Interpretation of the Simplest Integral Invariant of by Wilczynski E. J.

By Wilczynski E. J.

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Any derived neighbourhood of X in M collapses to X « Proof . By Lemma 14 it suffices to prove for one particular derived neighbourhood. , A^ of J - K that meet K in order of decreasing dimension . Each A^ meets N in a convex cell B. , with a face C. -A. ^ f^^ . There is 1' l i e an elementary collapse of B^ from C^ , and the sequence of collapses i = 1 , r determines the collapses N ^ X. Lemma 15 . Let h ; K -»K be a homeomorphism of a complex that maps each simplex onto itself^ and keeps a subcomplex L fixed .

Let A ^ denote an n-simplex . Theorem 2 . If B^"^ < b"^ and A < A then anv homeomorphism B^ ^ — y A ^ ^ can be extended to a homeomorphism - 4 - Corollary . If two balls meet in a cogimon face • then their union is a ball . (For by Theorem 2 the union is homeomorphic to the suspension of a simplex) . Theorem 3 . If B^ C S^ then s'^-B^ is a ball Remark 1 The original proofs of Theorem 2 and 3 were given by Newman and Alexander in the 1920's and 30's and used "stellar theory" instead of combinatorial theory .

Or B,P-1 ' Therefore cS q-1 which is principal in K , remains principal in K, . Also D vfc A or B , 1 q-1 because A , B do not lie in K^ , and so D cannot be a face of A or Since p < q. j C is not a face of A - 11 - B (again since p ^ q-l) . Therefore D remains a free face of C in K^ . Therefore , if K* = - (C L> D) , then there is an elementary collapse K^ ^ K^ across C from D . Meanwhile A remains principal in K^ , and B remains a free face . Therefore there is an elementary collapse K* N. K^ across A from B , The lemma is proved .

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