Homotopy types of subspace arrangements via diagrams of spaces

Ziegler, Günter M.; Živaljević, Rade T.

doi:10.1007/bf01444901

Cited by 107 publications

(131 citation statements)

References 18 publications

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“…Proposition 2 (see [28]). The induced filtration of the one-point compactificationL of the space L ,F…”

Section: 2mentioning

confidence: 99%

Untitled

2005

Trans. Moscow Math. Soc.

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Abstract. We construct a homology algebraic algorithm for computing combinatorial formulas of all finite degree knot invariants. Its input is an arbitrary weight system, i.e., a virtual principal part of a finite degree invariant, and the output is either a proof of the fact that this weight system actually does not correspond to any knot invariant or an effective description of some invariant with this principal part, i.e., a finite collection of easily described singular chains of full dimension in the space of spatial curves such that the value of this invariant on any generic knot is equal to the sum of multiplicities of these chains in a neighborhood of the knot. (In examples calculated by now, the former possibility never occurred.) This algorithm is formally realized over Z 2 , but its generalization to the case of arbitrary coefficients is just a technical task. The algorithm is based on the study of a complex of chains in the space of smooth curves in the three-dimensional space with a fixed flag of directions, and also in the discriminant variety of this space of curves.

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“…Proposition 2 (see [28]). The induced filtration of the one-point compactificationL of the space L ,F…”

Section: 2mentioning

confidence: 99%

Untitled

2005

Trans. Moscow Math. Soc.

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“…The results of [77] imply in particular that this filtration is homotopically split: there is a homotopy equivalence…”

Section: Order Complexes Of Discrete Posets and Simplicial Resolutionmentioning

confidence: 98%

“…[24], [28], [62], [77]. The 'continuous' version of the method will be demonstrated in the next section.…”

Section: Order Complexes Of Discrete Posets and Simplicial Resolutionmentioning

confidence: 99%

Resolutions of discriminants and topology of their complements

Vassiliev

2001

New Developments in Singularity Theory

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Abstract. We study topological invariants of spaces of nonsingular geometrical objects (such as knots, operators, functions, varieties) defined by the linking numbers with appropriate cycles in the complementary discriminant sets of degenerate objects. We describe the main construction of such classes (based on the conical resolutions of discriminants) and list the results for a number of examples.The discriminant subsets of spaces of geometric objects are the sets of all objects with singularities of some chosen type. The important examples are: spaces of polynomials with multiple roots, resultant sets of polynomial systems having common roots, spaces of functions with degenerate singular points, of non-smooth algebraic varieties, of linear operators with zero or multiple eigenvalues, of smooth maps S 1 → M n (n ≥ 3) having singular or self-intersection points, of non-generic plane curves, and many others.The discriminants are usually singular varieties, whose stratifications correspond to the classification of degenerations of the corresponding objects. E.g., the discriminant subset in the space of polynomials x 3 + ax + b is the semicubical parabola (a/3) 3 + (b/2) 2 = 0: its regular points correspond to polynomials with a root of multiplicity exactly 2, and the vertex to the polynomial x 3 . The discriminant in the space of polynomials x 4 +ax 2 +bx+c is the swallowtail, i.e. the surface shown in the right-hand part of Fig. 1: its self-intersection curve consists of polynomials having two double roots, and the semicubical edges correspond to the polynomials with one triple root; the most singular point is the polynomial x 4 with a root of multiplicity 4. Similar stratifications hold for polynomials of all higher degrees: their strata are indexed by the multiplicities and orders in R 1 of all corresponding multiple roots.Usually one is interested in the space of non-singular objects which is the complement of the discriminant Σ, e.g. in the space of polynomials without multiple roots, of smooth varieties, of non-degenerate operators, or of knots, i.e. maps S 1 → R 3 having no self-intersection or singular points.If the total space F of geometric objects is an N -dimensional vector space then the homology groups of these complementary spaces are related by the Alexander duality formula

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“…Since the appearance of [16], there have been a number of substantial improvements to Theorem 3.11; see [2], [3], [7], and [44].…”

Section: Complements Of Affine Subspacesmentioning

confidence: 99%