Geometry and Topology by James C. Alexander, John L. Harer

By James C. Alexander, John L. Harer

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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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