### Relative simplicial approximation

E. C. Zeeman
1964 Mathematical proceedings of the Cambridge Philosophical Society (Print)
The absolute simplicial approximation theorem, which dates back to Alexander (l), states that there is a simplicial approximation g to any given continuous map/between two finite simplicial complexes (see for instance (2), p. 37 or (3), p. 86). The relative theorem given here permits us to leave / unchanged on any subcomplex, on which / happens to be already simplicial. The only previous mention of this modification that I have seen* in the literature is a remark ((3), Remark I, p. 87) to the
more » ... I, p. 87) to the effect that the relative case is an immediate generalization of the absolute case. In fact a strict generalization of the absolute case is not true, as is shown by the counter-example at the end of the paper. It is evidently necessary to give special treatment to the neighbourhood of the subcomplex to be kept fixed. Presumably the relative theorem has been somewhat neglected, because simplicial approximation has only been used in the context of algebraic topology. If L denotes the subcomplex, then the absolute approximation theorem ensures that gL = fL, and that g\L is an approximation to f\L: L -+fL, which is sufficient for homological applications. However, for recent applications in geometric topology the stronger result g\L = f\L is necessary (as, for example, in the proof of (4), Lemma 2-7). Notation. Let K,L,... denote finite simplicial complexes, and let \K\ denote the polyhedron underlying K. We shall assume simplexes to be closed. Any point a; e j_ST| lies in the interior of a unique simplex in K, called the carrier of x. If A is a simplex of K let st(^4, K) denote the star of A in K, which is the open subset of \K\ consisting of the union of the interiors of all simplexes having A as a face. The stars of all the vertices of K form an open covering of |isT|, called the star covering of K. The double star of a simplex is defined