Novikov conjectures, index theorems, and rigidity: by Steven C. Ferry, Andrew Ranicki, Jonathan M. Rosenberg

By Steven C. Ferry, Andrew Ranicki, Jonathan M. Rosenberg

The Novikov Conjecture is the one most crucial unsolved challenge within the topology of high-dimensional non-simply hooked up manifolds. those volumes are the outgrowth of a convention held on the Mathematisches Forschungsinstitut Oberwolfach (Germany) in September, 1993, near to `Novikov Conjectures, Index Theorems and Rigidity'. they're meant to provide a photo of the prestige of labor at the Novikov Conjecture and similar issues from many issues of view: geometric topology, homotopy concept, algebra, geometry, research.

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Extra info for Novikov conjectures, index theorems, and rigidity: Oberwolfach, 1993

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The covering {Bx ∩ U say Bx1 , . . , Bxs . Now s ¯ = ∅} ⊆ {σ | im(σ) ∩ f −1 U {σ | im(σ) ∩ Bxi = ∅} , i=1 and the right hand side is a finite set. This gives the result. D. 8. The construction X → h f (X; W ) is functorial for proper maps of locally compact topological spaces. Proof: Let f : X → Y be a proper map between locally compact spaces and let fk! : Lk X → Lk Y be the order preserving map A → f ◦ A. Then we have the natural map lim h f (A; W ) → lim h f (B; W ) −→ A∈Lk X −→ B∈Lk Y induced by fk!

11. This gives the result. D. One now defines the analogous theory b h f (−, A), for A any spectrum, to be the realization of the simplicial spectrum k → lim h f (A; A). One −→ A∈Bk has the following results about this theory, analogous to the results about h f (X; A). 20 (a) Let A = K (G, 0) be an Eilenberg–MacLane spectrum. Then there is a natural isomorphism πi (b h f (X; A)) ∼ = Hi (b Cˆ∗ (X; G)) compatible with the analogous isomorphism πi (h f (X; A)) ∼ = Hi (Cˆ∗ (X; G)) . (b) Let A → B → C be a fibre sequence of spectra up to homotopy.

K, k+1}. (Ns ×Σn ) Ψ can be written as lim k (Ns (k)× → Ns (k) Ψ Σn Pn , n s Σn ) Ψn . But (N (k) × Σn ) Ψn has nerve equivalent to since both have nerves weakly equivalent to E Σn ×Σn Ψ({k, k+1}) , as one easily checks using the hypothesis that Ψ({i, i + 1} → {i}) induces an equivalence on nerves for all i. D. We finally need to examine group actions on inverse limits. Let C be a category with an action by group Γ. We may view this action as a functor Γ → CAT , where Γ denotes Γ viewed as a category with one object.

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