By Kevin Walker

This e-book describes an invariant, l, of orientated rational homology 3-spheres that is a generalization of labor of Andrew Casson within the integer homology sphere case. enable R(X) denote the gap of conjugacy sessions of representations of p(X) into SU(2). allow (W, W, F) be a Heegaard splitting of a rational homology sphere M. Then l(M) is asserted to be an effectively outlined intersection variety of R(W) and R(W) inside of R(F). The definition of this intersection quantity is a fragile activity, because the areas concerned have singularities. A formulation describing how l transforms below Dehn surgical procedure is proved. The formulation comprises Alexander polynomials and Dedekind sums, and will be used to provide a slightly common facts of the life of l. it's also proven that once M is a Z-homology sphere, l(M) determines the Rochlin invariant of M

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**Example text**

1 5. Consider a partial order o n the set {a, b, c , d } where the strict inequalities are: c --< a, d --< c , d --< a, and d --< b. J described in Problem 2. 3 ( 1 ) . 7. 1 6. Describe the closure o f a point i n a poset topology. 7. 1 7. Which singletons are dense in a poset topology? J, by which we denote the space introduced in Problem 2. 3( 1 ) . It describes the partial order on {a, b, c, d} that determines the topology of this space by 7. 1 5. e. , the elements of the set under consideration, at vertices of the graph of the pictogram, as shown in the b picture, then the vertices marked by comparable elements c are connected by a segment or ascending broken line, and d the greater element corresponds to the higher vertex.

Each set has at most one greatest and at most one smallest element . An element b of a set A is a maximal element of A if A contains no element c such that b -< c . An element b is a minimal element of A if A contains no element c such that c -< b. 7. M. An element b of A is maximal iff A n CX: ( b ) = b; an element b of A is minimal iff A n C_i ( b ) = b. 7. 6. Ridd le. 1) How are the notions of maximal and greatest elements related? 2) What can you say about a poset in which these notions coincide for each subset?

4 . Consider the space-time IR4 of special relativity theory, where points represent moment-point events and the first three coordinates x 1 , x2 and x 3 are the spatial coordinates, while the fourth one, t, is the time. This space carries a relation, "the event (x 1 , x2 , x 3 , t ) precedes (and may influence) the event (x 1 , x2 , x3 , t)" . The relation is defined by the inequality 2- ( X,-2-_-X-: X -c ) 2 -,- . ( t - t ) 2:: v'r-c(X 1 ):-= +� Is this a partial order? If yes , then what are the upper and lower cones of an event?