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Search: All articles in the CJM digital archive with keyword torsion

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1. CJM 2013 (vol 65 pp. 843)

Jonsson, Jakob
3-torsion in the Homology of Complexes of Graphs of Bounded Degree
For $\delta \ge 1$ and $n \ge 1$, consider the simplicial complex of graphs on $n$ vertices in which each vertex has degree at most $\delta$; we identify a given graph with its edge set and admit one loop at each vertex. This complex is of some importance in the theory of semigroup algebras. When $\delta = 1$, we obtain the matching complex, for which it is known that there is $3$-torsion in degree $d$ of the homology whenever $\frac{n-4}{3} \le d \le \frac{n-6}{2}$. This paper establishes similar bounds for $\delta \ge 2$. Specifically, there is $3$-torsion in degree $d$ whenever $\frac{(3\delta-1)n-8}{6} \le d \le \frac{\delta (n-1) - 4}{2}$. The procedure for detecting torsion is to construct an explicit cycle $z$ that is easily seen to have the property that $3z$ is a boundary. Defining a homomorphism that sends $z$ to a non-boundary element in the chain complex of a certain matching complex, we obtain that $z$ itself is a non-boundary. In particular, the homology class of $z$ has order $3$.

Keywords:simplicial complex, simplicial homology, torsion group, vertex degree
Categories:05E45, 55U10, 05C07, 20K10

2. CJM 2001 (vol 53 pp. 780)

Nicolaescu, Liviu I.
Seiberg-Witten Invariants of Lens Spaces
We show that the Seiberg-Witten invariants of a lens space determine and are determined by its Casson-Walker invariant and its Reidemeister-Turaev torsion.

Keywords:lens spaces, Seifert manifolds, Seiberg-Witten invariants, Casson-Walker invariant, Reidemeister torsion, eta invariants, Dedekind-Rademacher sums
Categories:58D27, 57Q10, 57R15, 57R19, 53C20, 53C25

3. CJM 2000 (vol 52 pp. 695)

Carey, A.; Farber, M.; Mathai, V.
Correspondences, von Neumann Algebras and Holomorphic $L^2$ Torsion
Given a holomorphic Hilbertian bundle on a compact complex manifold, we introduce the notion of holomorphic $L^2$ torsion, which lies in the determinant line of the twisted $L^2$ Dolbeault cohomology and represents a volume element there. Here we utilise the theory of determinant lines of Hilbertian modules over finite von~Neumann algebras as developed in \cite{CFM}. This specialises to the Ray-Singer-Quillen holomorphic torsion in the finite dimensional case. We compute a metric variation formula for the holomorphic $L^2$ torsion, which shows that it is {\it not\/} in general independent of the choice of Hermitian metrics on the complex manifold and on the holomorphic Hilbertian bundle, which are needed to define it. We therefore initiate the theory of correspondences of determinant lines, that enables us to define a relative holomorphic $L^2$ torsion for a pair of flat Hilbertian bundles, which we prove is independent of the choice of Hermitian metrics on the complex manifold and on the flat Hilbertian bundles.

Keywords:holomorphic $L^2$ torsion, correspondences, local index theorem, almost Kähler manifolds, von~Neumann algebras, determinant lines
Categories:58J52, 58J35, 58J20

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