Syzygies in Equivariant Cohomology for Non-abelian Lie Groups

Franz, Matthias

doi:10.1007/978-3-319-31580-5_14

Cited by 11 publications

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References 33 publications

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“…Syzygies therefore interpolate between torsion-freeness and freeness. In [1] and [2], Allday, Puppe and the author initiated the study of syzygies in the context of torus-equivariant cohomology; this was extended in [15] to actions of compact connected Lie groups. Because restriction to a maximal torus does not change the order of a syzygy in equivariant cohomology [15,Prop.…”

Section: Introductionmentioning

confidence: 99%

“…Note that the empty set is not acyclic sinceH −1 (∅) = R. From the next section on, we will only consider T -manifolds. However, in this section we more generally allow the same T -spaces X as in [15]. This means that X is a closed T -stable subset of a T -manifold or T -orbifold such that H * (X) is a finite-dimensional R-vector space.…”

Section: Introductionmentioning

confidence: 99%

“…Also, only finitely many subtori of T are allowed to occur as the identity component of an isotropy group in X. This latter condition is automatically satisfied if X is a manifold or locally orientable orbifold [15,Thm. 7.7].…”

Section: Introductionmentioning

confidence: 99%

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A Quotient Criterion for Syzygies in Equivariant Cohomology

Franz

2016

Transformation Groups

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Let X be a manifold with an action of a torus T such that all isotropy groups are connected and satisfying some other mild hypotheses. We provide a necessary and sufficient criterion for the equivariant cohomology H * T (X) with real coefficients to be a certain syzygy as module over H * (BT ). It turns out that, possibly after blowing up the non-free part of the action, this only depends on the orbit space X/T together with its stratification by orbit type. Our criterion unifies and generalizes results of many authors about the freeness and torsion-freeness of equivariant cohomology for various classes of T -manifolds.in equivariant cohomology. The first map in (1.3) is induced by the inclusion X 0 ֒→ X, and the others are the connecting maps in the long exact sequences for the triples (X i+1 , X i , X i−1 ). We call H * +i T (X i , X i−1 ) the i-th position of this sequence; H * T (X) is at position −1. One of the main achievements of [1] was to relate the exactness of the Atiyah-Bredon sequence to the syzygy order of H * T (X) [1, Thm. 5.7]:2010 Mathematics Subject Classification. Primary 57R91; secondary 13D02, 55N91. The author was supported by an NSERC Discovery Grant.Here ι QO : Q ֒→ O denotes the inclusion and rank Q the common dimension of the orbits in X lying over the interior of Q. The signs in (1.6) are determined by an ordering of the facets (corank-1 faces) of X/T , see Section 5. Theorem 1.3. Let X be a locally standard T -manifold, and let j ≥ 0. Then H * T (X) is a j-th syzygy if and only if H i (B * (P )) = 0 for all faces P of X/T and all i > max(rank P − j, 0). This characterization unifies and extends results of many authors concerning the freeness and torsion-freeness of torus-equivariant cohomology. This includes work of Barthel-Brasselet-Fieseler-Kaup [5], Masuda-Panov [24], Masuda [23] and Goertsches-Rollenske [17]. We also generalize a result of Bredon [7] about the cohomology of X/T and one of Ayzenberg-Masuda-Park-Zeng [4] about the equivariant cohomology of certain torus manifolds, and we recover the calculation of Ext modules of Stanley-Reisner rings done by Mustaţă [26] and Yanagawa [28] as well as Munkres' formula for the depth of such rings [25].Compact orientable T -manifolds of dimension 2r have received a lot of attention so far, for example smooth complete toric varieties or torus manifolds. We can now also explain why these spaces do not provide interesting examples of syzygies.Corollary 1.4. Let X be a compact orientable T -manifold of dimension 2r. Then H * T (X) is torsion-free if and only if it is free over R.The paper is organized as follows: In Section 2 we review background material, and in Section 3 we characterize syzygies by a depth condition on the equivariant cohomology of fixed point sets of subtori. Then we study blow-ups of characteristics manifolds (Section 4). Theorem 1.3 is proved in Section 5, based on a certain decomposition of the Atiyah-Bredon sequence for compact supports. Multiplicative aspects are discussed in Section 6. We present several applicatio...

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Quotient Criterion for Syzygies in Equivariant Cohomology

Franz

2016

Transformation Groups

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“…In a different direction, the notion of equivariant formality has been extended to that of a syzygy in equivariant cohomology by Allday-Franz-Puppe [2] (G a torus) and Franz [10] (G a compact connected Lie group). Let r be the rank of such a G, so that H * (BG; R) is a polynomial algebra in r variables of even degrees.…”

Section: Statement Of the Resultsmentioning

confidence: 99%

Symmetric Products of Equivariantly Formal Spaces

Franz

2018

Can. math. bull.

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Abstract. Let X be a CW complex with a continuous action of a topological group G. We show that if X is equivariantly formal for singular cohomology with coe cients in some eld k, then so are all symmetric products of X and in fact all its Γ-products. In particular, symmetric products of quasi-projective M-varieties are again M-varieties.is generalizes a result by Biswas and D'Mello about symmetric products of Mcurves. We also discuss several related questions.

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“…It is not hard to find examples of T -manifolds such that (1.1) is exact without H * T (X) being free over R, see below. This phenomenon was studied in detail by Allday-Franz-Puppe [1], [2], who characterized those T -manifolds for which the Chang-Skjelbred sequence is exact; in [9] this is generalized to non-abelian Lie groups. Allday-Franz-Puppe actually proved a more general theorem that involves higher equivariant skeletons.…”

Section: Introductionmentioning

confidence: 99%

The syzygy order of big polygon spaces

Franz,

Huang

2019

Preprint

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Big polygon spaces are compact orientable manifolds with a torus action whose equivariant cohomology can be torsion-free or reflexive without being free as a module over H * (BT ). We determine the exact syzygy order of the equivariant cohomology of a big polygon space in terms of the length vector defining it. The proof uses a refined characterization of syzygies in terms of certain linearly independent elements in H 2 (BT ) adapted to the isotropy groups occurring in a given T -space.

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Syzygies in Equivariant Cohomology for Non-abelian Lie Groups

Cited by 11 publications

References 33 publications

A Quotient Criterion for Syzygies in Equivariant Cohomology

A Quotient Criterion for Syzygies in Equivariant Cohomology

Symmetric Products of Equivariantly Formal Spaces

The syzygy order of big polygon spaces

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