Zdzisław Wojtkowiak

Cosimplicial Objects in Algebraic Geometry

Let X be an arc-connected and locally arc-connected topological space and let I be the unit interval. Applying the connected component functor to each fibre of the fibration of the total space map(I, X) over X × X, P(w) = (w(0), w(1)), we get a local system of sets (Poincare groupoid) over X × X. This construction does not have a straightforward generalization to algebraic varieties over any field. Using cosimplicial objects, we propose a generalization for smooth, algebraic varieties over an arbitrary field of characteristic zero. This leads to a definition of an algebraic fundamental group of De Rham type. We partly calculate the Betti lattice in the algebraic fundamental group for the projective line minus three points.

ON

We continue to study l-adic iterated integrals introduced in the first part. We shall show that the l-adic iterated integrals satisfy essentially the same functional equations as the classical complex iterated integrals. Next we are studying l-adic analogs of classical polylogarithms.

l

We investigate maps between p-completed classifying spaces of compact connected Lie groups. Let G and G′ be two connected compact Lie groups. For a space X, let Xp be a p-completion of X. If p does not divide the order of the Weyl group of G, we give descriptions of the set of homotopy classes [(BG)p, (BG′)p] in terms of K-theory and in terms of “admissible” maps of Adams and Mahmud.

-adic iterated integrals, II. Functional equations and

Polylogarithms are special cases of more general iterated integrals. One can hope that the known results about functional equations of polylogarithms hold also for more general iterated integrals. In fact in [5] we have proved some general results about functional equations of iterated integrals on P ( ) minus several points. In this paper we generalize our results from [5] to functional equations of iterated integrals on any smooth, quasiprojective algebraic variety. Our principal tool is the universal unipotent connection with logarithmic singularities. First we prove our results for a complement of a divisor with normal crossings in a smooth, projective variety. Next, using results of Hironaka about resolution of singularities we extend our results to smooth, quasi-projective varieties. The proofs (for a complement of a divisor with normal crossings) are straightforward generalizations of methods from [5]. These results are in the first three sections of this paper. In the fourth section of the paper we are dealing with the dilogarithm. It is well known that any functional equation of the logarithm on P ( ) can be obtained by successive applications of the functional equation

l

Let \({z}\;\in \;\mathbb{Q}\) and let γ be an l-adic path on \(\mathbb{P}\frac{1}{\mathbb{Q}}\backslash \{0,1,\infty\}\;\mathrm{from}\;\vec{01}\;\mathrm{to}\;z\). For any \(\sigma\in Gal(\bar{\mathbb{Q}}/\mathbb{Q})\), the element \(x^{-\kappa(\sigma)}f_{\gamma}(\sigma)\;\in\;\pi_{1}(\mathbb{P}\frac{1}{\mathbb{Q}}\backslash \{0,1,\infty\}, \vec{01})_{pro-l}\). After the embedding of π1 into \(\mathbb{Q}\{\{X,Y\}\}\) we get the formal power series \(\Delta_\gamma(\sigma)\;\in\;\mathbb{Q}\{\{X,Y\}\}\). We shall express coefficients of \(\Delta_\gamma(\sigma)\)as integrals over \((\mathbb{Z}_l)^r\) with respect to some measures \(K_r(z)\). The measures \(K_r(z)\) are constructed using the tower \((\mathbb{P}\frac{1}{\mathbb{Q}}\backslash(\{0,\infty\}\cup\mu_{l^n})_{n\in\mathbb{N}}\) of coverings of \(\mathbb{P}\frac{1}{\mathbb{Q}}\backslash\{0,1,\infty\}\). Using the integral formulas we shall show congruence relations between coefficients of the formal power series \(\Delta_\gamma(\sigma)\). The measures allow the construction of l-adic functions of non-Archimedean analysis, which however rest mysterious. Only in the special case of the measures \(K_1(\vec{10})\;\mathrm{and}\;K_1(-1)\)we recover the familiar Kubota–Leopoldt l-adic L-functions. We recover also l-adic analogues of Hurwitz zeta functions. Hence we get also l-adic analogues of L-series for Dirichlet characters.

-adic iterated polylogarithms

The purpose of this paper is to show equivalence of two criteria for functional equations of (complex and `-adic) iterated integrals, one given by D. Zagier in the case of polylogarithms which we generalize to arbitrary iterated integrals and the other given by the second named author. We establish a device for computing a functional equation from a family of morphisms on fundamental groups of varieties, and present some examples showing how our device works commonly both in complex and `-adic cases. Some of our `-adic examples already supply non-trivial arithmetic relations between “`-adic polylogarithmic characters” – functions on the absolute Galois group Gal(Q/Q) defined by Kummer properties along towers of certain arithmetic sequences – which were introduced in [NW] as generalization of the so-called Soulé characters studied by Ch. Soulé [S1], [S2]. Let V := P1 − {several points} be a punctured projective line defined over a subfield K of C. In [W0,W2,W5], the second named author gave conditions to have functional equations of iterated integrals on V in terms of induced morphisms on fundamental groups. In fact, in [W0], he formulated a complex iterated integral as the image of the (universal) unipotent period along a chain from x to z on V(C) by a 1-form on the Lie algebra of the pro-unipotent fundamental group of V . Also in [W5], introduced is an `-adic iterated integral using the action of the absolute Galois group Gal(K/K) on the torsor of paths from x

Maps between p-completed classifying spaces II

We show that two maps between classifying spaces of compact, connected Lie groups are homotopic after inverting the order of the Weyl group of the source if and only if they induce the same maps on rational cohomology. We shall also give some results on maps from classifying spaces of finite groups to classifying spaces of compact Lie groups. Among other things we construct a map from B(Z/2+Z/2+Z/3) into BSO(3) which is not induced by a homomorphism.

Functional equations of iterated integrals with regular singularities

We present in this note a definition of zeta function of the field \(\mathbb{Q}\) which incorporates all p-adic L-functions of Kubota-Leopoldt for all p and also so called Soule classes of the field \(\mathbb{Q}\). This zeta function is a measure, which we construct using the action of the absolute Galois group \(G_{\mathbb{Q}}\) on fundamental groups.

On l-adic Galois L-functions

In a series of papers we have introduced and studied l-adic polylogarithms and l-adic iterated integrals which are analogues of the classical complex polylogarithms and iterated integrals in l-adic Galois realizations. In this note we shall show that in the generic case l-adic iterated integrals are linearly independent over \({\mathbb{Q}}_{\mathcal{l}}\). In particular they are non trivial. This result can be viewed as analogous of the statement that the classical iterated integrals from 0 to z of sequences of one forms \(\frac{dz} {z}\) and \(\frac{dz} {z-1}\) are linearly independent over \(\mathbb{Q}\). We also study ramification properties of l-adic polylogarithms and the minimal quotient subgroup of the absolute Galois group G K of a number field K on which l-adic polylogarithms are defined. In the final sections of the paper we study l-adic sheaves and their relations with l-adic polylogarithms. We show that if an l-adic sheaf has the same monodromy representation as the classical complex polylogarithms then the action of G K in stalks is given by l-adic polylogarithms.

Non-abelian Fundamental Groups and Iwasawa Theory: Tensor and homotopy criteria for functional equations of ℓ-adic and classical iterated integrals

We begin with the following definitibn due to G . Mislin . Definition 1 . A CW complex X is of type FP if the singular chain complex of the universal cover of X is Z [ni (X))-chain homotopy equivalent to a complex of finite length 0 a Pn _*n-1 _* , . . . _# PO 4 0 where each P i is a finitely generated and projective Z(n1 (X))-module . If X is of type FP, the finiteness obstruction of C .T .C . Wall is defined by