By P. Hoffman, R. Piccinini, D. Sjerve

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Sm(~) complex, then for any We say has no torsion. (Sm(B)) ~ 0. CW-complex, such that H*(X;Z) Corollary 5 Let there exists X be a finite ~ ~ K2q(X) does not embed in Sm(~) odd, then q ~ ~(m). If q < ~(m) If then X s4m-4~Cm)+2q+l. Remark 6. vanishes for all with has no torsion. If q = a(m), we have to add the hypothesis 8 ~ A2m_2a(m) , and then we obtain Remark 7. X e:H 2~(m) (X)* H2m(x) does not embed in S4m-2a(m)+l. Corollary (5) should be compared with the results of Atiyah- Hirzebruch for manifolds, [2].

See. 75 (1969), 1293-1298. Zahler, The Adams-Novikov spectral sequence for the spheres, Thesis, University of Chicago, Chicago, 1970. ~103 , The Adams - Novikov spectral sequence for the spheres, Bulletin Amer. Math. Soc. 77. (1971), 169-17@. SURGERY~ BI~ ,,~ ~ p AND THE PL BONDISM RINGS R. James Mil~ram The development of differential topology over the past twenty years has shown the importance of the classifying spaces B 0 , BpL , BTo p and B G . l], [24]), G/O and the author ( [25], [19]), and G/Top , Top/FL by by Kirby, Lashof, Rothenberg, and Sibenmann ( [14], [16]).

Fm dim X = 2, can be found by means of the following According to [g; p. 149], the "telescope" has the homotopy type of write i. SI Since [ ~ - l , ~ ] ) c X x [0,®) cat X~ _< k, we may is closed and contractible in X~, hence 25 in X. , k. Y = ^