Combinatorial Foundation of Homology and Homotopy: by Hans-Joachim Baues

By Hans-Joachim Baues

This ebook considers deep and classical result of homotopy concept just like the homological Whitehead theorem, the Hurewicz theorem, the finiteness obstruction theorem of Wall, the theorems on Whitehead torsion and easy homotopy equivalences, and characterizes axiomatically the assumptions less than which such effects carry. This ends up in a brand new combinatorial origin of homology and homotopy. various specific examples and purposes in numerous fields of topology and algebra are given.

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Additional info for Combinatorial Foundation of Homology and Homotopy: Applications to Spaces, Diagrams, Transformation Groups, Compactifications, Differential Algebras, Algebraic Theories, Simplicial Objects, and Resolutions

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Here Xo is the coproduct of D and a free A-set and X n + 1 is obtained from Xn by attaching (71 + I)-cells for n 2 o. Let X = lim(X::,:o) be the direct limit of the sequence. We say that (X, D) is reduced if Xo = D and that (X, D) is normalized if the attaching maps in : Zn X sn-1 ---+ Xn- 1 (1) carry Z" x * to X o, 71 2 1. Here the free A-set Zn is called the set of n-cells of the A-CW-complex (X, D). We point out that for a space U in Top and an A-space Y we have 2 Homotopy Theory of Diagrams of Spaces A Top(A( -, a) xU, Y) = Top(U, Y(a)).

10) Here the functor II carries a space U to the fundamental groupoid of U. We use the A-groupoid II X to define the following category fA II X which we call the integrated fundamental groupoid of the A-space X (compare § 2 in MoerdijkSvenson [D]). 10) into one large category. The objects are pairs (a,x) where a E Ob(A) and x E X(a) = Ob(II X)(a). An arrow (a, x) ---+ (a', x') between such objects is a pair (a, t) where a : a ---+ a' is an arrow in A and t : x ---+ X(a)(x' ) E X(a) is an arrow in II X (a).

Let D be a G-space and let A be a set of functions a : Za ~ D in GTop where Za is a G-orbit set. We say that a function a : Z ~ D in GTop is A-finite if {31, ... , {3k E A together with a commutative diagram 38 Chapter A: Examples and Applications in Topological Categories in GTop are given where Xc> is an isomorphism. 14) are A-finite and (X, D) is finite dimensional. Now let (X, D) and (Y, D) be normalized reduced relative G-CW-complexes. 24) and a G-homotopy H : gf ~ 1 reID. The domination has dimension::::; n if dim(X, D) ::::; n and the domination is A-finite if (X, D) is A-finite.

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