Continuously controlled algebraic $K$ -theory of spaces and the Novikov conjecture

Abstract

Here the L-polynomial is a certain rational polynomial in the Pontrjagin classes. This theorem is surprising from many points of view: the left hand side is obviously a homotopy invariant and an integer, whereas the right hand side a priory is only a smooth invariant and a rational number. This led Novikov to the following conjecture: If M has fundamental group π and x is an element in H∗(Bπ), f the classifying map, could it be that the higher signatures, x ∪ L(M)[M ] are homotopy invariants of the manifold M? From this point of view Hirzebruch’s theorem is a verification of the Novikov conjecture for simply connected manifolds. Later Wall [17, section 17H] realized that the Novikov conjecture can be expressed using the assembly map h∗(Bπ;L(Z)) → L∗(Zπ). The Novikov conjecture is equivalent to the assembly map being a rational monomorphism. Over the years it has turned out that there are lots of assembly maps. In algebraic Ktheory, in C∗-theory and in A-theory among others. It has become common practice to call the statement that the assembly map is a rational monomorphism, the Novikov conjecture in that theory. In the case of C∗-theory, monicity of the assembly map implies, but is not equivalent to the classical Novikov conjecture. In this paper we treat the A-theory case. We wish to extend the results in [6] and [7] to A-theory, using a variation of the continuously controlled A-theory in [13, 15] to replace the continuously controlled K-theory in [1]. One of the main problems here is, that as a computational device, slightly discontinuous maps were allowed in [1] and [6], and Atheory does not respond nicely to that. The answer is to work with spaces that are so locally contractible, that the slightly discontinuous maps (eventually continuous maps) can be replaced by continuous maps. Otherwise the strategy is to follow [7, 6] and [13, 15], and we shall assume the reader has some familiarity with these papers. We prove the following theorem