Analytic Novikov for topologists

Abstract

Abstract. We explain for topologists the "dictionary" for understanding the analytic proofs of the Novikov conjecture, and how they relate to the surgery-theoretic proofs. In particular, we try to explain the following points:(1) Why do the analytic proofs of the Novikov conjecture require the introduction of C * -algebras? (2) Why do the analytic proofs of the Novikov conjecture all use Ktheory instead of L-theory? Aren't they computing the wrong thing? (3) How can one show that the index map µ or β studied by operator theorists matches up with the assembly map in surgery theory? (4) Where does "bounded surgery theory" appear in the analytic proofs?Can one find a correspondence between the sorts of arguments used by analysts and the controlled surgery arguments used by topologists?The literature on the Novikov conjecture (see [FRR]) consists of several different kinds of papers. Most of these fall into two classes: those based on topological arguments, usually involving surgery theory, and those based on analytic arguments, usually involving index theory. The purpose of this note is to "explain" the second class of papers to those familiar with the first class. I do not intend here to give a detailed sketch of the Kasparov KK-approach to the Novikov conjecture (for which the key details appear in [HsR]. Rather, I intend to concentrate on explaining the "dictionary" for relating the two main classes of papers on the Novikov conjecture, the topological and the analytic, and on trying

Higher ρ -invariants and the surgery structure set

“…e * B ∞ + · · · and an index theoretic map sign L : L n+1 (Zπ 1 (N )) → K * (C * π 1 (N )) ⊗ Z[ 1 2 ] and show that they are compatible with each other. Conjecturally, this latter map agrees (possibly up to a constant factor) with the wellknown map defined using quadratic forms [24].…”

Higher ρ -invariants and the surgery structure set

“…e * B ∞ + · · · and an index theoretic map sign L : L n+1 (Zπ 1 (N )) → K * (C * π 1 (N )) ⊗ Z[ 1 2 ] and show that they are compatible with each other. Conjecturally, this latter map agrees (possibly up to a constant factor) with the wellknown map defined using quadratic forms [24].…”

The Local Index Formula for SUq(2)

“…The aforementioned applications based on the spectral localizer suggests that one should look for invertible, or non-degenerate elements instead. In fact, this is nicely aligned with the elegant description of C * -algebraic K-theory K 0 (A) as the Witt group of hermitian forms over A (see [25,39]). Indeed, these are given by invertible self-adjoint elements in M n (A).…”

Assembly maps

“……”

Higher ρ -invariants and the surgery structure set

“…e * B ∞ + · · · and an index theoretic map sign L : L n+1 (Zπ 1 (N )) → K * (C * π 1 (N )) ⊗ Z[ 1 2 ] and show that they are compatible with each other. Conjecturally, this latter map agrees (possibly up to a constant factor) with the wellknown map defined using quadratic forms [24].…”

The Local Index Formula for SUq(2)

“…The aforementioned applications based on the spectral localizer suggests that one should look for invertible, or non-degenerate elements instead. In fact, this is nicely aligned with the elegant description of C * -algebraic K-theory K 0 (A) as the Witt group of hermitian forms over A (see [25,39]). Indeed, these are given by invertible self-adjoint elements in M n (A).…”

Assembly maps

“……”

Higher ρ -invariants and the surgery structure set

“…e * B ∞ + · · · and an index theoretic map sign L : L n+1 (Zπ 1 (N )) → K * (C * π 1 (N )) ⊗ Z[ 1 2 ] and show that they are compatible with each other. Conjecturally, this latter map agrees (possibly up to a constant factor) with the wellknown map defined using quadratic forms [24].…”

The Local Index Formula for SUq(2)

“…The aforementioned applications based on the spectral localizer suggests that one should look for invertible, or non-degenerate elements instead. In fact, this is nicely aligned with the elegant description of C * -algebraic K-theory K 0 (A) as the Witt group of hermitian forms over A (see [25,39]). Indeed, these are given by invertible self-adjoint elements in M n (A).…”

Assembly maps

“……”