By John R. Harper

Even supposing the idea and functions of secondary cohomology operations are a massive a part of a complicated graduate-level algebraic topology path, there are few books at the topic. The AMS fills that hole with the booklet of the current quantity. The author's major function during this publication is to advance the idea of secondary cohomology operations for singular cohomology idea, that's handled when it comes to straightforward buildings from normal homotopy conception. between many functions thought of are the Hopf invariant one theorem (for all primes $p$, together with $p = 2$), Browder's theorem on greater Bockstein operations, and cohomology thought of Massey-Peterson fibrations. various examples and workouts aid readers to achieve a operating wisdom of the speculation. A precis of extra complicated elements of the middle fabric is incorporated within the first bankruptcy. Prerequisite is uncomplicated algebraic topology, together with the Steenrod operations. The ebook is aimed at graduate scholars and study mathematicians drawn to algebraic topology and will be used for self-study or as a textbook for a sophisticated path at the subject. it's to be had in either hardcover and softcover versions

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The canonical map MSp —> MSU induces an isomorphism Image p^ = Image [//' : S —> MSU], and Image p! was computed by Anderson, Brown and Peterson [3]. 2, we identify elements of Image p which project to Image p^. $ I * > 0} • PROOF. Let x G 'Eoo be a nonzero element of positive degree. T. Then P2{X) G E2 is a representative of fioo(x) £ Eoo. Select a representative X £ 'E2 Recall that 'E2 = C o t o r ^ ^ ^ , Z2), £2 = Cotorof(//*(A^S;;; Z 2 ), Z 2 ) and p2 = Cotor(/**, 1). ^(X) has a representative in C(^»(Z2),2l*, Z2) C(Hm{MSp\ ^2), 21", ^2).

In Section 3, we generalize the First Reduction heorem of [13] to a general algebraic setting. Then we apply this theorem to H*MSp* to replace the cobar construction with a quotient 5 such that Hm$ = H*MSpm and £ is a torsion 2-group. In Section 4 we choose specific representatives for the key elements of MSpm. Then we determine a mumber of very important boundaries in £. These boundaries will determine the admissibility relations in Aj p , the analogue of the relations of A*. In Section 5, we generalize the Second Reduction Theorem of [13] to construct Aj p as a quotient of 5- We show that A*Sp splits into a direct sum of subcomplexes, one subcomplex for each group generator of the M5p*A/5p-primitive elements of MSpm.

PROOF. Since £ *f % MSp*, X(g> MSp*X. A') ^ Moreover, the composite of these isomorphisms is J - 1 . 3 For any spectrum X, define A' as the composite map A': MSpmX -£> £ ® M 5 p . A ' - ^ T MSpmX. 2] we defined an isomorphism between HmBP and the analogue of % in that context. That proof generalizes directly to our context to give the following isomorphism between HmMSp and %. 4 6 : HMSp -^ £®H+MSp - ^ % H*MSp i ^ > % Z{2) = % is an isomorphism of algebras and %-comodules. 6 in HmMSp.

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