Handbook of Algebraic Topology by I.M. James

By I.M. James

Algebraic topology (also often called homotopy conception) is a flourishing department of recent arithmetic. it's very a lot a global topic and this is often mirrored within the heritage of the 36 major specialists who've contributed to the guide. Written for the reader who already has a grounding within the topic, the quantity includes 27 expository surveys overlaying the main energetic parts of study. they supply the researcher with an updated assessment of this intriguing department of arithmetic

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PROPOSITION. Let X be a simply connected CW-space. Then (A) and (B) hold, (A) The Hurewicz homomorphism hnX is split injectivefor all n if and only if X has the homotopy type of a product of Eilenberg-MacLane spaces. (B) Moreover hnX is split surjective for all n if and only if X has the homotopy type of a one point union of Moore spaces. Properties (A) and (B) form a further nice example of if 7r-duality. 17). Then the collection {/n} defines a map PROOF. be a map with f:X-^l[K{7rnX,n) which is a homotopy equivalence by the Whitehead theorem.

4) up to homotopy equivalence. For this compare Kan [57] who proved that GX —• AX induces the Hurewicz homomorphism. C. 5) TT^X^) where the homomorphism is induced by the inclusion X^~^ C X^. 7. ^J. 5). 6. PROPOSITION. Let X be a simply connected CW-complex. Then there are natural isomorphisms (a) FnX = iTnrX. (h)rn{AX) = 7rn{Arx), (C) HnX = TTnSPoc^X, {d)Hn{A,X) = irn{A,SPooX), The isomorphisms which we shall use as identifications are compatible with A — fi exact sequences above. Here (a) and (c) are due to Kan [57] and Dold and Thorn [30], respectively.

We associate with an A^-system S the exact T-sequence H3 - ^ G{ri) -> 7r2 -> if2 - ^ Ho^Z/l ^ TTI ^ if 1 -^ 0. (12) Here H\ = cok{rj) is the cokernel of rj and the extension 00^(63) >-> 7r2 ->• ker{b2) is obtained by the element /3, that is, the group n2 is given by the extension element /Jf € Ext{ker{b2),cok{b7)) defined by p^ = A'\un0). Here j : ker{b2) C /f2 is the inclusion. 11. CLASSinCATlON THEOREM. For n ^ 4 there exist detecting functors A',X' for which the following diagram offunctors commutes up to natural isomorphism spacesl^ • A^-Systems types^ • A^-systems Section 10 Homotopy types 51 Moreover for S = A'{X)^ X E spaces^y the F-sequence of S describes part of the r-sequence of X, that is HQ = HnX and H3- - ^ G{v) • ^H2 -^//o®Z/2- Hi -TTn i + l-X" ^^ Hn+lX In addition G(i4,77) = Tn+i (A, X).

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