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

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 used to be held on the collage of Washington. The path notes via Emmanuel Dror Farjoun and by means of Frederick R. Cohen contained during this quantity are conscientiously written graduate point expositions of yes elements of equivariant homotopy idea and classical homotopy idea, respectively. M.E. Mahowald has integrated the various fabric from his extra papers, characterize a variety of modern homotopy thought: the Kervaire invariant, solid splitting theorems, computing device calculation of volatile homotopy teams, and stories of L(n), Im J, and the symmetric groups.

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Extra resources for Algebraic Topology: Proceedings of a Workshop held at the University of Washington, Seattle, 1985

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X,Y) is the space of pointed maps from X to Y. In Lemma 19,4 a map k: ~S9 ÷ S5 is given which induces an epimorphism on ~8$5. (p6(2)vS4,S9 ) is null-homotopic. (z2A3,S9). (pk(2),S n) where pk(2) is the cofibre of [ 2 ] : Sk-I ÷ Sk - l . (pk(2),S n) product for most values of n and that there are product decompositions for some values of n. These sections are based completely on work in [[CCPS],[Cl]]. (pk(2),sn), i t is necessary to compute i t ' s homology as a Hopf algebra over the Steenrod algebra.

Proposition I I . I . (1) is null-homotopic i f and only i f n=O,l, or 3. (2) ~@ is null-homotopic i f and only i f W2n+l is divisible by 2. The proof of I I . 2. I. @ is null homotopic on the 4n-skeleton of ~S2n+l i f and only i f n=O,l, or 3. 2. ~@ is null homotopic on the (4n-l)-skeleton of ~2s2n+I i f and only i f W2n+l is d i v i s i b l e by 2. 3. ~2@ is null-homotopic on the (4n-2)-skeleton of ~3s2n+I i f and only i f a. nzO(2) and W2n+l=2x+Yn4n or b. nzl(2) and W2n+l=2X. 3. The map ~q@ is null-homotopic on the (4n-q+l)-skeleton of ~qs 2n+l, n~q, i f and only i f there is a homotopy commutative diagram 43 s4n+1 W2n+l> s2n+l i / z2n+2p2n-I 2n-q-l where the cofibre of e is z2np 2n 2n-q-I There are several well-known equivalent formulations of this l a s t question and these can be found in work of Barratt, Mahowald, Jones, and Selick [M,BJM,S4].

Y(dy) p - , ~0, by commutation with the coproduct, f((dy)pk)~o and the lemma follows. 4. 2 that the mod-p homology of ~2S3<3> is isomorphic to k~l . A[Y2pk . 1] k~l . ~/P[X2pk_ . 2] and BY2pk l--X2pk-2 By inspection there is a most one primitive in any degree and a basis for pJ the module of primitives is {Y2p k_l,(x2pk_2) Ik>l, j>_O}. h. I t suffices to check that f , (primitive) is non-zero. Since P~(X2pk+l _2 ) PJ-- -(x 2pk_2)p j+1 and f , commutes with p~, i t suffices to check pJ that f,(x)iO for x=Y2p_l or X2p_2.

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