Combinatorial Equivalence Versus Topological Equivalence
Transactions of the American Mathematical Society
In , Milnor exhibited an example of two finite complexes Xy,X2 which are homeomorphic, but combinatorially distinct. That was the first example of the disparity between combinatorial notions versus topological ones on finite complexes. Previously the only results were positive in nature. Papakyriakopoulos had proved the Hauptvermutung for 2-dimensional complexes , and Moise  (and later Bing) had proved the Hauptvermutung for 3-dimensional manifolds. The object of this paper is to
... ide an example (see §5) of a finite complex K and two simplicial imbeddings, a,ß:K-*Sm which are combinatorially inequivalent, yet topologically equivalent. The nature of the argument is such that the minimum m it yields is m = 23. The construction of this example uses relative forms of the strong and weak stability theorem (see  ). It also uses recent results of Haefliger, Bott, and Milnor. To show that the two imbeddings a,ß:K->Sm are not combinatorially equivalent, I notice that Sm/0L(K) * Xy, Sm/ß(K) x X2 (the isomorphisms being combinatorial). Since Xy is combinatorially distinct from X2, the result follows that there is no combinatorial homeomorphism K-.Sm £ Sm such that ko a = ß. In the course of the construction, a general theorem is proved ( §6) which provides the possibility of producing many pairs of (combinatorially distinct) combinatorial imbeddings which are topologically equivalent. An unsettled question is the following : Are there combinatorial imbeddings /, g :Sk-> Sm which are combinatorially distinct yet topologically equivalent? By recent results of Zeeman  , all combinatorial imbeddings of Sk in Sm(m ^ k + 3) are equivalent combinatorially. By the results of Smale  , all combinatorial imbeddings of Sm_1 in Sm which are combinatorially locally trivial are combinatorially equivalent (for m sufficiently large). It is yet possible, however, for there to be two combinatorial imbeddings Received by the editors February 20,1962.