Volumes of the German-Ianguage subseries 'Äspekte der Mathematik" are listed at the end of the book.

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Abstract. We use bordism theory to construct periodic cohomology theories, which we call elliptic cohomology, for which the cohomology of a point is a ring of modular functions. These are complex-oriented multiplicative cohomology theories, with formal groups associated to the universal elliptic genus studied by a number of authors ([CC,LS,O,W1,Z]). We are unable to find a geometric description for these theories.

For real projective spaces, (a) the Euclidean immersion dimension, (b) the existence of axial maps and (c) the topological complexity are known to be three facets of the same problem. But when it comes to embedding dimension, the classical work of Berrick, Feder and Gitler leaves a small indeterminacy when trying to identify the existence of Euclidean embeddings of these manifolds with the existence of symmetric axial maps. As an alternative we show that the symmetrized version of (c) captures, in a sharp way, the embedding problem. Extensions to the case of even-torsion lens spaces and complex projective spaces are discussed. 55M30, 57R40Dedicated to the memory of Bob Stong Main resultThe Euclidean immersion and embedding questions for projective spaces were topics of intense research during the beginning of the second half of the last century. In the case of real projective spaces, the immersion problem has recently received a fresh push, partly in view of a surprising reformulation in terms of a basic concept arising in robotics, namely, the motion planning problem of mechanical systems. In more detail, Farber, Tabachnikov and Yuzvinsky [14] showed that for r ¤ 1; 3; 7, the immersion dimension of P r -the r -dimensional real projective space-agrees with TC.P r / 1, one unit less than the topological complexity of P r (see Definition 1.1 and Theorem 4.2 below). In this paper we accomplish a completely analogous goal by connecting the Euclidean embedding dimension of P r with Farber-Grant's notion of symmetric motion planning. Before stating our main results, we recall the relevant definitions. Definition 1.1 The topological complexity of a space X , TC.X /, is defined as the genus of the endpoints evaluation map evW P .X / ! X X , where P .X / is the free path space X OE0;1 with the compact-open topology.TC.X / is a homotopy invariant of X . Thinking of X as the space of configurations of a given mechanical system, TC.X / gives a measure of the topological instabilities in a motion planning algorithm for X -a perhaps discontinuous (but global) section of the map ev. We refer the reader to Farber [10] for a very useful survey of results in this area, and to the Farber's book [11] for a thorough introduction to the new mathematical discipline of topological robotics.We now come to the main definition (introduced and explored by Farber and Grant [12]). For a topological space X , let X be the diagonal in X X and ev 1 W P 1 .X / ! X X X be the restriction of the fibration ev in Definition 1.1. Thus P 1 .X / is the subspace of P .X / consisting of paths W OE0; 1 ! X with .0/ ¤ .1/. Note that ev 1 is a Z=2-equivariant map, where Z=2 acts freely on both P 1 .X / and X X X , by running a path backwards in the former, and by switching coordinates in the latter. Let P 2 .X / and B.X; 2/ denote the corresponding orbit spaces, and let ev 2 W P 2 .X / ! B.X; 2/ denote the resulting fibration.Definition 1.2 With the above conditions, the symmetric topological complexity of X , TC S .X /, is defined by TC S .X / D genus.ev 2...

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