By Hans-Joachim Baues

ISBN-10: 3642084478

ISBN-13: 9783642084478

ISBN-10: 3662113384

ISBN-13: 9783662113387

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

**Read or Download Combinatorial Foundation of Homology and Homotopy: Applications to Spaces, Diagrams, Transformation Groups, Compactifications, Differential Algebras, Algebraic Theories, Simplicial Objects, and Resolutions PDF**

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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**

**Sample text**

The "partial suspension" shows that the category mod(ox) does not depend on the choice of n with n ~ 2. Therefore we omit n in the notation and we write S~ V X 2 = So V X 2 E mod(8x ) for an object in the additive category mod(ox). 14). Each map u : ax ---. f)y induces a functor u* : mod(ox) ---. mod(f)y) which carries S:;. V X 2 to S:;. 14) to the map Now we choose a set A of elements a where a : Zo ---. D is an oo-map defined on an oo-set Zo. Then the enveloping functor U,A: ooCoef(D) ---. 16) carries ax to the full subcategory of mod( ax) consisting of objects S:;.

17) Definition. Let D be a G-space and let (X, D) be a relative G-CW-complex which is normalized and reduced. : 1 where Zn is the G-orbit set of n-cells. Let Then there is a well defined chain complex { C*(X,D) in mOd( r H) = 0 for n ::; o. : 1 Cn(X, D) = Lan (H) and Cn(X, D) with JOr(G) If D is a G-orbit set then we define ao: Zo =D by the identity of D. : 0 JOr(G) with (2) 36 Chapter A: Examples and Applications in Topological Categories and CnX = 0 for n < 0 is defined. 25). 5). We get C*(X, D) by the general procedure in (V, § 2).

We get C*(X, D) by the general procedure in (V, § 2). The augmentation functor aug used in (2) is described in (II, § 6). 18) Here Hom is defined by the abelian category Mod(for(G) H) with H = II(XO, DO) = II (X2 , DO). This is a twisted version of the cohomology of Bredon [Ee]; see Moerdijk-Svenson [D] where this cohomology is studied if G is discrete. 30»). We now are ready to formulate the homological Whitehead theorem for G-spaces. 20) Theorem. Let D be a G-space and let f : (X, D) ---+ (Y, D) be a cellular map between normalized reduced relative G -CW-complexes in (GTop)?

### Combinatorial Foundation of Homology and Homotopy: Applications to Spaces, Diagrams, Transformation Groups, Compactifications, Differential Algebras, Algebraic Theories, Simplicial Objects, and Resolutions by Hans-Joachim Baues

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