AGNT Ical Atom

In the talk, I will discuss the Turner-Turaev formalism for unoriented Topological Quantum Field Theory (TQFT). Building upon this formalism, I will introduce an analogous version for (d+1)-dimensional TQFT for pro-p Poincare duality groups. In the case of d = 1, this enables us to study cobordisms and TQFTs for both the maximal pro-p quotient of absolute Galois groups of p-adic fields and pi_1(X)^p, the pro-p completions of fundamental groups of surfaces. This generalisation gives a framework for arithmetic TQFTs and strengthens the analogies within arithmetic topology, which relates p-adic fields to surfaces (oriented mod p^r). I will explain the classification of TQFTS for the (1+1)-dimensional case, in terms of Frobenius algebras with some extra structure.

If time permits, I will explain how we define a Dijkgraaf Witten like theory, to get formulas for counting G-covers of X, where X is either a surface, or a p adic field, and G is a p-group (these formulas are similar to the ones given by Mednykh for surfaces using TQFTs, and by Masakazu Yamagishi using a more algebraic approach). I will also try to outline how we plan to also get similar formulas for Hom(\pi_1(X)^p,G), where G=GL_n(k) for k=F_{p^r} or Z/p^rZ.

The talk is based on joint work with Oren Ben-Bassat.

For a number field K admitting a real embedding, it is impossible to construct a functorial in G group structure in the Galois cohomology pointed set H^1(K,G) for all connected reductive K-groups G. However, over an arbitrary number field K, we define a diamond (or power) operation of raising to power n (x,n) \mapsto x^{\Diamond n}: H^1(K,G) \times Z -> H^1(K,G). We show that this operation has many functorial properties. When G is a torus, the set H^1(K,G) has a natural group structure, and x^{\Diamond n} coincides with the n-th power of x in this group.

For a cohomology class x in H^1(K,G), we define the period per(x) to be the greatest common divisor of n>0 such that x^{\Diamond n}=1, and the index ind(x) to be the greatest common divisor of the degrees [L:K] of finite separable extensions L/K splitting x. These period and index generalize the period and index of a central simple algebra over K (in the special case where G is the projective linear group PGL_n, the elements of H^1(K, G) can be represented by central simple algebras). For an arbitrary reductive group G defined over a local or global field K, we show that per(x) divides ind(x), that per(x) and ind(x) have the same prime factors, but the equality per(x)=ind(x) may not hold.

The talk is based on a joint work with Zinovy Reichstein.

Please note the special time

tbd