Download Geometric Topology: Localization, Periodicity and Galois by Dennis P. Sullivan, Andrew Ranicki PDF

By Dennis P. Sullivan, Andrew Ranicki

The seminal `MIT notes' of Dennis Sullivan have been issued in June 1970 and have been largely circulated on the time, yet simply privately. The notes had an enormous impression at the improvement of either algebraic and geometric topology, pioneering the localization and crowning glory of areas in homotopy conception, together with P-local, profinite and rational homotopy thought, the Galois motion on gentle manifold buildings in profinite homotopy conception, and the K-theory orientation of PL manifolds and bundles. this can be the 1st time that this significant paintings has truly been released, and made on hand to a person attracted to topology.

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Geometric Topology: Localization, Periodicity and Galois Symmetry: The 1970 Mit Notes

The seminal `MIT notes' of Dennis Sullivan have been issued in June 1970 and have been generally circulated on the time, yet merely privately. The notes had an immense impact at the improvement of either algebraic and geometric topology, pioneering the localization and of entirety of areas in homotopy thought, together with P-local, profinite and rational homotopy concept, the Galois motion on delicate manifold constructions in profinite homotopy thought, and the K-theory orientation of PL manifolds and bundles.

Groups with the Haagerup Property: Gromov’s a-T-menability

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1 profinite -completion: G = {finite -quotients of G} lim ← − “subgroups of index ” ¯ =G⊗Z = formal -completion: G dir lim − −−− → {H } . finitely generated subgroups H ⊂ G Let G be an Abelian group and let be a non-void set of primes. Proposition There is a natural commutative diagram fffP G ≡ G ⊗ Z fffff f f f f ff fffff fffff f f f f  ff fffff G G ¯ ≡G⊗Z G fˆˆˆˆˆˆˆˆ ˆˆˆˆˆ ˆˆˆˆˆ ˆˆˆˆˆ ˆˆˆˆˆ ˆˆˆˆˆ  ˆˆˆD G ≡ lim finite -quotients ← of G Proof: First observe there is a natural diagram GG €€€ €€€ €€ c profinite completion €€€€  €9 localization G €€€ G 25 Algebraic Constructions For taking direct limits over G  G/H Hα · · ∼ = prime to using GG  G G/H Hα , Hα of finite -index in G implies G maps canonically to each finite -quotient of G.

We discuss the reassembly of the classical homotopy types from these pieces using an Adele type and a homotopy analogue of the “arithmetic square” of Chapter 1. Construction of the Profinite Completion X We outline the construction. We begin with the following observation. Let F denote a space with finite homotopy groups. Then the functor defined by F , [ ,F] may be topologized in a natural way. This (compact) topology arises from the equivalence ∼ = → [Y, F ] − lim ← [Y Yα , F ] finite subcomplexes Yα and is characterized by the separation property – Hausdorff.

E. ) The decomposition ∗ ∗ Zp Z = p tells one how AQ is related to the fields ApQ = {Q with all pα roots of unity adjoined} , ∗ for Gal (ApQ , Q) = Zp , the group of units in the ring of p-adic integers. 19 Algebraic Constructions iii) If k = Q and K = Q, an algebraic closure of Q, then G = “the Galois group of Q” = Gal (Q/ Q) = Gal (K/ Q) lim ← Galois number fields K is a profinite group of great importance. G has very little torsion, only “complex conjugations”, elements of order 2. These are all conjugate, and each one commutes with no element besides itself and the identity.

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