Download e-book for kindle: Casson's Invariant for Oriented Homology Three-Spheres: An by Selman Akbulut

By Selman Akbulut

ISBN-10: 0691085633

ISBN-13: 9780691085630

Within the spring of 1985, A. Casson introduced an attractive invariant of homology 3-spheres through structures on illustration areas. This invariant generalizes the Rohlin invariant and offers remarkable corollaries in low-dimensional topology. within the fall of that very same yr, Selman Akbulut and John McCarthy held a seminar in this invariant. those notes grew out of that seminar. The authors have attempted to stay on the subject of Casson's unique define and continue by way of giving wanted info, together with an exposition of Newstead's effects. they've got usually selected classical concrete methods over normal equipment. for instance, they didn't try and supply gauge conception factors for the result of Newstead; as an alternative they his unique ideas.

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Additional resources for Casson's Invariant for Oriented Homology Three-Spheres: An Exposition. (MN-36) (Mathematical Notes)

Example text

Recall that the projective plane was defined as P2 (Fq ) = (F3q \ {(0, 0, 0)})/ ∼, where (X0 , Y0 , Z0 ) ∼ (X1 , Y1 , Z1 ) if and only if there is some α ∈ k × with X1 = αX0 , Y1 = αY0 , and Z1 = αZ0 . In the same spirit, we have: 37 38 6. 1. The projective line P1 (Fq ) is defined to be (F2q \ {(0, 0)})/ ∼, where (X0 , Y0 ) ∼ (X1 , Y1 ) if and only if there is some α ∈ F× q with X1 = αX0 and Y1 = αY0 . Writing (X0 : Y0 ) for the equivalence class of the point (X0 , Y0 ), we have that P1 (Fq ) = {(α : 1) | α ∈ Fq } ∪ {(1 : 0)} We may think of P1 as the line defined by the equation Z = 0 in P2 .

A cyclic group is a group C which has an element a such that C = {ak | k ∈ Z}. In this case we write C = a . The order of a group is the number of elements it has. 21 below) there is only one cyclic group of order n for each positive integer n. We will use Cn to denote this group. 2. (Fundamental Theorem of Finite Abelian Groups) Let G be a finite abelian group. Then G can be written as a direct sum of cyclic groups. In fact, there are two canonical ways of doing this: • There are primes p1 , .

11) applies. Then it is ¯ q ) guaranteed not hard to see that the de points of intersection in P2 (F by Bezout’s theorem cluster into points of varying degrees over Fq , with the sum of those degrees being de. As an example of a curve with points of higher degree, let C0 be the projective plane curve over F3 corresponding to the affine equation y 2 = x3 + 2x + 2. 5. Check that C0 is nonsingular and show that it has genus 1. By plugging in the values 0, 1, 2 for x, we see that there are no F3 rational affine points on C.

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