By Dennis P. Sullivan, Andrew Ranicki

The seminal `MIT notes' of Dennis Sullivan have been issued in June 1970 and have been greatly circulated on the time, yet in basic terms privately. The notes had a huge effect at the improvement of either algebraic and geometric topology, pioneering the localization and finishing touch of areas in homotopy thought, together with P-local, profinite and rational homotopy concept, the Galois motion on delicate manifold constructions 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 really been released, and made on hand to a person drawn to topology.

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Of p-adic numbers where all but ﬁnitely many of the rp are actually p-adic integers. ¯ as the diagonal sequences Note that Q is contained in Q n/m → (n/m, n/m, . . , n/m, . . ) . If we combine this embedding with the embedding of Q in the reals we obtain an embedding ¯ × {real completion of Q} Q →Q as a discrete subgroup with a compact quotient. g. Q (ξ) = Q (x)/(xp − 1) and GL (n, Z). These Adele groups have natural measures, and the volumes of the corresponding compact quotients have interesting number theoretical signiﬁcance.

In spirit we follow Artin and Mazur1 , who ﬁrst conceived of the proﬁnite completion of a homotopy type as an inverse system of homotopy types with ﬁnite homotopy groups. We “complete” the Artin-Mazur object to obtain an actual homotopy type X for each connected CW complex X. This proﬁnite completion X has the additional structure of a natural compact topology on the functor, homotopy classes of maps into X, [ , X] . The compact open topology on the functor [ , X] allows us to make inverse limit constructions in homotopy theory which are normally impossible.

17 Algebraic Constructions ordered set of open subgroups of G of ﬁnite index. Then ∗ ∼ lim {G/H} = “continuous completion of G” . G = ← {H} It sometimes happens however that every subgroup of ﬁnite index in G is open. This is true if G = Z, in fact for the proﬁnite completion of any ﬁnitely generated Abelian group. Thus in these cases the topology of G can be recovered from the algebra using the isomorphism ∗. The topology is essential for example in ∞ ∞ Z/p = proﬁnite completion Z/p . Examples from Topology and Algebra Now we consider some interesting examples of “proﬁnite groups”.