By the classical result of Milnor and Novikov, the unitary cobordism ring is isomorphic to a graded polynomial ring with countably many generators: $Ω^U_*\simeq \mathbb Z[a_1,a_2,\dots]$, ${\rm deg}(a_i)=2i$. In this paper we solve a well-known problem of constructing geometric representatives for $a_i$ among smooth projective toric varieties, $a_n=[X^{n}], \dim_\mathbb C X^{n}=n$. Our proof uses a family of equivariant modifications (birational isomorphisms) $B_k(X)\to X$ of an arbitrary smooth complex manifold $X$ of (complex) dimension $n$ ($n\geq 2$, $k=0,\dots,n-2$). The key fact is that the change of the Milnor number under these modifications depends only on the dimension $n$ and the number $k$ and does not depend on the manifold $X$ itself.
Toric orbifolds are a generalization of simplicial projective toric varieties. In this paper, we show that there is a resolution of singularities of a toric orbifold. In a different category, the class of quasi-contact toric manifolds contains the class of good contact toric manifolds. We prove that a quasi-contact toric manifold is equivariantly a boundary. Moreover, we conclude that good contact toric manifolds and generalized lens spaces are equivariantly boundaries.
A general problem in complex cobordism theory is to find useful representatives for cobordism classes. One particularly convenient class of complex manifolds consists of smooth projective toric varieties. The bijective correspondence between these varieties and smooth polytopes allows us to examine which complex cobordism classes contain a smooth projective toric variety by studying the combinatorics of polytopes. These combinatorial properties determine obstructions to a complex cobordism class containing a smooth projective toric variety. However, the obstructions are only necessary conditio…
A smooth closed 3-manifold $M$ fibered by tori $T^2$ is characterized by an element $ϕ\in GL(2,\mathbb{Z})$. We show that $M$ is the boundary of a 4-manifold fibered by tori over a surface such that the bundle structure on $M$ is the restriction of the bundle structure on the 4-manifold if and only if $ϕ$ is from the commutator subgroup $(GL(2,\mathbb{Z}))'$. The notions of oriented and unoriented cobordisms in the class of closed 3-manifolds fibered by tori are introduced. It turns out that in this case the cobordisms form a group, namely $\mathbb{Z}_{12}$ in the oriented case and $\mathbb{Z}…
The present paper generalises the results of Ray and Buchstaber-Ray, Buchstaber-Panov-Ray in unitary cobordism theory. I prove that any class $x\in Ω^{*}_{U}$ of the unitary cobordism ring contains a quasitoric totally normally and tangentially split manifold.
Torus orbifolds are topological generalization of symplectic toric orbifolds. We give a construction of smooth orbifolds with torus actions whose boundary is a disjoint union of torus orbifolds using toric topological method. As a result, we show that any orientable locally standard torus orbifold is equivariantly cobordant to some copies of orbifold complex projective spaces. We also discuss some further equivariant cobordism results including the cases when torus orbifolds are actually torus manifolds.
Although it is well-known that the complex cobordism ring is a polynomial ring $Ω_{*}^{U}\cong\mathbb{Z}\left[α_{1},α_{2},\ldots\right]$, an explicit description for convenient generators $α_{1},α_{2},\ldots$ has proven to be quite elusive. The focus of the following is to construct complex cobordism polynomial generators in many dimensions using smooth projective toric varieties. These generators are very convenient objects since they are smooth connected algebraic varieties with an underlying combinatorial structure that aids in various computations. By applying certain torus-equivariant blo…
We apply results of Harada, Holm and Henriques to prove that the Atiyah-Segal equivariant complex $K$-theory ring of a divisive weighted projective space (which is singular for nontrivial weights) is isomorphic to the ring of integral piecewise Laurent polynomials on the associated fan. Analogues of this description hold for other complex-oriented equivariant cohomology theories, as we confirm in the case of homotopical complex cobordism, which is the universal example. We also prove that the Borel versions of the equivariant $K$-theory and complex cobordism rings of more general singular tori…
Baker and Richter construct a remarkable $A_\infty$ ring-spectrum $MΞ$ whose elements possess characteristic numbers associated to quasisymmetric functions; its relations, on one hand to the theory of noncommutative formal groups, and on the other to the theory of omnioriented (quasi)toric manifolds [in the sense of Buchstaber, Panov, and Ray], seem worth investigating.
This paper presents a commutative complex oriented cohomology theory with coefficients the quotient ring of complex cobordism MU$^*[1/2]$ modulo the ideal generated by any subsequence of any polynomial generators in special unitary cobordism MSU$^*[1/2]$ viewed as elements in MU$^*[1/2]$ by forgetful map.
Our aim is to bring the theory of analogous polytopes to bear on the study of quasitoric manifolds, in the context of stably complex manifolds with compatible torus action. By way of application, we give an explicit construction of a quasitoric representative for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection. We suggest a systematic description for omnioriented quasitoric manifolds in terms of combinatorial data, and explain the relationship with non-singular projective toric varieties (otherwise known as toric manifolds). By exp…
The cobordism ring of symplectic manifolds defined by V.L. Ginzburg is shown to be isomorphic to the Pontrjagin ring of complex-oriented manifolds with free circle actions. This suggests an interpretation of the formal group law of complex cobordism, in terms of a composition-law on semiclassical expansions. An appendix discusses related questions about cobordism of toric varieties.
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 $\mathbb{Z}_2^n$ 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 $\mathcal{Z}_n(\mathbb{Z}_2^n)$ of 2-torus manifolds, and determine the dimention of $\mathcal{Z}_n(\mathbb{Z}_2^n)$ as a $\mathbb{Z}_2$-vector space. With the help of toric topology, Lü and Tan proved that the bordism groups $\mathcal{Z}…
We extend work of Davis and Januszkiewicz by considering {\it omnioriented} toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an element of an equivariant cobordism ring. As an application, we compute the complex bordism groups and cobordism ring of an arbitrary omnioriented toric manifold. We consider a family of examples $B_{i,j}$, which are toric manifolds over products of simplices, and verify that their natural stably c…
We construct a new family of toric manifolds generating the unitary bordism ring. Each manifold in the family is the complex projectivisation of the sum of a line bundle and a trivial bundle over a complex projective space. We also construct a family of special unitary quasitoric manifolds which contains polynomial generators of the special unitary bordism ring with 2 inverted in dimensions >8. Each manifold in the latter family is obtained from an iterated complex projectivisation of a sum of line bundles by amending the complex structure to make the first Chern class vanish.
We consider compact homogeneous spaces G/H, where G is a compact connected Lie group and H is its closed connected subgroup of maximal rank. The aim of this paper is to provide an effective computation of the universal toric genus for the complex, almost complex and stable complex structures which are invariant under the canonical left action of the maximal torus T^k on G/H. As it is known, on G/H we may have many such structures and the computations of their toric genus in terms of fixed points for the same torus action give the constraints on possible collections of weights for the correspon…
Our primary aim is to develop a theory of equivariant genera for stably complex manifolds equipped with compatible actions of a torus T^k. In the case of omnioriented quasitoric manifolds, we present computations that depend only on their defining combinatorial data; these draw inspiration from analogous calculations in toric geometry, which seek to express arithmetic, elliptic, and associated genera of toric varieties in terms only of their fans. Our theory focuses on the universal toric genus Φ, which was introduced independently by Krichever and Loeffler in 1974, albeit from radically diffe…
A quasitoric manifold is a smooth 2n-manifold M^{2n} with an action of the compact torus T^n such that the action is locally isomorphic to the standard action of T^n on C^n and the orbit space is diffeomorphic, as manifold with corners, to a simple polytope P^n. The name refers to the fact that topological and combinatorial properties of quasitoric manifolds are similar to that of non-singular algebraic toric varieties (or toric manifolds). Unlike toric varieties, quasitoric manifolds may fail to be complex; however, they always admit a stably (or weakly almost) complex structure, and their co…
This paper shows that the integral equivariant cohomology Chern numbers completely determine the equivariant geometric unitary bordism classes of closed unitary $G$-manifolds, which gives an affirmative answer to the conjecture posed by Guillemin--Ginzburg--Karshon in [20, Remark H.5, $\S3$, Appendix H], where $G$ is a torus. As a further application, we also obtain a satisfactory solution of [20, Question (A), $\S1.1$, Appendix H] on unitary Hamiltonian $G$-manifolds. Our key ingredients in the proof are the universal toric genus defined by Buchstaber--Panov--Ray and the Kronecker pairing of …
We describe Bott towers as sequences of toric manifolds M^k, and identify the omniorientations which correspond to their original construction as toric varieties. We show that the suspension of M^k is homotopy equivalent to a wedge of Thom complexes, and display its complex K-theory as an algebra over the coefficient ring. We extend the results to KO-theory for several families of examples, and compute the effects of the realification homomorphism; these calculations breathe geometric life into Bahri and Bendersky's recent analysis of the Adams Spectral Sequence. By way of application we inves…
This paper presents a commutative complex-oriented cohomology theory that realizes the Buchstaber formal group law localized away from 2. Also, the restriction of the classifying map of FB on special unitary cobordism ring localized away from 2 defines a four parameter genus, studied by Hoehn and Totaro.
Continuing work begin in arXiv:1910.12609, we interpret the Hurewicz homomorphism for Baker and Richter's noncommutative complex cobordism spectrum $Mξ$ in terms of characteristic numbers (indexed by quasi-symmetric functions) for complex-oriented quasitoric manifolds, and show that automorphisms or cohomology operations on this representation are defined by a `renormalization' Hopf algebra of formal diffeomorphisms at the origin of the noncommutative line, previously considered (over $Q$) in quantum electrodynamics. The resulting structure can be presented in purely algebraic terms, as a grou…
We investigate geometrical interpretations of various structure maps associated with the Landweber-Novikov algebra S^* and its integral dual S_*. In particular, we study the coproduct and antipode in S_*, together with the left and right actions of S^* on S_* which underly the construction of the quantum (or Drinfeld) double D(S^*). We set our realizations in the context of double complex cobordism, utilizing certain manifolds of bounded flags which generalize complex projective space and may be canonically expressed as toric varieties. We discuss their cell structure by analogy with the class…
Toric topology emerged in the end of the 1990s on the borders of equivariant topology, algebraic and symplectic geometry, combinatorics and commutative algebra. It has quickly grown up into a very active area with many interdisciplinary links and applications, and continues to attract experts from different fields. The key players in toric topology are moment-angle manifolds, a family of manifolds with torus actions defined in combinatorial terms. Their construction links to combinatorial geometry and algebraic geometry of toric varieties via the related notion of a quasitoric manifold. Discov…
For any (n+1)-dimensional weight vector χ of positive integers, the weighted projective space P(χ) is a projective toric variety, and has orbifold singularities in every case other than CP^n. We study the algebraic topology of P(χ), paying particular attention to its localisation at individual primes p. We identify certain p-primary weight vectors π for which P(π) is homeomorphic to an iterated Thom space over S^2, and discuss how any P(χ) may be reconstructed from its p-primary factors. We express Kawasaki's computations of the integral cohomology ring H^*(P(χ);Z) in terms of iterated Thom is…
Batyrev constructed a family of Calabi-Yau hypersurfaces dual to the first Chern class in toric Fano varieties. Using this construction, we introduce a family of Calabi-Yau manifolds whose SU-bordism classes generate the special unitary bordism ring $\varOmega^{SU}\otimes\mathbb{Z}[\frac{1}{2}]\cong\mathbb{Z}[\frac{1}{2}][y_{i}\colon i\ge 2]$. We also describe explicit Calabi-Yau representatives for multiplicative generators of the SU-bordism ring in low dimensions.
