Invariance of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>o</mml:mi></mml:math>-minimal cohomology with definably compact supports

Edmundo, Mário J.; Prelli, Luca

doi:10.5802/cml.17

Cited by 5 publications

(10 citation statements)

References 23 publications

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“…In the paper [22, Section 4] a generalization to the cartesian product J = Π m i=1 J i of definable group-intervals of a construction done in o-minimal expansions of ordered groups in [1,Section 7] is presented. The construction is similar but, while in [1] one needs to consider only one parameter in [22] one considers m parameters, one for each J i . For the readers convenience we recall this construction in Theorem 4.13 below as well as a couple of results that we will be using.…”

Section: Cohomology With Definably Compact Supports Of J-bounded Cellsmentioning

confidence: 99%

“…(A0) follows from that fact that a product of locally closed definable subsets of a cartesian product of definably compact definable groups is also a locally closed definable subset of a definably compact definable group; (A1) follows from Corollary 3.15; (A2) follows since: (i) a definably compact group is definably normal ([24, Corollary 2.3]) and (ii) a locally closed definable subset of a cartesian product of a given definably compact definable group has a definably normal completion, namely its closure; (A3) was proved in [22,Theorem 1.1].…”

Section: Applications To Definably Compact Abelian Groupsmentioning

confidence: 99%

“…See Section 3 and Subsection 4.3. These techniques from [1] were already used in [22,Theorem 1.1] to prove (A3) for definably compact definable groups.…”

Section: Introductionmentioning

confidence: 99%

“…From the properties of m, it is standard to show that µ is a co-multiplication making H * (G; k G ) into a Hopf algebra over k. Compare with [20, Corollary 3.5].Since G is definably normal ([24, Corollary 2.3]), by[17, Proposition 4.2] we have H p (G; k G ) = 0 for all p > dim G and therefore H * (G; k G ) is a bounded Hopf algebra. By[22, Theorem 1.2], it is of finite type.In the case char(k) = 0, the description of the Hopf algebra H * (G; k G ) follows from the Hopf-Leray theorem. See[20, Corollary 3.6] for details.…”

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confidence: 99%

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On Pillay's conjecture in the general case

Edmundo

Mamino

Prelli

et al. 2017

Advances in Mathematics

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Let M be an arbitrary o-minimal structure. Let G be a definably compact, definably connected, abelian definable group of dimension n. Here we compute: (i) the new intrinsic o-minimal fundamental group of G; (ii) for each k > 0, the k-torsion subgroups of G; (iii) the o-minimal cohomology algebra over Q of G. As a corollary we obtain a new uniform proof of Pillay's conjecture, an o-minimal analogue of Hilbert's fifth problem, relating definably compact groups to compact real Lie groups, extending the proof already known in o-minimal expansions of ordered fields.

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Section: Cohomology With Definably Compact Supports Of J-bounded Cellsmentioning

confidence: 99%

Section: Applications To Definably Compact Abelian Groupsmentioning

confidence: 99%

“…See Section 3 and Subsection 4.3. These techniques from [1] were already used in [22,Theorem 1.1] to prove (A3) for definably compact definable groups.…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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On Pillay's conjecture in the general case

Edmundo

Mamino

Prelli

et al. 2017

Advances in Mathematics

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“…where In these examples, (A3) is obtained in [25] after extending an invariance result for closed and bounded definable sets in o-minimal expansions of ordered groups from [2].…”

Section: Proper Direct Imagementioning

confidence: 99%