By Shiing-Shen Chern

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**Example text**

Homomorphism from the symmetric group to the orthogonal group. 3 we can construct an m-dimensional vector bundle T(m) over R' (m) K (S (m) , 1) and over all spaces Ian (m) , with fiber over a collection C c ]Rn being the space of functions on its points. In the case n = oo this bundle coincides with the one induced by the trivial bundle over BO(m) via the right map of (5). THEOREM. The ith Stiefel- Whitney class of the bundle T (m) is equal to the sum of all cells corresponding to standard (n , m)-trees of depth 2 and codimension i.

P be a collection of standard trees. Place them on the same plane so that they do not intersect and the roots lie in the correct order on the line Lo , and then act on the strip 1-1[0, 1] by a transformation shrinking each line 1_1(,e) by 1 /e , so that all the roots get identified to one point. The resulting tree is denoted by {T1, ... , F'}. Obviously, the codimension of the cell corresponding to this tree is equal to the sum of codimensions of the cells corresponding to T1, ... , F'. For U E S(t) denote by c(T1, ...

THEOREM. The obstruction to the existence of a section of p;,;ri-1 is equal to the class of the element -b2) E X®(m-1)' Cm-1(®) in the quotient group Cm-1(e)/oCm_2(e). 1. In fact, under the obvious transn(m-I) ®(m-1) formation of coefficients ±Z the above obstruction is mapped into a generator, while for m = pk all elements of the subgroup ac m-2 (®) c Cm-1(®) are mapped to multiples of p (by formula (14) of Chapter I). 2 we will construct a section of co* m-1) over the complement to the cell a (m) .