Conjugacy classes of affine automorphisms of K^n and linear automorphisms of P^n in the Cremona groups

We describe the conjugacy classes of affine automorphisms in the group Aut(n,\K) (respectively Bir(\K^n)) of automorphisms (respectively of birational maps) of \K^n. From this we deduce also the classification of conjugacy classes of automorphisms of \Pn in the Cremona group Bir(\K^n).Comment: 17 pages, no figure

The number of conjugacy classes of elements of the Cremona group of some given finite order

This note presents the study of the conjugacy classes of elements of some given finite order n in the Cremona group of the plane. In particular, it is shown that the number of conjugacy classes is infinite if n is even, n=3 or n=5, and that it is equal to 3 (respectively 9) if n=9 (respectively 15), and is exactly 1 for all remaining odd orders. Some precise representative elements of the classes are given.Comment: 14 page

On realizing diagrams of Pi-algebras

Given a diagram of Pi-algebras (graded groups equipped with an action of the primary homotopy operations), we ask whether it can be realized as the homotopy groups of a diagram of spaces. The answer given here is in the form of an obstruction theory, of somewhat wider application, formulated in terms of generalized Pi-algebras. This extends a program begun in [J. Pure Appl. Alg. 103 (1995) 167-188] and [Topology 43 (2004) 857-892] to study the realization of a single Pi-algebra. In particular, we explicitly analyze the simple case of a single map, and provide a detailed example, illustrating the connections to higher homotopy operations.Comment: This is the version published by Algebraic & Geometric Topology on 21 June 200

On the inertia group of elliptic curves in the Cremona group of the plane

We study the group of birational transformations of the plane that fix (each point of) a curve of geometric genus 1. A precise description of the finite elements is given; it is shown in particular that the order is at most 6, and that if the group contains a non-trivial torsion, the fixed curve is the image of a smooth cubic by a birational transformation of the plane. We show that for a smooth cubic, the group is generated by its elements of degree 3, and prove that it contains a free product of Z/2Z, indexed by the points of the curve.Comment: 14 pages, no figur

Simple relations in the Cremona group

We give a simple set of generators and relations for the Cremona group of the plane. Namely, we show that the Cremona group is the amalgamated product of the de Jonqui\`eres group with the group of automorphisms of the plane, divided by one relation which is $\sigma\tau=\tau\sigma$, where $\tau=(x:y:z)\mapsto (y:x:z)$ and \sigma=(x:y:z)\dasharrow (yz:xz:xy)

Algebraic invariants for homotopy types

We define inductively a sequence of purely algebraic invariants - namely, classes in the Quillen cohomology of the Pi-algebra \pi_* X - for distinguishing between different homotopy types of spaces. Another sequence of such cohomology classes allows one to decide whether a given abstract Pi-algebra can be realized as the homotopy Pi-algebra of a space in the first place. The paper is written for a relatively general "resolution model category", so it also applies, for example, to rational homotopy types