Torus manifolds in equivariant complex bordism

Darby, Alastair

doi:10.1016/j.topol.2015.03.014

Cited by 6 publications

(7 citation statements)

References 22 publications

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“…consists of all classes that can be represented by unitary T n -manifolds with finite fixed point set. In his Ph.D's thesis [6], Darby refined Hanke's result to the following pull back square with all maps injective.…”

Section: Equivariant Bordism Of Unitary Torus Manifoldsmentioning

confidence: 97%

“…To eliminate the sign ε(p) of a fixed point p and absorb the effect of orientations in the image ϕ Ω ([M ]), consider the exterior algebra Λ * Z (J C n ) as in Darby [6].…”

Section: Theorem 31 ([6]mentioning

confidence: 99%

“…Let Z U * (T n ) = m≥0 Z U m (T n ) denote the equivariant unitary bordism ring of unitary T n -manifolds with finite fixed point set, then Z U n (T n ) is exactly the equivariant unitary bordism group of unitary toric manifolds. By analogy with the 2-torus manifolds, Darby [6] generalized the methods in [14] to the study of Z U n (T n ). To deal with the effect of orientations and stably complex structures of the unitary manifolds, Darby considered the exterior algebra Λ * Z (J C n ) which is generated by the exterior product of irreducible T n -representations.…”

Section: Introductionmentioning

confidence: 99%

“…For n = 1 and 2, the group Z U n (T n ) was shown to be generated by quasitoric manifolds with omniorientations, and each class of Z U n (T n ) contains a quasitoric manifold with omniorientation as its representative. But for n > 2, it can only be conjectured that each class of Z U n (T n ) contains a quasitoric manifold with omniorientation as its representative (see [6]).…”

Section: Introductionmentioning

confidence: 99%

“…The homotopy type of X(Z n ) and Theorem 1.4 inspire that Z U 2n (T n ) is generated by quasitoric manifold with omniorientations (see Proposition 3.13). The generalized connect sums introduced by Buchstaber-Ray [2] and Darby [6] suggest that the sum of two class [M 1 ] and [M 2 ] can still be represented by a quasitoric manifold with omniorientations, if [M 1 ] and [M 2 ] are represented by quasitoric manifolds with omniorientations. So we can prove the following proposition and solve the conjecture that Z U n (T n ) is generated by quasitoric manifolds with omniorientations .…”

Section: Introductionmentioning

confidence: 99%

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Equivariant Bordism of 2-Torus Manifolds and Unitary Toric Manifolds

Chen¹,

Li²,

Tan³

2022

Preprint

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The equivariant bordism classification of manifolds with group actions is an essential subject in the study of transformation groups. We are interesting in the action of 2-torus group Z n 2 and torus group T n , and study the equivariant bordism of 2-torus manifolds and unitary toric manifolds. In this paper, we give a new description of the group Zn(Z n 2 ) of 2-torus manifolds, and determine the dimention of Zn(Z n2 ) as a Z 2 -vector space. With the help of toric topology, Lü and Tan proved that the bordism groups Zn(Z n 2 ) are generated by small covers. We will give a new proof to this result. These results can be generalized to the equivariant bordism of unitary toric manifolds, that is, we will give a new description of the group Z U n (T n ) of unitary torus manifolds, and prove that Z U n (T n ) can be generated by quasitoric manifolds with omniorientations.

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Section: Equivariant Bordism Of Unitary Torus Manifoldsmentioning

confidence: 97%

“…To eliminate the sign ε(p) of a fixed point p and absorb the effect of orientations in the image ϕ Ω ([M ]), consider the exterior algebra Λ * Z (J C n ) as in Darby [6].…”

Section: Theorem 31 ([6]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Equivariant Bordism of 2-Torus Manifolds and Unitary Toric Manifolds

Chen¹,

Li²,

Tan³

2022

Preprint

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Number of fixed points for unitary Tn−1-manifold

Wen

2019

Front. Math. China

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Almost complex torus manifolds - a problem of Petrie type

Jang

2024

Proc. Amer. Math. Soc.

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The Petrie conjecture asserts that if a homotopy C P n \mathbb {CP}^n admits a non-trivial circle action, its Pontryagin class agrees with that of C P n \mathbb {CP}^n . Petrie proved this conjecture in the case where the manifold admits a T n T^n -action. An almost complex torus manifold is a 2 n 2n -dimensional compact connected almost complex manifold equipped with an effective T n T^n -action that has fixed points. For an almost complex torus manifold, there exists a graph that encodes information about the weights at the fixed points. We prove that if a 2 n 2n -dimensional almost complex torus manifold M M only shares the Euler number with the complex projective space C P n \mathbb {CP}^n , the graph of M M agrees with the graph of a linear T n T^n -action on C P n \mathbb {CP}^n . Consequently, M M has the same weights at the fixed points, Chern numbers, cobordism class, Hirzebruch χ y \chi _y -genus, Todd genus, and signature as C P n \mathbb {CP}^n , endowed with the standard linear action. Furthermore, if M M is equivariantly formal, the equivariant cohomology and the Chern classes of M M and C P n \mathbb {CP}^n also agree.

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Torus manifolds in equivariant complex bordism

Cited by 6 publications

References 22 publications

Equivariant Bordism of 2-Torus Manifolds and Unitary Toric Manifolds

Equivariant Bordism of 2-Torus Manifolds and Unitary Toric Manifolds

Number of fixed points for unitary Tn−1-manifold

Almost complex torus manifolds - a problem of Petrie type

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