Differentiable Periodic Maps

1. The bordism groups. This note presents an outline of the authors' efforts to apply Thorn's cobordism theory [ó] to the study of differentiable periodic maps. First, however, we shall outline our scheme for computing the oriented bordism groups of a space [ l ] . These preliminary remarks bear on a problem raised by Milnor [4]. A finite manifold is the finite disjoint union of compact connected manifolds with boundary each of which carries a O-differential structure. The boundary of a finite manifold, B, is denoted by dB. A closed manifold is a finite manifold with void boundary. We now define the oriented bordism groups of a pair (X, ^4). An oriented singular manifold in (X, A) is a map ƒ: (B} dB ) —»(X, A) of an oriented finite manifold. Such a singular manifold bords in (X, A) if and only if there is a finite oriented manifold W and a map F: W—->X such that BC.dW as a finite regular submanifold whose orientation is induced by that of W and such that F\ jB=/, F(dW— B) C.A. From two such oriented singular manifolds (Bl fx) and (£?, /2) a disjoint union (B\\JB n 2l fxKJf2) is formed with B\C\B% = 0 and / i U / 2 | £?==ƒ,, * = 1 , 2. Obviously ( £ » , ƒ ) = ( J 3 n , ƒ). We £ay that two singular manifold (5J, /i) and (J5J,/2) are bordant in (X, yl) if and only if the disjoint union (JB*U -~B1,f\\Jf<ï) bords in (X, ^4). By the well-known angle straightening device [5] this is shown to form an equivalence relation. The oriented bordism class of (B,f) is written \B, ƒ] and the collection of all such bordism classes is On(X, A). An abelian group structure is imposed on £2n(X, A) by disjoint union, and then following Atiyah we refer to fin(X, A) as an oriented bordism group of (X, ^4). The weak direct sum fi*(X, A) = ^2Q 0n(X, A) is a graded right module over the oriented Thorn cobordism ring £2. For any ƒ: (B, dB)—>(X, ^4) and any closed oriented manifold V the module product is given by [B, / ] [ F W ] = [BX V, g] where g(x9 y) =ƒ(*). For any map : (X, A)-*(Y, B) there is an induced homomorphism <£*: ön(X, A)-JÇln(Y, B) given by <£*([£, ƒ]) = [B, f]. There is also d*: Qn(X, A)-*Qn-i(A) given by d*([5», ƒ ] ) = [3B», f\dB-*A]. Actually 0*: &*(X, i4)-*Q*(F, 5 ) and d*: J2*(X, ^4)~>fts|c(^4) are fl-module homomorphisms of degree 0 and 1 .