By A. S. Smogorzhevski
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Extra info for Acerca de la Geometría de Lobachevski
L Exanple 2 . Let h" denote the set of polyhomeomorphians of E^ onto itself having compact support . The hypothesis of compact support enables us to define on H^ , as above , both a function ^ace polystructure and a linear polystructure , Let H^ , H^ be the resulting topological spaces, both having H^ as underlying set . Then it appears that H^ , H^ have different homotopy structures . By Alexander's Lemma on isotopy , it is easy to show that H^ is contractible . However Kuiper has used the queer 7 6 differential structures on S to show that either TT° (H„) 0 5 or 0 .
0 Corollary 2 . If X C K , and N , N^ are regular neighbourhoods of X in M , such that N ^ : N , then N - N^ = N x I . Proof . Construct two derived neighbourhoods as in the proof of Lemma 14 . N* = f"^ [ 0 ,6 J , N* ^ f"^ [O , S where 0 < 6< 1 . 1 Then / - N* = f"^ [ S , = f'^ t X I = ii* x I . Therefore the result is true for N* , N* . By Theorem 8 (2) choose a - 25 - homeomorphism h ! N*—^, N keeping X fixed . Now h N* , N 0 1 J- are both regular neighbourhoods of X in N , and so by Theorem 8 (3) we can ambient isotope h N* onto N^ keeping N fixed .
Let X be the space obtained by identifying the intervals x y = xy' , xz of y z in X . xz' , and let Y be the image - 26 - X Then X 0 conewise , axid Y 0 because Y is an arc . But X Y because any initial elementary simplicial collapse of any triangulation of X must have its free face in Y , and so must remove part of B . Similarly can build examples to show that X Y ^ /v XuY 0 XnYRemark 2 . Corollary 4 is useful for simplifying ^ines . For example the spine of a bounded 3-manifold can be normalised in the following sense : we can find a spine , which is a 2-dimensional cell complex in which every edge bounds exactly 3 faces , and every vertex bounds exactly 4 edges and 6 faces .