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Extra info for Seminar on Combinatorial Topology

Example text

There is an isomorphism ^ V ^ determined by the vertex map A C — C - for all C J^ . Under this isomorphism P n Q (a B , J'^ ) . Now a B is collapsible , being a cone j and (a B)' is full in J^ , which is an (n-l)-sphere or ball , by Lemma 9 . Therefore N (a B , J'^) is a derived neighbourhood of a collapsible polyhedron , and is an (n-l)-ball by induction on n . Hence P n Q is an (n-l)-ball c Similarly (PuQ)ri R c R , and if we now choose J^ = Ik (B , J') , - 19 - then there is an isomorphism R _ J ' ^ , throwing (P U Q) n R onto A , N (A B , J'^) .

But for simply-connected spaces homotopy type = simply homotopy type , and there are simply-connected nonhomeomorphic manifolds of the same homotopy type . The Dunce Hat If we preserve the order X then the relation between X , Y is much sharper . Trivially if X is collapsible then X is contractible (homotopywise) . But the converse is not true. For example consider the Dunce Hat D which is defined to be a triangle with its sides identified ab = ac = be . Then D is contractible (although the contraction is hard to visualise) , and so D is the same simple homotopy type as a pointj but D is not collapsible because there is nowhere to start.

12 - *Lemma_13. If K L is an elementary collapse , then there exists a subdivision such that K' L' and L' is stellar (but K' may not be) . Proof . Let A = K - L , and B n-ball land B a face . Let By Theorem A , P be an n-simplex and an (n-l)-face . choose a homeomorphism h : A,B — A K A n L . Then A is an , r . / 'A Choose subdivisions so that h is a simplicial isomorphism h : p' Let IT : A — > T be the linear projection , mapping the vertex opposite ^ to the barycentre of I"" . Choose subdivisions A " , P " of A ' , P ' so that Tf A" r" is simplicial .