Homology of spaces of knots in any dimensions

Abstract

I shall describe the recent progress in the study of cohomology rings of spaces of knots in Rn, H*({knots in Rn}), with arbitrary n ⩾ 3. ‘Any dimensions’ in the title can be read as dimensions n of spaces Rn, as dimensions i of the cohomology groups Hi, and also as a parameter for different generalizations of the notion of a knot. An important subproblem is the study of knot invariants. In our context, they appear as zero–dimensional cohomology classes of the space of knots in R3. It turns out that our more general problem is never less beautiful. In particular, nice algebraic structures arising in the related homological calculations have equally (or maybe even more) compact description, of which the classical ‘zero–dimensional’ part can be obtained by easy factorization. There are many good expositions of the theory of related knot invariants. Therefore, I shall deal almost completely with results in higher (or arbitrary) dimensions.