Topology of Real Algebraic Sets

Akbulut, Selman; King, Henry C.

doi:10.1007/978-1-4613-9739-7

Cited by 96 publications

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“…By applying point b) of Lemma 2.8.1 of [3] with V := R n , K := B q (2s) × B p (2r), P = L := N and f := h i , we obtain a regular function f i : R n −→ R such that…”

Section: A Symmetric Version Of Wallace's Trickmentioning

confidence: 99%

“…In the polyhedral setting, a complete topological classification of polyhedra admitting algebraic models is known only in the case of isolated singularities and in dimension ≤ 3 (see [1,3,7,16]). …”

Section: Introductionmentioning

confidence: 99%

“…Denote by Λ : V −→ V the restriction of λ n from V to V . The map Λ is dominating, because its rank at ξ is equal to v. In this way, by applying Proposition 2.3.2 of [3] to Λ, we obtain a real algebraic subset…”

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confidence: 98%

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Algebraic models of symmetric Nash sets

Ghiloni

Tancredi

2013

Rev Mat Complut

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The aim of this paper is to prove the existence of algebraic models for Nash sets having suitable symmetries. Given a Nash set M ⊂ R n , we say that M is specular if it is symmetric with respect to an affine subspace L of R n and M ∩ L = ∅. If M is symmetric with respect to a point of R n , we call M centrally symmetric. We prove that every specular compact Nash set is Nash isomorphic to a specular real algebraic set and every specular noncompact Nash set is semialgebraically homeomorphic to a specular real algebraic set. The same is true replacing "specular" with "centrally symmetric", provided the Nash set we consider is equal to the union of connected components of a real algebraic set. Less accurate results hold when such a union is symmetric with respect to a plane of positive dimension and it intersects that plane.The algebraic models for symmetric Nash sets M we construct are symmetric. If the local semialgebraic dimension of M is constant and positive, then we are able to prove that the set of birationally nonisomorphic symmetric algebraic models for M has the power of continuum.

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“…By applying point b) of Lemma 2.8.1 of [3] with V := R n , K := B q (2s) × B p (2r), P = L := N and f := h i , we obtain a regular function f i : R n −→ R such that…”

Section: A Symmetric Version Of Wallace's Trickmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Algebraic models of symmetric Nash sets

Ghiloni

Tancredi

2013

Rev Mat Complut

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“…For other treatments of this foundational material, consult Akbulut and King [1], Dovermann and Schultz [4], Goresky and MacPherson [5], Verona [17], and Mather [11], [12]. Definition 2.1.…”

Section: Background On Spaces With Stratificationsmentioning

confidence: 99%

Stratified path spaces and fibrations

Hughes¹

1999

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Abstract. The main objects of study are the homotopically stratified metric spaces introduced by Quinn. Closed unions of strata are shown to be stratified forward tame. Stratified fibrations between spaces with stratifications are introduced. Paths which lie in a single stratum except possibly at their initial points form a space with a natural stratification, and the evaluation map from that space of paths is shown to be a stratified fibration. Applications to mapping cylinders and to the geometry of manifold stratified spaces are expected in future papers.

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“…But there is another candidate for the name real algebraic geometry which is the study of real zero loci of real polynomials. The development in this branch of geometry shows that it has many interesting problems and people start indicating it the real algebraic geometry, for example in [1], [3]. Since we deal with both of these geometries in this paper, to distinguish them, we will keep the name real algebraic geometry for the classical involutional geometry and totally real algebraic geometry for the other one.…”

Section: Introductionmentioning

confidence: 99%

Complexification of real cycles and the Lawson suspension theorem

Teh¹

2007

Journal of the London Mathematical Society

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We embed the space of totally real r-cycles of a totally real projective variety into the space of complex r-cycles by complexification. We provide a proof of the holomorphic taffy argument in the proof of Lawson suspension theorem by using Chow forms and this proof gives us an analogous result for totally real cycle spaces. We use Sturm theorem to derive a criterion for a real polynomial of degree d to have d distinct real roots and use it to prove the openness of some subsets of real divisors. This enables us to prove that the suspension map induces a weak homotopy equivalence between two enlarged spaces of totally real cycle spaces.

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Topology of Real Algebraic Sets

Cited by 96 publications

References 0 publications

Algebraic models of symmetric Nash sets

Algebraic models of symmetric Nash sets

Stratified path spaces and fibrations

Complexification of real cycles and the Lawson suspension theorem

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