Geometry & Topology

Solvable groups, free divisors and nonisolated matrix singularities II: Vanishing topology

James Damon and Brian Pike

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In this paper we use the results from the first part to compute the vanishing topology for matrix singularities based on certain spaces of matrices. We place the variety of singular matrices in a geometric configuration of free divisors which are the “exceptional orbit varieties” for representations of solvable groups. Because there are towers of representations for towers of solvable groups, the free divisors actually form a tower of free divisors n, and we give an inductive procedure for computing the vanishing topology of the matrix singularities. The inductive procedure we use is an extension of that introduced by Lê–Greuel for computing the Milnor number of an ICIS. Instead of linear subspaces, we use free divisors arising from the geometric configuration and which correspond to subgroups of the solvable groups.

Here the vanishing topology involves a singular version of the Milnor fiber; however, it still has the good connectivity properties and is homotopy equivalent to a bouquet of spheres, whose number is called the singular Milnor number. We give formulas for this singular Milnor number in terms of singular Milnor numbers of various free divisors on smooth subspaces, which can be computed as lengths of determinantal modules. In addition to being applied to symmetric, general and skew-symmetric matrix singularities, the results are also applied to Cohen–Macaulay singularities defined as 2×3 matrix singularities. We compute the Milnor number of isolated Cohen–Macaulay surface singularities of this type in 4 and the difference of Betti numbers of Milnor fibers for isolated Cohen–Macaulay 3–fold singularities of this type in 5.

Article information

Geom. Topol., Volume 18, Number 2 (2014), 911-962.

Received: 12 September 2012
Revised: 16 April 2013
Accepted: 3 August 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32S30: Deformations of singularities; vanishing cycles [See also 14B07]
Secondary: 17B66: Lie algebras of vector fields and related (super) algebras 14M05: Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) [See also 13F15, 13F45, 13H10] 14M12: Determinantal varieties [See also 13C40]

matrix singularity determinantal variety vanishing cycles Milnor number singular Milnor number Cohen–Macaulay singularities free divisors deformation codimension


Damon, James; Pike, Brian. Solvable groups, free divisors and nonisolated matrix singularities II: Vanishing topology. Geom. Topol. 18 (2014), no. 2, 911--962. doi:10.2140/gt.2014.18.911.

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