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

7roX is surjective. It is easy to see that homology and cohomology of (X, D) does not depend on the choice of Y.

C,T) (V. , T) is a homological cofibration category if D is non empty. Here T is defined above. 2) This is the subcategory of (ATop)? consisting of normalized reduced relative A-CW-complexes (X, D) and cellular maps. c. 25). 1) (VIII. 3) (ATop)? 37). Here D is allowed to be empty. 5) For X in (ATop)? this is the push out of B x X ~ B x D -E:.. 37) (3). It will be convenient to have these examples in mind in order to visualize the abstract and categorical theory in the second part of the book below.

Here U : Xl -+ Y is the restriction of g. This is the infinity-analogue of a classical theorem of obstruction theory. , V X 2 , C*(X, D)) = Hn Hom(Sa V X 2 ) C*X) Hn(X, D) (S·. V X 2 ) Hn(X)(Sa V X 2 ) 4 Homotopy Theory Controlled at Infinity 47 Here Hom denotes the set of morphisms in mod(ax). 30). We use them for the following homological Whitehead theorem which is a special case of (VI, § 7). 24) Theorem. Let f : (X, D) -+ (Y, D) be a cellular map between normalized reduced relative oo-CW-complexes in (ooEnd)p.

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