EXERCISES FOR MATHEMATICS 205C SPRING 2011 File Number 05 DEFAULT HYPOTHESES Unless specifically stated otherwise all spaces are assumed to be Hausdorff and locally arcwise connected In the first two exercises we shall use the concept of local homology groups to develop criteria for showing that certain pairs of graphs cannot be homeomorphic Recall that if x X then the local homology groups of X at x are defined as H X X x and by excision these are isomorphic to the groups obtained if one replaces X by an arbitrary open neighborhood U of x in X 1 Let X E be a connected graph and suppose that v is a vertex of E The supplement of v written Supp v E is defined to be the subcomplex of all vertices except v and all edges which do not have v as one of their vertices and the open star OpSt v is defined to be the complement of this subcomplex Geometrically this is just a finite union of half open intervals sharing a common end point a Prove that if x X is not a vertex then the local homology group H 1 X X x is isomorphic to Z Hint Let Ex be the unique edge containing x and let O x be obtained from Ex by deleting its enspoints Then Ox is open because its complement is the finite union of all vertices and all edges except Ex also Ox is homeomorphic to an open interval and x O x By excision the local homology group given above is isomorphic to H 1 Ox Ox x note that the deleted neighborhood Ox x is homeomorphic to a disjoint union of two open intervals b If v is a vertex of E define the branching number B v E to be the number of edges which have v as one of their vertices Prove that H 1 X X v is a free abelian group on B v E 1 generators Hint As noted above by excision the local homology group is isomorphic to H1 OpSt x OpSt x x note that OpSt x x is homeomorphic to a disjoint union of B v E open intervals Notational convention If x X is not a vertex then we shall say that the branching number B x E is equal to 2 With this convention the conclusion of b extends to all points in X c If k 6 2 is a positive integer explain why the number n k E of points x X with B x E k is finite and that if Y E 0 is another such that X and Y are homeomorphic then nk E nk E 0 Hint If f X Y is a homeomorphism such that f x y then we have H X X x H Y Y y Another Notational convention If X E is a graph then by c we can define n k X nk E because this number does not depend upon the particular graph structure E this number is finite if k 6 2 and infinite if k 2 Similarly if k 0 define V k X to be the set of all points with 1 branching number k Finally we may also define B x X B x E because the latter does not depend upon the choice of E 2 a Suppose that X0 E0 and X1 E1 are graphs and let f X0 X1 be a homeomorphism Prove that for all positive integers n the map h sends V n X0 to Vn X1 In particular show that V2 X0 and V2 X1 have the same finite numbers of components b Using the notion of n fold branch points show that there are at least 7 homeomorphism types represented by the standard hexadecimal digits as written below in sans serif type 0 1 2 3 4 5 6 7 8 9 A B C D E F Are new homeomorphism types added if we consider the remaining letters of the alphabet Explain Obviously one can formulate similar questions for a more or less arbitrary set of printed characters c The Figure 8 and Figure Theta spaces corresponding to 8 and respectively turn out to have the same homotopy type see the comments below but neither is a deformation retract of the other and in fact neither is homeomorphic to a subspace of the other Prove the last assertion in the preceding sentence Hint Suppose more generally that we have 1 dimensional graphs A and X such that A is homeomorphic to a subset of X and let x A Modify arguments from the previous exercises to show that B x A B x X and explain why this shows that the Figure Eight cannot be a subset of the Figure Theta and vice versa by describing the sets V n Figure Eight and Vn Figure Theta for n 2 Note In fact each of these spaces is a deformation retract of R 2 two points A proof of this for the Figure Eight is described on page 462 of Munkres and as noted in Example 3 on that page one can give a similar argument for the theta space See also the discussion and drawings on pages 132 133 of Lee Introduction to Topological Manifolds 3 Suppose that a space X is the union of two open arcwise connected subsets U and V and the intersection U V is nonempty but not arcwise connected Choose a base point p U V Prove that both H1 X H1 X p and 1 X p have infinite order and hence the conclusion of the Seifert van Kampen Theorem fails very badly and systematically if one drops the assumption that the intersection be arcwise connected 4 Suppose that X S 2 is a union of two simple closed curves C 1 and C2 homeomorphic 1 to S such that their intersection is a single point hence X is homeomorphic to a Figure Eight Prove that S 2 X has three components U V W such that the boundary of U is C 1 the boundary of V is C2 and the boundary of W is X 5 a If X S 2 is homeomorphic to a tree prove that the reduced homology groups of f S 2 X are trivial Hint We know this if X has a single edge Proceed by induction and H use the fact that X X0 E where E is an edge such that exactly one vertex lies in X 0 b Suppose that X is a connected graph whose fundamental group is a free group on m generators and suppose that A S 2 is homeomorphic to X Prove that S 2 A has m 1 connected components 6 Suppose that U Rm is a nonempty open subset and m n Prove that there is no continuous 1 1 mapping from U into R n Hint Explain why it suffices to consider the case where U V V 0 where V and V 0 are open in Rn and Rm n respectively and each contains 0 Let v be …