Monotone simplicial functions on combinatorial spheres

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1968

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Abstract

Let f be a monotone simplicial function from a triangulated combinatorial n-sphere. S[raised n] onto a triangulated, combinatorial n-manifold M[raised n]. It is shown that f is point-like if and only if fraised -1 is algebraically [n/2] - connected for each vertex v of M[raised n], provided that n=3. The proof is accomplished along lines which it is hoped will lead to a proof for other dimensions, perhaps for all n, in at least a modified version of the statement. Several related conjectures are investigated by showing that some of the. lemmas used to prove the main theorem are true in all dimensions, or at least in all but two. The particular difficulties encountered by the author in trying to prove these conjectures are explained.

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