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Xn on the manifolds v and v . I therefore call it f (M, M ). I can change the variables by replacing y1 , y2 , . . , yp by holomorphic functions of p variables z1 , z2 , . . , zp chosen in such a way that a system of values of the y corresponds to a single system of values of the z. For this it is necessary that the Jacobian of the y with respect to the z never vanishes, and I can always assume that the z are arranged in an order which makes the determinant positive. The function f (M, M ) is then multiplied by this Jacobian, and consequently retains its sign.

Vλ . We express this fact by the notation v1 + v2 + · · · + vλ ∼ 0 More generally, the notation k1 v1 + k2 v2 ∼ k3 v3 + k4 v4 where the k are integers and the v are manifolds of q − 1 dimensions will denote that there exists a manifold W of q dimensions forming part of V , the boundary of which is composed of k1 manifolds similar to v1 , k2 manifolds similar to v2 , k3 manifolds similar to v3 but oppositely oriented, and k4 manifolds similar to v4 but oppositely oriented. Relations of this form will be called homologies.

We shall see examples later. §10. Geometric representation §10. Geometric representation 49 There is a manner of representing manifolds of three dimensions situated in a space of four dimensions which considerably facilitates their study. , Pn . We may assume that in the space of four dimensions there are three-dimensional manifolds Q1 , Q2 , . . , Qn homeomorphic to the P1 , P2 , . . , Pn respectively. Let F1 be a face of the polyhedron P1 , and Φ the set of points on the boundary of Q1 which correspond to the points of F1 .

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