Notation \\x — y\\ denotes the distance between points x, y G W1 \\v\\ denotes the length of the vector v G Mn d(x,y) denotes the distance between points x and y inf denotes the greatest lower bound X UY denotes the disjoint union of X and Y s u p p / = {x : f(x) 7^ 0} denotes the support of the function / X * Y denotes the join of the spaces X and Y SP n (X ) denotes the n-fold symmetric product of X f: (X, Y) —• (Xi, YI) denotes the map of pairs which takes Y C X to Yx C Xi 7ri(X, XQ) denotes the fundamental group of the space X with base point XQ G X 7Tn(X, xo) denotes the n-dimensional homotopy group of the space X with base point XQ G X degf denotes the degree of a map / rank/(x ) denotes the rank of / at the point x G(n, k) denotes the Grassmann manifold GLfc(R) denotes the group of k x k nonsingular matrices with real entries U(n) denotes the group of unitary matrices of order n SU(n) denotes the group of unitary matrices of order n with de- terminant 1 O(n) denotes the group of orthogonal matrices of order n SO(n) denotes the group of orthogonal matrices of order n with determinant 1 TxMn denotes the tangent space at the point x G Mn TMn denotes the tangent bundle Qfr(n-\-k) denotes the set of classes of framed cobordant ^-manifolds in Rn+k.

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