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**

**Example text**

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.