Patrick M. Gilmer

Integrality for TQFTs

We discuss ways that the ring of coefficients for a TQFT can be reduced if one restricts somewhat the allowed cobordisms. When we apply these methods to a TQFT associated to SO(3) at an odd prime p, we obtain a functor from a somewhat restricted cobordism category to the category of free finitely generated modules over a ring of cyclotomic integers : Z[ζ2p], if p ≡ −1 (mod 4), and Z[ζ4p], if p ≡ 1 (mod 4), where ζk is a primitive kth root of unity. We study the quantum invariants of prime power order simple cyclic covers of 3-manifolds. We define new invariants arising from strong shift equivalence and integrality. Similar results are obtained for some other TQFTs but the modules are only guaranteed to be projective.

Integral lattices in TQFT

Abstract We find explicit bases for naturally defined lattices over a ring of algebraic integers in the SO ( 3 ) -TQFT-modules of surfaces at roots of unity of odd prime order. Some applications relating quantum invariants to classical 3-manifold topology are given.

Discriminants of Casson-Gordon invariants

Casson–Gordon invariants were first used to prove that certain algebraically slice knots in S 3 are not slice knots [2, 3]. Since then they have been applied to a wide range of problems, including embedding problems and questions relating to boundary links [2, 10, 21, 25]. The most general Casson–Gordon invariant takes its value in L 0 (ℚ(ζ d )( t )) ⊗ ℚ; here ζ d denotes a primitive d th root of unity. Litherland [20] observed that one could usually tensor with ℤ (2) instead of ℚ, and in this way preserve the 2-torsion in the Witt group. He then constructed new examples of non-slice genus two knots which were detected with torsion classes in L 0 (ℚ(ζ d )) ⊗ ℤ (2) modulo the image of L 0 (ℚ(ζ d )) ⊗ ℤ (2) .

On the slice genus of knots

Given a knot K in the 3-sphere, the genus of K, denoted g(K), is defined to be the minimal genus for a Seifert surface for K. The slice genus gs(K) is defined to be the minimal genus of an oriented surface G admitting a smooth proper embedding in the 4-ball which maps CG to K. If we insist the embedding of G have no local maximum with respect to the radial function, we obtain the ribbon genus gr(K) instead. Thus a knot is slice (ribbon) if and only if gs(K) = 0 (gr(K)= 0). It is clear that g.~(K)<g,.(K)<=g(K). There are well known lower bounds on gs(K) given by invariants of a Seifert matrix for K. These are all included in the invariant re(K) [8] defined by Taylor. It gives the best possible bound based on a Seifert matrix, re(K) vanishes if and only if the Seifert pairing is metabolic. If this is the case, K is called algebraically slice. The work of Casson and Gordon [1, 2, 5] showed that certain algebraically slice knots are not in fact slice. We generalize the main theorem of [1]. As an application, we give a sequence of algebraically slice knots Q, such that gs(Q,)=g(Q,,)=n. We also study the slice genus of K, 41=K, where K, denotes the t twisted double of the unknot. We show for example that g.~(K 12 =~ K 1 z ) = 2 . K I2 is algebraically slice but not slice by [ l ] . Section 1 has some preliminaries on the linking form. In Sect. 2, we state and prove our main theorem. In Sect. 3 we give our examples. In this paper, all manifolds are oriented. We use e to denote the Euler characteristic.

Maslov index, lagrangians,mapping class groups and TQFT

Abstract. Given a mapping class f of an oriented surface and a lagrangian λ in the first homology of , we define an integer . We use to describe a universal central extension of the mapping class group of as an index-four subgroup of the extension constructed from the Maslov index of triples of lagrangian subspaces in the homology of the surface. We give two descriptions of this subgroup. One is topological using surgery, the other is homological and builds on work of Turaev and work of Walker. Some applications to TQFT are discussed. They are based on the fact that our construction allows one to precisely describe how the phase factors that arise in the skein theory approach to TQFT-representations of the mapping class group depend on the choice of a lagrangian on the surface.

Integral bases for TQFT modules and unimodular representations of mapping class groups

Abstract We construct integral bases for the

On the Witten-Reshetikhin-Turaev representations of mapping class groups

SO(3)

LINK COBORDISM IN RATIONAL HOMOLOGY 3-SPHERES

-TQFT-modules of surfaces in genus one and two at roots of unity of prime order and show that the corresponding mapping class group representations preserve a unimodular Hermitian form over a ring of algebraic integers. For higher genus surfaces the Hermitian form sometimes must be non-unimodular. In one such case, genus three at a fifth root of unity, we still give an explicit basis.

3-manifold invariants and periodicity of homology spheres.

Abstract.We consider a central extension of the mapping class group of a surface with acollection of framed colored points. The Witten-Reshetikhin-Turaev TQFTs associ-ated to SU(2) and SO(3) induce linear representations of this group. We show thatthe denominators of matrices which describe these representations over a cyclotomicfield can be restricted in many cases. In this way, we give a proof of the known resultthat if the surface is a torus with no colored points, the representations have finiteimage. Recall that an object in a cobordism category of dimension 2+1 is a closedoriented surfaces Σ perhaps with some specified further structure. A morphismMfrom Σ to Σ ′ is (loosely speaking) a compact oriented 3-manifold perhaps withsome specified further structure, called a cobordism, whose boundary is the disjointunion of −Σ and Σ ′ .A morphism M ′ from Σ to Σ ′′ is composed with a morphismfrom Σ to Σ ′ by gluing along Σ ,inducing any required extra structure from thestructures on Mand M ′

Integral topological quantum field theory for a one-holed torus

We define the 2-signatures, 2-nullities and Arf invariants (when possible) for links which are null-homologous modulo two in a rational homology three-sphere. We define these invariants using the Goeritz form on non-oriented spanning surfaces. We develop their cobordism properties from this point of view. We give a good way to index these invariants. We also define d-signatures and d-nullities for links which are null-homologous modulo d in a rational homology sphere from the point of view of branched covers. We index d-signatures and d-nullities and develop their cobordism properties. Finally we define Arf invariants (when possible) in a general closed 3-manifold using spin structures.