Abstract. In this paper, we consider the versal deformations of three dimensional Lie algebras. We classify Lie algebras and study their deformations by using linear algebra techniques to study the cohomology. We will focus on how the deformations fasten the space of all such structures together. This space is known as the moduli space. We will give a geometric description of this space, derived from deformation theory, in order to illustrate general features of Lie algebras' moduli spaces.
Abstract. We establish several new results about both the (n)-solvable filtration, {F m n }, of the set of link concordance classes and the (n)-solvable filtration of the string link concordance group. We first establish a relationship between Milnor's invariants and links, L, with certain restrictions on the 4-manifold bounded by ML. Using this relationship, we can relate (n)-solvability of a link (or string link) with its Milnor's µ-invariants. Specifically, we show that if a link is (n)-solvable, then its Milnor's invariants vanish for lengths up to 2 n+2 − 1. Previously, there were no known results about the "other half" of the filtration, namely F m n.5 /F m n+1 . We establish the effect of the Bing doubling operator on (n)-solvability and using this, we show that F m n.5 /F m n+1 is nontrivial for links (and string links) with sufficiently many components. Moreover, we show that these quotients contain an infinite cyclic subgroup. We also show that links and string links modulo (1)-solvability is a nonabelian group. We show that we can relate other filtrations with Milnor's invariants. We show that if a link is n-positive, then its Milnor's invariants will also vanish for lengths up to 2 n+2 − 1. Lastly, we prove that the Grope filtration, {G m n }, of the set of link concordance classes is not the same as the (n)-solvable filtration.
Abstract. In this paper, we study the moduli space of 2|1-dimensional complex associative algebras, which is also the moduli space of codifferentials on the tensor coalgebra of a 1|2-dimensional complex space. We construct the moduli space by considering extensions of lower dimensional algebras. We also construct miniversal deformations of these algebras. This gives a complete description of how the moduli space is glued together via jump deformations.
Cochran, Orr, and Teichner developed a filtration of the knot concordance group indexed by half integers called the solvable filtration. Its terms are denoted by F n . It has been shown that F n /F n.5 is a very large group for n ≥ 0. For a generalization to the setting of links the third author showed that F n.5 /F n+1 is non-trivial. In this paper we provide evidence that for knots F 0.5 = F 1 . In particular we prove that every genus 1 algebraically slice knot is 1-solvable.Proposition ([COT03], Theorem 8.9). If L is a derivative for K and L is n-solvable then K is (n + 1)-solvable. If L is n.5-solvable, then K is (n + 1.5)-solvable.
In groundbreaking work from 2004, Cimasoni gave a geometric computation of the multivariable Conway potential function in terms of a generalization of a Seifert surface for a link called a C-complex [2]. Lemma 3 of that paper provides a family of moves which relates any two C-complexes for a fixed link. This allows for an approach to studying links from the point of view of C-complexes and in following papers it has been used to derive invariants. This lemma is false. We present counterexamples, a correction with detailed proof, and an analysis of the consequences of this error on subsequent works that rely on this lemma.
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