The Topology of Real Algebraic Sets with Isolated Singularities

Akbulut, Selman; King, Henry C.

doi:10.2307/2006992

Cited by 65 publications

(56 citation statements)

References 14 publications

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“…Proof. This is just a minor modification of results proved by Akbulut and King [1], and Benedetti and Tognoli [10,12,28]. Then C is nonsingular, irreducible, and has two connected components, say Cx and C2, diffeomorphic to Sx .…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 56%

“…Then M is said to admit an algebraic approximation of type G if there exists a C°° embedding e : M -y E" , arbitrarily close in the C°° topology to the inclusion mapping M <-» E" , such that X = e(M) is a nonsingular algebraic subset of E" and h*(G) = Hkx%(X, Z/2), where h: X -> M is the diffeomorphism defined by h(e(m)) = m for m in M. Now let us turn to the case Fal with k>2. Akbulut and King [1,2,4,23] conjectured that every compact C°° manifold M is diffeomorphic to an affine nonsingular real algebraic variety X with totally algebraic homology, that is, a variety satisfying Hf%(X, Z/2) = Hk(X, Z/2) for k > 0. They also described how this conjecture would allow us to simplify several of their proofs and how it would be useful in further work on a topological characterization of real algebraic sets.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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Algebraic Cycles and Approximation Theorems in Real Algebraic Geometry

Bochnak¹,

Kucharz²

1993

Transactions of the American Mathematical Society

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Abstract. Let Af be a compact C°° manifold. A theorem of Nash-Tognoli asserts that M has an algebraic model, that is, M is diffeomorphic to a nonsingular real algebraic set X . Let FV^AfX, Z/2) denote the subgroup of Hk(X, Z/2) of the cohomology classes determined by algebraic cycles of codimension k on X . Assuming that M is connected, orientable and dim M > 5 , we prove in this paper that a subgroup G of H2(M, Z/2) is isomorphic to H^ (X, Z/2) for some algebraic model X of M if and only if w2(TM) is in G and each element of G is of the form W2K) for some real vector bundle £ over M , where w2 stands for the second Stiefel-Whitney class. A result of this type was previously known for subgroups G of HX(M, Z/2).

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 56%

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Algebraic Cycles and Approximation Theorems in Real Algebraic Geometry

Bochnak¹,

Kucharz²

1993

Transactions of the American Mathematical Society

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“…Suppose that M is symmetric with respect to the origin and 0 ∈ V \ M . In this case, we are able to prove a symmetric version of Wallace's trick, which allows to assume again (1). Since M = −M , the above procedure implies that Φ induces a Nash isomorphism from M to M .…”

Section: The Reader Observes That Ifmentioning

confidence: 97%

“…Later, Tognoli [22] proved that every compact smooth manifold is diffeomorphic to a nonsingular real algebraic set; that is, it admits a nonsingular algebraic model. Afterwards, Akbulut and King [1] completed this result showing that a noncompact smooth manifold admits a nonsingular algebraic model if and only if it is diffeomorphic to the interior of a compact smooth manifold with boundary. In [2,4], Akbulut, King and Taylor proved the existence of algebraic models for every compact PL manifolds.…”

Section: Introductionmentioning

confidence: 99%

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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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Non‐recursive functions, knots “with thick ropes,” and self‐clenching “thick” hyperspheres

Nabutovsky

1995

Comm Pure Appl Math

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We introduce an approach to certain geometric variational problems based on the use of the algorithmic unrecognizability of the n-dimensional sphere for n 2 5 . Sometimes this approach allows one to prove the existence of infinitely many solutions of a considered variational problem. This recursion-theoretic approach is applied in this paper to a class of functionals on the space of C1,'-smooth hypersurfaces diffeomorphic to S" in R"", where n is any fixed number 2 5. The simplest of these functionals K?) is defined by the formula K&") = (vol(Z"))l/"/r(Y'). where r ( P ) denotes the radius of injectivity of the normal exponential map for Z" C R"". We prove the existence of an infinite set of distinct locally minimal values of K~ on the space of C'.'-smooth topological hyperspheres in Rn+l for any n 2 5.The functional K?, naturally arises when one attempts to generalize knot theory in order to deal with embeddings and isotopies of "thick' circles and, more generally, "thick" spheres into Euclidean spaces. We introduce the notion of knot "with thick rope" types. The theory of knot "with thick rope" types turns out to be quite different from the classical knot theory because of the following result: There exists an infinite set of non-trivial knot "with thick rope" types in codimension one for every dimension greater than or equal to five. In this paper we start an investigation of (generalized) knots "with thick ropes." Let k C R3 be a knot. If x 5 r(k) (i.e., there exists a non-selfintersecting tube of radius x around k), then we can regard k as an axis of a knot with the rope of thickness x . Since rescalings do not change the shape of knots from any intuitive point of view, we consider length(k)/r(k) (i.e., K & ) ) as an invariant of the knot. We say that two knots kl and kZ have the same knot "with thick rope" x-type if there exists an isotopy between kl and k2 passing only through knots k such that K&) 5 x .These definitions can be straightforwardly generalized for the case of greater dimensions/codimensions.One of the main results of this paper is that the theory of (generalized) knot "with thick rope" types differs drastically from the usual theory of multidimensional knots. Namely, for any n L 5 there exists an infinite increasing sequence x; such that I ) lim;++x x; = +GO; 2) For any x; there exists a non-trivial knot "with

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The Topology of Real Algebraic Sets with Isolated Singularities

Cited by 65 publications

References 14 publications

Algebraic Cycles and Approximation Theorems in Real Algebraic Geometry

Algebraic Cycles and Approximation Theorems in Real Algebraic Geometry

Algebraic models of symmetric Nash sets

Non‐recursive functions, knots “with thick ropes,” and self‐clenching “thick” hyperspheres

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