A Second Order Algebraic Knot Concordance Group
نویسنده
چکیده
Let Knots be the abelian monoid of isotopy classes of knots S ⊂ S under connected sum, and let C be the topological knot concordance group of knots modulo slice knots. Cochran-OrrTeichner [COT03] defined a filtration of C: C ⊃ F(0) ⊃ F(0.5) ⊃ F(1) ⊃ F(1.5) ⊃ F(2) ⊃ . . . The quotient C/F(0.5) is isomorphic to Levine’s algebraic concordance group AC1 [Lev69]; F(0.5) is the algebraically slice knots. The quotient C/F(1.5) contains all metabelian concordance obstructions. The Cochran-Orr-Teichner (1.5)-level two stage obstructions map the concordance class of a knot to a pointed set (COT (C/1.5), U). We define an abelian monoid of chain complexes P , with a monoid homomorphism Knots → P . We then define an algebraic concordance equivalence relation on P and therefore a group AC2 := P/ ∼, our second order algebraic knot concordance group. The results of this thesis can be summarised in the following diagram:
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