The unifying theme of this book is the Eckmann-Hilton duality theory, not to be found as the motif of any other text. Since many topics occur in dual pairs, this provides motivation for the ideas and reduces the amount of repetitious material. This carefully written text moves at a gentle pace, even with fairly advanced material. In addition, there is a wealth of illustrations and exercises. The more difficult exercises are starred, and hints to them are given at the end of the book.

Key topics include:

*basic homotopy

*H-Spaces and Co-H-Spaces;

*cofibrations and fibrations;

*exact sequences;

*applications of exactness;

*homotopy pushouts and pullbacks and

the classical theorems of homotopy theory;

*homotopy and homology decompositions;

*homotopy sets; and

*obstruction theory.

The book is written as a text for a second course in algebraic topology, for a topics seminar in homotopy theory, or for self instruction.

(see [39, pp. 318–319]). Therefore to ensure the existence of homotopy groups with coefficients for any abelian group G, they have been defined in terms of Moore spaces. We carry this discussion of co-Moore spaces a bit further. For a finitely generated abelian group G, write G ✏ F ❵ T, where F is a free-abelian group and T is a finite abelian group. If C ♣G, nq denotes a co-Moore space of type ♣G, nq, then a simple calculation of cohomology shows that M ♣F, nq ❴ M ♣T, n ✁ 1q is a C ♣G, nq. In

homeomorphism. ❭❬ This shows that the cofibration property and the based homotopy extension property are essentially the same. More precisely, if f : X Ñ Y is a cofiber map, then the inclusion of f ♣X q, the image of f, in Y has the based homotopy extension property and X ✕ f ♣X q. We then obtain the following consequence of Proposition 1.5.18. f / X q / Q is a cofiber sequence and A is conProposition 3.2.7 If A tractible, then q is a homotopy equivalence. Next we consider pushouts. Definition

O♣nq Ñ O♣nq④O♣n ✁ k q has a local section at e. By Lemma 3.4.10, it suffices to show that q : Vn,n Ñ Vk,n has a local section at en✁k 1 , . . . , en . Lemma 3.4.11 The map q : Vn,n Ñ Vk,n that assigns to an n-frame the last k vectors has a local section at en✁k 1 , . . . , en . Proof. We seek a continuous function s that assigns to a k-frame v1 , . . . , vk in Rn an n-frame u1 , . . . , un✁k , v1 , . . . , vk in Rn . Let Mn,l be the space of n ✂ l matrices with real entries. By considering the

familiarity with the tensor product A ❜ B and the group of homomorphisms Hom♣A, B q of two abelian groups A and B. In addition, we refer to basic facts about the torsion product Tor♣A, B q ✏ A ✝ B and the group of extensions Ext♣A, B q, that can be found in Appendix C. We begin by noting the following extension of Proposition 2.5.9 which is an immediate consequence of Lemma 2.5.13. Lemma 5.2.1 If G and H are abelian groups and f, g : K ♣G, nq Ñ K ♣H, nq are two maps, then f✝ ✏ g✝ : πn ♣K ♣G, nqq

✂ ΩS d♣n 1q✁1 in each of these three cases. In addition, the result holds for the fibration S7 / S 15 / S8 obtained from the Cayley numbers (see Example 3.4.5), and so ΩS 8 ΩS 15 . ✔ S7 ✂ i / E p / B is a fiber sequence with i ✔ ✝, then Remark 5.4.9 If F by Corollary 5.4.5, ΩB ✔ F ✂ ΩE, and F is an H-space. Thus ΩB and F ✂ ΩE are H-spaces, and it is reasonable to ask if there is a homotopy equivalence between them which is an H-map (such an equivalence is called an H-equivalence). For the