Topology of spaces of functions without compound singularities

Vasil'ev, V. A.

doi:10.1007/bf01078942

Cited by 11 publications

(11 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Similar problems for K = R were also considered by Arnold in [7], where in particular the cohomology groups of F d \ Σ k were calculated. The Smale-Hirsch principle for stabilizations of these spaces was found in [58] in answering Arnold's question on the multiplicative structure in these groups. The unique difference for items I, J is that for K = C the limit of spaces Syst(k, m) \ Res(k, m) as m → ∞ is homotopy equivalent to Ω 2 S 2k−1 (and not just stably homotopy equivalent as in the previous case); this is a theorem of G. Segal [48].…”

Section: Monic Polynomials Withoutmentioning

confidence: 99%

“…In our case of the discriminant Σ k ⊂ F d the spectral sequence again degenerates at the first term, E 1 ≡ E ∞ (see [55], [58]). Moreover, in this case we also have homotopy splittings of discriminants similar to (2).…”

Section: 1mentioning

confidence: 99%

“…The spectral sequence coincides with one calculating the cohomology group of the corresponding space of all continuous sections of the jet bundle not intersecting our singular subset. This allows us to prove the homological Smale-Hirsch principle stating the homology (and even stable homotopy) equivalence of these spaces (and hence, if σ ≥ 3, even the usual homotopy equivalence), see [58], [55], [56]. Moreover, in the most classical case of codimension 2, when n = 1 and the forbidden singularity class consists of all germs more complicated than Morse and A 2 , ordinary homotopy equivalence also holds: it was proved up to dimension m − 1 in [32] and in all dimensions in [25].…”

Section: Monic Polynomials Withoutmentioning

confidence: 99%

“…Stabilization. For K = R the spectral sequences stabilize to one calculating the cohomology groups of the space F ∞ \ Σ k of functions R 1 → R 1 with fixed behavior at infinity and without k-fold zeros, and also the cohomology group of the loop space Ω(R k \0) ∼ ΩS k−1 , see [58], [55], [56]. For K = C and k = 2 the stabilized spectral sequence calculates the cohomology group of the stable braid group (with infinitely many strings).…”

Section: Monic Polynomials Withoutmentioning

confidence: 99%

“…These formulas can be induced via the same embedding from the known expressions for the space ΩS k−1 . For K = R, k = 3, the generator of the group H 1 (F d \ Σ 3 ) π 1 (ΩS 2 ) π 2 (S 2 ) (with index equal to 1) is identified in [3], the generator of π 2 (F d \ Σ 3 ) d≥9 π 3 (S 2 ) (with Hopf invariant 1) in [58], [55], [56]. L. Multiplication.…”

Section: Monic Polynomials Withoutmentioning

confidence: 99%

See 4 more Smart Citations

Resolutions of discriminants and topology of their complements

Vassiliev

2001

New Developments in Singularity Theory

View full text Add to dashboard Cite

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

show abstract

Section: Monic Polynomials Withoutmentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: Monic Polynomials Withoutmentioning

confidence: 99%

Section: Monic Polynomials Withoutmentioning

confidence: 99%

Section: Monic Polynomials Withoutmentioning

confidence: 99%

See 3 more Smart Citations

Resolutions of discriminants and topology of their complements

Vassiliev

2001

New Developments in Singularity Theory

View full text Add to dashboard Cite

show abstract

Comments

Arnold's Problems

View full text Add to dashboard Cite

Topology of spaces of hyperbolic polynomials and combinatorics of resonances

Kozlov

2002

Isr. J. Math.

View full text Add to dashboard Cite

Abstract. In this paper we study the topology of the strata, indexed by number partitions λ, in the natural stratification of the space of monic hyperbolic polynomials of degree n. We prove stabilization theorems for removing an independent block or an independent relation in λ. We also prove contractibility of the one-point compactifications of the strata indexed by a large class of number partitions, including λ = (k m , 1 r ), for m ≥ 2. Furthermore, we study the maps between the homology groups of the strata, induced by imposing additional relations (resonances) on the number partition λ, or by merging some of the blocks of λ.

show abstract

Topology of spaces of functions without compound singularities

Cited by 11 publications

References 6 publications

Resolutions of discriminants and topology of their complements

Resolutions of discriminants and topology of their complements

Comments

Topology of spaces of hyperbolic polynomials and combinatorics of resonances

Contact Info

Product

Resources

About