Vol. 18, No. 3, 1966

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Relative general position

David Wilson Henderson

Vol. 18 (1966), No. 3, 513–523

In this paper we will say that a piecewise linear map f : K M from a finite complex into an n-manifold is a general position (gp) map, if for every pair of simplexes, A, B, contained in K,

(dimension of the singularities of f | A + B )
≦ (dimension of A)+ (dimension of B )− n.

By letting B = , we see that a gp map into an n-manifold is an embedding on each simplex of dimension less than or equal to n. Also note that the restriction of a gp map to a subcomplex is again a gp map. It is well known that every map f of a complex into a combinatorial manifold can be homotopically approximated by a gp map, g, on some subdivision of the complex. One might suppose that, if L is a subcomplex on which f is already a gp map, then gL could be made equat to fL. However, this cannot be done, in general, even if the manifold is a Euclidean space and the complex is a subdivision of a cell. (See the Remark at the end of §3.)

In §3 are two general position theorems which fix the map on a subcomplex on which it is already a gp map, but not without some severe restrictions. These theorems are stated in terms of relative general position (rgp) which applied to maps from a pair into a pair. Section 4 considers maps f : (D,BdD) (M,N) of a 2-manifold, D, into a 3-manifold, M, with 2-submanifold, N, with the added restriction that f(BdD) f(D BdD) = . It is, in general, impossible in this setting to make f into an rgp map while keeping fBdD fixed. However, two “relative normal position theorems” are proved which make the singularities “nice” while not considering a particular subdivision.

Mathematical Subject Classification
Primary: 57.20
Received: 30 January 1965
Published: 1 September 1966
David Wilson Henderson