O-Minimal Cech Cohomology

Edmundo, Mário J.; Peatfield, Nicholas J.

doi:10.1093/qmath/ham036

Cited by 7 publications

(6 citation statements)

References 11 publications

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“…for 0 ≤ i < p. Note that, in the case R = R, the validity of the Vietoris-Begle theorem can be seen as a corollary of the existence of a semi-algebraic cohomology that satisfies the Eilenberg-Steenrod axioms for aČech theory (see [21]). …”

Section: Definition 211mentioning

confidence: 99%

Polynomial Hierarchy, Betti Numbers, and a Real Analogue of Toda’s Theorem

Basu

Zell

2010

Found Comput Math

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Toda (in SIAM J. Comput. 20(5):865-877, 1991) proved in 1989 that the (discrete) polynomial time hierarchy, PH, is contained in the class P #P , namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #P. This result, which illustrates the power of counting, is considered to be a seminal result in computational complexity theory. An analogous result in the complexity theory over the reals (in the sense of Blum-Shub-Smale real machines in Bull. Am. Math. Soc. (NS) 21(1): 1-46, 1989) has been missing so far. In this paper we formulate and prove a real analogue of Toda's theorem. Unlike Toda's proof in the discrete case, which relied on sophisticated combinatorial arguments, our proof is topological in nature. As a consequence of our techniques, we are also able to relate the computational hardness of two extremely well-studied problems in algorithmic semi-algebraic geometry: the problem of deciding sentences in the first-order theory of the reals with a constant number of quantifier alternations, and that of computing Betti numbers of semi-algebraic sets. We obtain a polynomial time reduction of 430 Found Comput Math (2010) 10: 429-454 the compact version of the first problem to the second. This latter result may be of independent interest to researchers in algorithmic semi-algebraic geometry.

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Section: Definition 211mentioning

confidence: 99%

Polynomial Hierarchy, Betti Numbers, and a Real Analogue of Toda’s Theorem

Basu

Zell

2010

Found Comput Math

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“…The following theorem relates the topology of J p (A) to that of the image of f in the case when f covers semi-algebraic compacts and is crucial for what follows. ; for 0 ≤ i < p. Note that in the case R = R, the validity of the Vietoris-Begle theorem can be seen as a corollary of the existence of a semi-algebraic co-homology that satisfies the Eilenberg-Steenrod axioms for aČech theory (see [21]).…”

Section: Compact Coveringsmentioning

confidence: 99%

“…Note that in the case R = R, the validity of the Vietoris-Begle theorem can be seen as a corollary of the existence of a semi-algebraic co-homology that satisfies the Eilenberg-Steenrod axioms for a Čech theory (see [21]).…”

Section: Topological Ingredientsmentioning

confidence: 99%

Polynomial Hierarchy, Betti Numbers and a Real Analogue of Toda's Theorem

Basu

Zell

2009

2009 50th Annual IEEE Symposium on Foundations of Computer Science

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Abstract. Toda [35] proved in 1989 that the (discrete) polynomial time hierarchy, PH, is contained in the class P #P , namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #P. This result which illustrates the power of counting is considered to be a seminal result in computational complexity theory. An analogous result in the complexity theory over the reals (in the sense of Blum-Shub-Smale real Turing machines [9]) has been missing so far. In this paper we formulate and prove a real analogue of Toda's theorem. Unlike Toda's proof in the discrete case, which relied on sophisticated combinatorial arguments, our proof is topological in nature. As a consequence of our techniques we are also able to relate the computational hardness of two extremely well-studied problems in algorithmic semi-algebraic geometry -namely the problem of deciding sentences in the first order theory of the reals with a constant number of quantifier alternations, and that of computing Betti numbers of semialgebraic sets. We obtain a polynomial time reduction of the compact version of the first problem to the second. This latter result might be of independent interest to researchers in algorithmic semi-algebraic geometry.

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“…With a good cohomology theory in arbitrary o-minimal structures which generalizes the o-minimal singular cohomology in o-minimal expansion of real closed fields ( [10] and [22]) one could obtain a uniform proof of the computation of m-torsion subgroups of abelian definably compact definable groups in arbitrary o-minimal structures which would include the three cases above. The authors already have made significant advances in this direction building on previous joint work with other authors ( [6], [8] and [9]). …”

Section: Introductionmentioning

confidence: 96%