L 2-Betti numbers of plane algebraic curves

Friedl, Stefan; Leidy, Constance; Maxim, Laurenţiu

doi:10.1307/mmj/1250169069

Cited by 4 publications

(7 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this note we prove an analogous statement for complements of complex affine hypersurfaces in general position at infinity. Furthermore, we recast and extend to this higher-dimensional setting results of [FLM09,LM06] about L 2 -Betti numbers of plane curve complements. …”

mentioning

confidence: 85%

“…Similar invariants were defined and studied in [LM06,LM07] in the case of plane curves. Moreover, as noted in [FLM09], the higher-order degrees δ i,m (X) can be regarded as L 2 -Betti numbers of the infinite cyclic cover M X , so the results of this note characterize the Cochran-Harvey invariants as well. At this point we want to emphasize that the higher-order degrees of a space M , hence also the L 2 -Betti numbers of the infinite cyclic cover M , may as well be infinite, since M is not in general a finite CWcomplex.…”

Section: Introductionmentioning

confidence: 91%

“…The following result of [FLM09] relates L 2 -Betti numbers to ranks of modules over skew fields. We now recall the definition of the Cochran-Harvey invariants (which in the context of complex algebraic geometry were first studied in [LM06], for plane curve complements).…”

Section: 2] and The References Therein)mentioning

confidence: 99%

“…Theorem 1.1 can be seen as an analogous statement for the complement of a hypersurface in C n which is in general position at infinity. In particular, we recast and generalize to arbitrary dimensions some results of [LM06,FLM09], where the case n = 2 of plane curve complements was considered.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

L 2-Betti Numbers of Hypersurface Complements

Maxim

2013

International Mathematics Research Notices

Self Cite

View full text Add to dashboard Cite

Abstract. In [DJL07] it was shown that if A is an affine hyperplane arrangement in C n , then at most one of the L 2 -Betti numbers b(2) i (C n \ A, id) is non-zero. In this note we prove an analogous statement for complements of complex affine hypersurfaces in general position at infinity. Furthermore, we recast and extend to this higher-dimensional setting results of [FLM09,LM06] about L 2 -Betti numbers of plane curve complements. IntroductionLet M be any topological space and α : π 1 (M ) → Γ an epimorphism to a group Γ (all groups are assumed countable). Then for i ∈ N ∪ {0} we can consider theWe recall the definition and some of the most important properties of L 2 -Betti numbers in Section 2. Let X ⊂ C n (n ≥ 2) be a reduced affine hypersurface defined by a polynomial equation f = f 1 · · · f s = 0, where f i are the irreducible factors of f . Denote by X i := {f i = 0}, i = 1, · · · , s, the irreducible components of X, and letbe the hypersurface complement. Then M X has the homotopy type of a finite CW complex of dimension n. It is well-known that H 1 (M X ; Z) is a free abelian group generated by the meridian loops γ i about the non-singular part of each irreducible component X i of X. Throughout the paper we denote by φ the map π 1 (M X ) → Z given by sending each meridian γ i to 1. This is the same map as the homomorphism f * : π 1 (M X ) → π 1 (C * ) = Z induced by f . We also refer to φ as the total linking number homomorphism. We call an epimorphism α : π 1 (M X ) → Γ admissible if the total linking number homomorphism φ factors through α.The main result of this note is the following "nonresonance-type" theorem.Theorem 1.1. Let X ⊂ C n be a reduced affine hypersurface in general position at infinity, i.e., whose projective completion intersects the hyperplane at infinity transversely. If α : π 1 (M X ) → Γ is an admissible epimorphism, then the L 2 -Betti numbers of the complement M X are computed by:

show abstract

mentioning

confidence: 85%

Section: Introductionmentioning

confidence: 91%

Section: 2] and The References Therein)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

L 2-Betti Numbers of Hypersurface Complements

Maxim

2013

International Mathematics Research Notices

Self Cite

View full text Add to dashboard Cite

show abstract

“…Furthermore, it can be also derived by using the corresponding vanishing statement for the L 2 ‐Betti numbers of such complements, see [, Theorem 1.1]. Indeed, it follows from [, Proposition 2.4] that we have the identification:

b_{i} false(M_{X}; ξ false) = b_{i}^{(2)} false(M_{X}, ξ : π_{1} (M_{X}) \to Im (ξ) false)

between the Novikov–Betti numbers and the L 2 ‐Betti numbers corresponding to ξ. However, to our knowledge, Novikov torsion numbers do not have such interpretation in terms of L 2 ‐invariants.…”

Section: Novikov Homology Of Complex Hypersurface Complementsmentioning

confidence: 98%

Twisted Novikov homology of complex hypersurface complements

Friedl

Maxim

2016

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

show abstract

A Survey of Twisted Alexander Polynomials

Friedl

Vidussi

2011

The Mathematics of Knots

Self Cite

107

View full text Add to dashboard Cite

Abstract. We give a short introduction to the theory of twisted Alexander polynomials of a 3-manifold associated to a representation of its fundamental group. We summarize their formal properties and we explain their relationship to twisted Reidemeister torsion. We then give a survey of the many applications of twisted invariants to the study of topological problems. We conclude with a short summary of the theory of higher order Alexander polynomials.

show abstract

L 2-Betti numbers of plane algebraic curves

Cited by 4 publications

References 22 publications

L 2-Betti Numbers of Hypersurface Complements

L 2-Betti Numbers of Hypersurface Complements

Twisted Novikov homology of complex hypersurface complements

A Survey of Twisted Alexander Polynomials

Contact Info

Product

Resources

About