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

During the wintry weather and spring of 1985 a Workshop in Algebraic Topology was once held on the collage of Washington. The path notes by way of Emmanuel Dror Farjoun and via Frederick R. Cohen contained during this quantity are rigorously written graduate point expositions of convinced points of equivariant homotopy thought and classical homotopy concept, respectively. M.E. Mahowald has incorporated the various fabric from his additional papers, characterize a variety of modern homotopy conception: the Kervaire invariant, sturdy splitting theorems, laptop calculation of risky homotopy teams, and reports of L(n), Im J, and the symmetric groups.

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**Example text**

3 follows. 4 which gives well-known and equivalent formulations on the d i v i s i b i l i t y are [M,BJM,S4]. of W2n+l=[12n+l,~2n+l ] . 4(2) is immediate. 4(5) was checked in section I I . (I) is equivalent to (3): Assume that W2n+l=2X; we construct a map f : p4n+2(2) ÷ ~S2n+2 which is non-zero in mod-2 homology. Since E(2x) is n u l l , there is a map f: p4n+2(2) ÷ ~S2n+2 which is E(x) on the bottom c e l l . To show that f , is non-zero on H4n+2, i t suffices to check that the Hopf i n v a r i a n t of f in [p4n+2(2),~s4n+2] is essential.

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. Apply Sql to get 2 f,(X4n_ I) = B(v2). Apply Sq{ to this last equation to get f,(X~n_l) = Bu2. 2 Notice that f,(X2n_l)=U and f,(X4n_l)=V. by commutation with the coproduct.

P=2. 1 i f By other means, we shall show that ~2S3<3> is a 2-1ocal r e t r a c t of map,(p3(2),S5). 3. 2S3 + p2p+l(p) which is onto in homology (where p is an odd prime)? 37 §I0. Desuspension and non-desuspension theorems In the l a s t section we saw that Selick's r e t r a c t i o n theorem is equivalent ot the existence of a map h: z3~2S3 ÷ z3P2p-I (p) in homology i f p is an odd prime. which is onto Thus we are interested in the same question at the prime 2. Lemma l O . l . There is a map h: z4~2s2n+l ÷ z4p4n-l(2) which is onto in homology.