By F. R. Cohen (auth.), Haynes R. Miller, Douglas C. Ravenel (eds.)

During the iciness and spring of 1985 a Workshop in Algebraic Topology was once held on the college of Washington. The direction notes via Emmanuel Dror Farjoun and by way of Frederick R. Cohen contained during this quantity are rigorously written graduate point expositions of yes features of equivariant homotopy concept and classical homotopy conception, respectively. M.E. Mahowald has incorporated a number of the fabric from his extra papers, signify a variety of modern homotopy conception: the Kervaire invariant, reliable splitting theorems, laptop calculation of volatile homotopy teams, and reviews of L(n), Im J, and the symmetric groups.

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Then f is a p-local equivalence. We start by constructing h. S2 _ _ ~ Since there is a cofibre sequence S2 ÷ p3(p), there is a fibration ~2X ~ g2X÷ map,(p3(p),X) where p denotes the pth power map on ~2X. 5 and show that pRh is null. Hence there is a map h: ~2S3 ÷ map,(p3(p),S 2p+I P l i f t i n g ~hp. 2. Notice that a choice of generator for H2p_I~2S3 suspends to the p-th power in H2p~S3. Thus (ghp), is onto H2p_l. Since H2p_2(map,(p3(p),s2p+I);~) i s ~ / p by the Hurewicz theorem, i t follows that h, is an isomorphism on H2p_2 by commutation with the homology Bockstein.

W2n+l^Y)=O ~(2x) in ~Sn+l s2n+l, and thus the claim follows. To finish, we must check that i f ~@ is null-homotopic, then W2n+l is 48 divisible by 2. 1, the (4n-l)-skeleton of ~2s2n+I, X, has partial cell diagram @ By inspection of the Serre Spectral sequence, we see that the composite, ~r, XC+ a2S 2n+l aH÷ a2s4n+1 is homotopic to the composite X p i n c h s4n-1 E2, a2S 4n+l. Since there is an isomorphism of groups [X,a2S 4n+l] ~ [p4n-l(2),a2s4n+l] and aW2n+l-~ is assumed to be null, we get a map p4n-l(2) --~ a2s2n+1 which is also null and which is the adjoint of W2n+l onp4n-l(2)/s4n-2.

This was proven in [CM] with the same method which we give below. Consider f: ~2s2n+I ÷ Qp4n-I(2) with f , onto H4n_2. Write v as the generator of H4n_IQp4n-I(2) and u=Sqlv. primitive in H2in_ 1 Write X21n_ I. for the unique ~2s2n+I 2k 1 2 and ~,X8n_l=O i f k>O• Then by the Nishida relations ~,X8n_l=X4n_l Since X8n_l an odd degree primitive, f,X8n_l = AQ3u+BQIV for some A and B. 2 2 Sq,x8n_l 2 =0 and ~,Q3u=~,QIV=QI u Next observe that because the degree of u is 4n-2. Thus by n a t u r a l i t y , A=B.