By R. Fröberg
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Additional info for Koszul Algebras
Then E1 ∪b W is dominated by E2 ∪c W . Proof. 38, we embed W × I into W × I and thus extend the given embedding i: E1 × I → E2 × I to an embedding i1 : (E1 ∪b W ) × I → i1 ((E1 ∪b W ) × I), (E2 ∪c W ) × I. To construct a collapse (E2 ∪c W ) × I we collapse W × I to i1 (W × I), and then apply the given collapse E2 × I i(E1 × I). 40 is a powerful tool for constructing new pairs of special polyP2 , one hedra such that one is dominated by the other. Given a pair P1 may attach, step by step, additional wings to P1 and P2 , each time getting a new pair.
13. (2-cell shifting) Let P be a simple polyhedron and f, g : S 1 → P two homotopic curves in general position. Then the simple polyhedra Q1 = P ∪f D2 and Q2 = P ∪g D2 are (T, U, L)-equivalent. Proof. 16. The only diﬀerence is that there is no 3-manifold, where D1 , D2 , and the trace of the homotopy between the curves could bound a proper 3-ball. 3 Special Polyhedra Which are not Spines 37 Let ft : S 1 → P be a homotopy between f and g. Deﬁne a map F : S 1 × I → P × I by the rule F (x, t) = (ft (x), t).
Creating a loop transform it to produce a new special polyhedron P1 that does not embed into a 3-manifold. Choose a point on a triple line and modify a neighborhood of this point by reattaching a sheet incident to this line such that there appears a new loop as shown in Fig. 32. 8. Indeed, we have a butterﬂy: The sheets B, C, D form a disc while the sheet A passes the point twice and thus produces two wings. Note that a regular neighborhood of the loop in the modiﬁed polyhedron obius band in the union of wings A and C.