By Michèle Audin (auth.), J. Aguadé, R. Kane (eds.)

**Contents:** M. Audin: sessions Caracteristiques Lagrangiennes.- A. Baker: Combinatorial and mathematics Identities in keeping with Formal workforce Laws.- M.C. Crabb: at the sturdy Splitting of U(n) and ÛU(n).- E. Dror Farjoun, A. Zabrodsky: The Homotopy Spectral series for Equivariant functionality Complexes.- W.G. Dwyer, G. Mislin: at the Homotopy kind of the elements of map*(BS3, BS3).- W.G. Dwyer, H.R. Miller, C.W. Wilkerson: The Homotopy strong point of BS3.- W.G. Dwyer, A. Zabrodsky: Maps among Classifying Spaces.- B. Eckmann: Nilpotent crew motion and Euler Characteristic.- N.D. Gilbert: at the primary Catn-Group of an n-Cube of Spaces.- H.H. Glover: Coloring Maps on Surfaces.- P. Goerss, L. Smith, S. Zarati: Sur les A-Algèbres Instables.- K.A. Hardie, K.H. Kamps: The Homotopy class of Homotopy Factorizations.- L.J. Hernández: right Cohomologies and the correct category Problem.- A. Kono, ok. Ishitoya: Squaring Operations in Mod 2 Cohomology of Quotients of Compact Lie teams by means of Maximal Tori.- J. Lannes; L. Schwartz: at the constitution of the U-Injectives.- S.A. Mitchell: The Bott Filtration of a Loop Group.- Z. Wojtkowiak: On Maps from Holim F to Z.- R.M.W. wooden: Splitting (CP x...xCP ) and the motion of Steenrod Squares Sqi at the Polynomial Ring F2 Äx1,...,xnÜ.

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To the (x,z) : > > (U(V) x collapsing Then identity. ) The a stable onto same to the map is the splitting the top cell. Pontrjagin-Thom standard tubular we o b t a i n a map onto Rn(V)/Rn-I(V) : ~(V)+ ^ ~(V) + way. neighbourhood s : identity. is the o n of the p r o j e c t i o n The general construction, the : ~(V) S ~(V) (This case Let c : identified + + ~(V) ^ 2(V) > it is i n d u c e d 3 b y (x/2 - I) (x/2 + 1 ) - l e x p ( - z ) in So s gives in the ~(V)) + = U(V) + ^ ~(V) +. Indeed, the Rn(V)/Rn-I(v) cell ~(V), (c A I)°S (0,0) of s i g n top construction at choice the = ~(V) @ and the d e r i v a t i v e same.

L. from the o r i g i n a l 1985-6. It is a p l e a s u r e in p a r t i c u l a r Gonqalves, for their algebra. R. Hubbuck, helpful comments. Notation Appropriate theory. The dimensional End(V) and and notation will take care of most of the e q u i v a r i a n t letters V and W will be r e s e r v e d ¢-Hilbert spaces. for n o n - z e r o Let n = dime V. ) for example, To s t u d y of U(V); PU(V) where s y m m e t r y we i n t r o d u c e ~ PU(V) Miller's F(V) the group F(V) to need to do so. of smooth theorem unitary direct space of elements necessarily) extends to Stiefel A superscript construction automorphisms Automorphisms We shall w r i t e embeddings As s y m m e t r y such that of U(V) isometries.

3) and the (This is c l e a r if space completion. and its i n d e x context is a ¢ [ z ] - s u b m o d u l e : ~(pE ) w i t h T h e n g, b e c o m e s under deformation. that the c o n s t r u c t i o n arose. a It was But we a r g u e from first p r i n c i p l e s . 7. Let g 6 Sk(V), T h e n w e h a v e an e x a c t h 6 Sz(V). sequence h,l 0 ~ ~ (h) ~ > ~ (gQh) O n l y the s u r j e c t i v i t y gm 6 H+(V), Since m a p Sk(V) t i o n of Sk(V) S k(v) has d i m e n s i o n and k )0. 8. (Mitchell > {M 6 G k(Ikv) if m 6 H(V), 6 H+(V).