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

ISBN-10: 3540184813

ISBN-13: 9783540184812

ISBN-10: 3540479864

ISBN-13: 9783540479864

During the wintry weather and spring of 1985 a Workshop in Algebraic Topology was once held on the collage of Washington. The direction notes by means of Emmanuel Dror Farjoun and through Frederick R. Cohen contained during this quantity are conscientiously written graduate point expositions of definite features of equivariant homotopy thought and classical homotopy conception, respectively. M.E. Mahowald has integrated a few of the fabric from his extra papers, symbolize quite a lot of modern homotopy idea: the Kervaire invariant, reliable splitting theorems, desktop calculation of risky homotopy teams, and reviews of L(n), Im J, and the symmetric groups.

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**Example text**

Hence D(f) l i f t s to Fl by a map D(f). This l i f t is not unique. Inductively, define Fq+l to be the fibre of the inclusion FqVFq+ FqXFq and Dq+l(f)=D(Dq(f)). Inductively define ~q+l: Fq+l ÷ X by setting ~q+l V~ equal to the composite Fq+l ÷ FqVFq q q> X where ~l is the composite Fl ÷ z2Bvs2B ~W> X. Notice that ~q+l has order 2 because it is given in terms of iterated Whitehead products. By definition ~s+l'Ds+l(f)=(2~s)'Ds(f)-2(~s'Ds(f)). If Hi{Z2A)=O for i~n2q-I and z2B is n-connected, then Dq(f) is null and thus 2q(~f)=O.

The space of pointed map from A to X. (B,~X). (sB,X). for the kth power map as before. 2. (zB,X)=O. Y. Write pn(k) for the mod-k Moore space which is the cofibre of the degree k map, [ k ] : Sn-I ÷ Sn - l . the i d e n t i t y for p3(k) is k [ N l ] . 3. (p3(p),s2p+l). (p3(p),S 2p+I) ÷ ~2S3<3> such that by (~h). 4. Let f: ~2S3<3> ÷ ~2S3<3> be any map which induces a non-zero ma~ on H2p_2. 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.

We add the remark given by I f X is atomic, then i t does not admit a product decomposition in the homotopy category. 3. (p3(2),S 2n+l) is atomic. (p3(2),S 2n) are also proven, but we shall not need these r e s u l t s here. 4. (p3(2),S 2n+l) be a map which induces an isomorphism on the module of primitives in dimensions 2n-2 and 4n-3. (p3(2),S 2n+l) as 62 a coalgebra over the Steenrod algebra, then f . is an isomorphism. (pk(2),S n) for n=5 or 9. (p4(2),S 17) are given in [CS]. (pk(2),s2n), n=2,4 s p l i t s i f k>3 because ~S2n is homotopy equivalent to s2n-lx~s 4n-l.

### Algebraic Topology: Proceedings of a Workshop held at the University of Washington, Seattle, 1985 by F. R. Cohen (auth.), Haynes R. Miller, Douglas C. Ravenel (eds.)

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