A generalization of Einstein's gravitational theory is discussed in which the spin of matter as well as its mass plays a dynamical role. The spin of matter couples to a non-Riemannian structure in space-time, Cartan's torsion tensor. The theory which emerges from taking this coupling into account, the U4 theory of gravitation, predicts, in addition to the usual infinite-rhnge gravitational interaction mediated by the metric field, a new, very weak, spin contact interaction of gravitatiorial origin. %'e summarize here all the available theoretical evidence that argues for admitting spin and torsion into a relativistic gravitational theory. Not least among this evidence is the demonstration that the U4 theory arises as a local gauge theory for the Poincare group in space-time. The deviations of the U" theory from standard general relativity are estimated, and the prospects for further theoretical development are assessed. CONTENTS

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In .order to t.ake fu~1 account of spin in general relativity, it is necessary to consider space-time as a !'letnc .space Wlt~ torSIOn, as was shown else.where. We tr~t a Dirac particle in such a space. The generalIzed Dirac equatIOn turns out to be of a Heisenberg-Pauli type. The nonlinear terms induced by torsion express a universal spin-spin interaction of range zero.

The Poincaré (inhomogeneous Lorentz) group underlies special relativity. In these lectures a consistent formalism is developed allowing an appropriate gauging of the Poincaré group. The physical laws are formulated in terms of points, orthonormal tetrad frames , and components of the matter fields with respect to these frames. The laws are postulated to be gauge invariant under local Poincaré transformations. This implies the existence of 4 translational gauge potentials e α ("gravitons") and 6 Lorentz gauge potentials Γ αβ ("rotons") and the coupling of the momentum current and the spin current of matter to these potentials, respectively. In this way one is led to a Riemann-Cartan spacetime carrying torsion and curvature, richer in structure than the spacetime of general relativity. The Riemann-Cartan spacetime is controlled by the two general gauge field equations (3.44) and (3.45), in which material momentum and spin act as sources. The general framework of the theory is summarized in a table in Section 3.6. -Options for picking a gauge field lagrangian are discussed (teleparallelism, ECSK). We propose a lagrangian quadratic in torsion and curvature governing the propagation of gravitons and rotons. A suppression of the rotons leads back to general relativity.Author's note. This work was originally published as a chapter in the book that is now long out of print. § The purpose of this arXiv version is to make these lectures more accessible to the current generation of students and researchers. I am extremely grateful to my colleague and friend Milutin Blagojević (Belgrade) for arranging the republication of my Erice lectures. Moreover, I'd like to thank his secretary Vanja Mihajlović for putting the text most carefully into latex. For a more modern look at that subject, see M.B. & F.W.H. (eds.), Gauge Theories of Gravitation, Imperial College Press, London (2013).

Recently, the study of three-dimensional spaces is becoming of great interest. In these dimensions the Cotton tensor is prominent as the substitute for the Weyl tensor. It is conformally invariant and its vanishing is equivalent to conformal flatness. However, the Cotton tensor arises in the context of the Bianchi identities and is present in any dimension n. We present a systematic derivation of the Cotton tensor. We perform its irreducible decomposition and determine its number of independent components as n(n 2 − 4)/3 for the first time. Subsequently, we exhibit its characteristic properties and perform a classification of the Cotton tensor in three dimensions. We investigate some solutions of Einstein's field equations in three dimensions and of the topologically massive gravity model of Deser, Jackiw, and Templeton. For each class examples are given. Finally we investigate the relation between the Cotton tensor and the energy-momentum in Einstein's theory and derive a conformally flat perfect fluid solution of Einstein's field equations in three dimensions. file cott16.tex,Recently, the study of three-dimensional spaces is becoming of great interest; for these spaces the Weyl tensor is always zero and the vanishing of the Cotton tensor depends on the type of relation between the Ricci tensor and the energy-momentum tensor of matter.Any three-dimensional space is conformally flat if the Cotton tensor vanishes. If matter is present, the Ricci tensor is related to the energy-momentum tensor of matter by means of the Einstein equations. Then the vanishing of the Cotton tensor imposes severe restrictions on the energy-momentum tensor. The Cotton tensor also plays a role in the context of the Hamiltonian formulation of general relativity, see [10].The outline of the article is as follows. First we derive the Cotton 2-form in the context of the Bianchi identities. Subsequently we describe its characteristic properties and perform