I have a real question now. Suppose S is a simplicial complex of dimension n with the property that every n-1 face of every n simplex is a face of at least two n simplexes. I want to conclude that H_n(S) is non-zero. Assume that the union of the n simplexes is connected (any connected component would inherit the hypothesis). Then it seems clear geometrically that the union of the faces would enclose one or more holes, but I don't see how to actually prove this. The space would appear to have no boundary, but it also is not a manifold since a point in one of the faces could have a branched neighborhood.
Dear Mike, You are asking for every strongly connected (finite) n-complex to have nonzero $H_n$, which I think you can find -- if your Russian suffices -- in P. S. Alexandrov Combinatorial Topology, OGIZ, 1947 (660 pp.) [MR 10, 55b]. It should be around Chapter 14 or 15. I have never tried to go that far in the book, nor to read the Russian at all. I have long owned, and have used as texts in classes, the English translations which go to Chapter 12 I think; Graylock, Rochester, vol. 1, 1956 [MR 17, 882a] and vol. 2, 1957 [MR 19, 759a]. With any luck you will strike an algebraic topologist who can give you an English reference. Yours, John
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John R Isbell -
Michael Barr