Get Novikov Conjectures, Index Theorems, and Rigidity PDF

Algebraic Geometry

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, with reference to `Novikov Conjectures, Index Theorems and Rigidity'. they're meant to offer a photograph of the prestige of labor at the Novikov Conjecture and similar issues from many issues of view: geometric topology, homotopy idea, algebra, geometry, research.

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Let F0 (X)op denote the full subcategory of F(X)op consisting of those finite sets which intersect each Γ–orbit of X in at most one element. F0 (X)op is closed under the Γ–action, and we obtain an equivariant restriction map X op holim ΦX A → holim ΦA | F0 (X) . 24 The construction X → holim ΦX A | F0 (X) ←− F0 (X)op rial for proper equivariant maps of free Γ–sets. The induced map on fixed point sets Γ Γ   op   →  holim ΦX  holim ΦX A | F0 (X) A ←− F0 (X)op ←− F (X)op of the above mentioned restriction map is a natural weak equivalence of functors from the category of free Γ–sets and proper equivariant maps to S.

This gives the isomorphism stated in the theorem. The naturality of the isomorphism follows from the naturality of the spectral sequence with respect to maps in S CAT . D. 0 We now extend the definition of h f (−; W ) to simplicial sets, and finally to spaces. 5. By a locally finite simplicial set, we mean a simplicial object in setsp . If X. is a locally finite simplicial set, then applying h f (−; W ) gives a simplicial spectrum for any spectrum W . We define the simplicial locally finite homology of X.

An object in (a, b) ↓ ρ · i consists of an object n[P1 , . . , Pn ] of P and a morphism (U, V, ϕ, Ψ) : (a, b) → ⊕n−1 i=1 Pi , Pn in T . Consider the full subcategory of (a, b) ↓ ρ · i consisting of those objects for which the reference map is an isomorphism. ((a, b) ↓ ρ · i) is equivalent to the nerve of this subcategory. But the subcategory has the terminal object (2[a, b], Id : (a, b) → ρ · i(2[a, b])), hence we have the result. D. We will also need to understand the symmetric monoidal category–theoretic versions of mapping telescopes of spectra.

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