structures in quantum topology. Thus our approach provides a weakened version of quantization in the category of oriented 3-manifolds in comparison to the deep results for cylinders over oriented surfaces due to Goldman [5] and Turaev [20]. Because of the lack of a geometric product structure in an arbitrary oriented 3-manifold such an extension will not be based on the deformation of algebra structures but the structures of modules over the Chas-Sullivan Lie algebra.In order to define a natural refinement of the Chas-Sullivan structure, isotopy has to be weakened to link homotopy of oriented links in M . Recall that in a deformation through link homotopy arbitrary self-crossings are allowed but the different components have to be disjoint during a deformation. This equivalence relation has first been considered by Milnor in 1952 [16] (also see the recent ...

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Section: Theorem 6 the Inclusion Zlmentioning

confidence: 99%

Deformation of string topology into homotopy skein modules

Kaiser

2003

Algebr. Geom. Topol.

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“…Before Low this way of defining linking numbers for nonzero homologous circles in ST R 2 was used by S. Tabachnikov [45]. The general theory of linking numbers when the linked objects are zero homologous in the homology group of the ambient manifold modulo boundary was developed by U. Kaiser [21]. When M is a closed manifold, the number lk defined using auxiliary negative fibers over some points is not an invariant of the link ( W x,M , W y,M ), since it changes when a link component passes through the auxiliary fiber corresponding to the other link component.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Linking and Causality in Globally Hyperbolic Space-times

Чернов

Rudyak

2008

Commun. Math. Phys.

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Abstract. The classical linking number lk is defined when link components are zero homologous. In [13] we constructed the affine linking invariant alk generalizing lk to the case of linked submanifolds with arbitrary homology classes. Here we apply alk to the study of causality in Lorentzian manifolds.Let M m be a spacelike Cauchy surface in a globally hyperbolic space-time (X m+1 , g). The spherical cotangent bundle ST * M is identified with the space N of all null geodesics in (X, g). Hence the set of null geodesics passing through a point x ∈ X gives an embedded (m− 1)-sphere Sx in N = ST * M called the sky of x. Low observed that if the link (Sx, Sy) is nontrivial, then x, y ∈ X are causally related. This motivated the problem (communicated by Penrose) on the Arnold's 1998 problem list to apply the machinery of knot theory to the study of causality. The spheres Sx are isotopic to the fibers of (ST * M ) 2m−1 → M m . They are nonzero homologous and the classical linking number lk(Sx, Sy) is undefined when M is closed, while alk(Sx, Sy) is well defined. Moreover, alk(Sx, Sy) ∈ Z if M is not an odd-dimensional rational homology sphere.We give a formula for the increment of alk under passages through Arnold dangerous tangencies. If (X, g) is such that alk takes values in Z and g is conformal to b g that has all the timelike sectional curvatures nonnegative, then x, y ∈ X are causally related if and only if alk(Sx, Sy) = 0. We prove that if alk takes values in Z and y is in the causal future of x, then alk(Sx, Sy) is the intersection number of any future directed past inextendible timelike curve to y and of the future null cone of x. We show that x, y in a nonrefocussing (X, g) are causally unrelated if and only if (Sx, Sy) can be deformed to a pair of S m−1 -fibers of ST * M → M by an isotopy through skies. Low showed that if (X, g) is refocussing, then M is compact. We show that the universal cover of M is also compact. PreliminariesWe work in the C ∞ -category, and the word "smooth" means C ∞ . An isotopy of a smooth embedding f : P → Q is a path in the space of smooth embeddings P → Q starting at f. Given an oriented manifold M m , consider its tangent bundle T M → M and put z : M → T M to be the zero section. Let R + be the group of positive real numbers under multiplication that acts on T M as (r, µ) → rµ, r ∈ R + , µ ∈ T M. We put ST M = T M \ z(M) /R + and note that the tangent bundle T M → M yields the spherical tangent bundle pr :For the reasons discussed right before Theorem 2.2, we will assume that dim M > 1. We denote by T * M → M the cotangent bundle over M, and we construct the spherical cotangent bundle pr : ST * M → M in a similar way. It is well known that ST * M possesses a canonical contact structure and that the S m−1 -fibers of ST * M are Legendrian submanifolds with respect to this contact structure, see [2] or Appendix A. Note also that the orientation of M yields canonical orientations on the fibers of spherical (co)tangent bundles. Namely, it is well known that every spherical (co)tangen...

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“…Assume that there is a mapping from a torus, which is not homologous into ∂M . It follows by Poincare duality (see [6], Appendix A) that there exists an oriented loop, which has a non-trivial intersection number with the singular torus. The map from the torus is the adjoint map of a representative of some element in π 1 (L(M ), f β ) for some β ∈π(M ).…”

Section: Proposition 1 S(m ) Is Isomorphic To Srπ(m ) (Equivalently mentioning

confidence: 99%

“…Obviously (see e.g. [6], Appendix A) each oriented embedded surface with a non-trivial intersection number with an oriented loop is non-separating. Thus Przytycki's theorem follows easily from the following consequence of D. Gabai's fundamental result.…”

Section: Computation Of the Monodromiesmentioning

confidence: 99%