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Zhi Lü

Publications and source records attributed to Zhi Lü.

At least 19 recordsLinked to original sources
arXiv2026

Equivariant Milnor map

The Milnor map is the homomorphism from the unitary bordism ring to the unoriented bordism ring, halving the dimension, that maps the unitary bordism classes of the complex Milnor hypersurfaces to the unoriented bordism classes of their real points. In this work, we propose to generalize this construction to the equivariant setup and we show the existence of such a map for the equivariant unitary groups of the circle and the cyclic group of order two. Furthermore, we relate the kernel of these Milnor maps to the magnetic unitary equivariant bordism groups of free conjugations.

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arXiv2026

Homological Description of Equivariant Geometric Bordism

Using the evenness criterion in \cite{LCLS} and duality principle in \cite{CLT}, we construct a finite chain complex $\mathfrak{B}_{\bullet}((\mathbb{Z}_2)^k)$ built from the universal complex $X((\mathbb{Z}_2)^k)$ and its links. We then show that the equivariant unoriented bordism group $\mathcal{Z}_{k+1}((\mathbb{Z}_2)^k)$ of all $(k+1)$-dimensional smooth closed connected manifolds with effective $(\mathbb{Z}_2)^k$-actions fixing isolated points, is naturally isomorphic to $H_{k-2}(\mathfrak{B}_{\bullet}((\mathbb{Z}_2)^k);\mathbb{Z}_2)$. The vertical spectral sequence associated with the natural double-complex structure on $\mathfrak{B}_\bullet((\mathbb{Z}_2)^k)$ collapses at $E^3$, yielding an explicit formula for $\dim_{\mathbb{Z}_2}\mathcal{Z}_{k+1}((\mathbb{Z}_2)^k)$ for every $k$.

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arXiv2026

Representation characterization of equivariant geometric bordism

Let $G_k=(\mathbb Z_2)^k$. We establish a representation characterization determining when finitely many faithful $G_k$-representations can serve as the fixed-point data of a smooth closed $G_k$-manifold. By partitioning representations via multiplicities of nonzero weights $ρ$ and isomorphic restrictions to $\ker ρ$, we formulate an evenness condition; realizability is equivalent to this condition, yielding a finite local restatement of the tom Dieck--Kosniowski--Stong integrality criterion. As an application, we prove that $\dim_{\mathbb Z_2}\mathcal Z_4(G_3)=32$ and exhibit 32 basis elements represented by $G_3$-actions on $\mathbb RP^2\times\mathbb RP^2$, $\mathbb RP^4$ and $\mathbb RP(γ\oplusγ\oplusγ\oplus\underline{\mathbb R})$, obtained from three basic actions by precomposition with automorphisms of $G_3$.

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arXiv2026

On the toric lifting properties for simplicial $3$-spheres

We study the lifting problem for mod $2$ characteristic maps over simplicial $3$-spheres. Using a bad-block partition of the universal complex $X(\mathbb{Z}_2^4)$, we prove an avoidance criterion for liftability. We show that every simplicial $3$-sphere with at most $20$ vertices has the toric lifting property. We also obtain image-size and join-type results, and prove sharpness of the image-size bound in the universal-complex sense.

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arXiv2026

Reduced characteristic number criteria for equivariant bordism of $T^k$- and $(\mathbb{Z}_2)^k$-manifolds with isolated fixed points

Classical equivariant bordism theories require computing the full collection of equivariant characteristic numbers to detect whether an equivariant manifold bounds equivariantly or not. This paper establishes simplified equivariant bordism characterizations for two families of equivariant manifolds with isolated fixed points: unitary $T^k$-manifolds and closed smooth $(\mathbb{Z}_2)^k$-manifolds. For any unitary $T^k$-manifold $M$ with isolated fixed points, we establish an equivariant unitary bordism criterion built entirely from a single polynomial of equivariant Chern classes. We further introduce the minimal distinguishing degree and obtain two key inequalities that capture the interplay between $\dim M$ and the Euler characteristic $χ(M)$ through this minimal distinguishing degree. These inequalities settle the existence problem of a linear lower bound for $χ(M)$ within the framework of Kosniowski's conjecture and partially verify the conjecture under natural admissible assumptions. We also provide an alternative proof settling the toric generalization of Kosniowski's conjecture when $\dim M=2k$. By contrast, for a closed smooth $(\mathbb{Z}_2)^k$-manifold with isolated fixed points, we derive a more concise equivariant bordism criterion relying solely on the powers of the top equivariant Stiefel-Whitney class. Our new criteria substantially reduce computational demands.

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arXiv2026

Non-finitely generated $(\mathbb{Z}_2)^k$-equivariant bordism ring

In 1998, Mukherjee and Sankaran posed two problems concerning the algebraic structure of the equivariant bordism ring of smooth closed $(\mathbb{Z}_2)^k$-manifolds with only isolated fixed points. One is the property of being finitely generated as a $\mathbb{Z}_2$-algebra, and the other is the existence of indecomposable elements. This paper definitively resolves both problems for the fully effective case. Specifically, let $\mathcal{Z}_*((\mathbb{Z}_2)^k)$ denote the equivariant bordism ring of smooth closed manifolds equipped with fully effective smooth $(\mathbb{Z}_2)^k$-actions having only isolated fixed points. We prove that $\mathcal{Z}_*((\mathbb{Z}_2)^k)$ is not finitely generated as a $\mathbb{Z}_2$-algebra for all $k\geqslant 3$. Moreover, the proof explicitly constructs an infinite family of indecomposable elements with unbounded degrees, thereby settling the second problem simultaneously.

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arXiv2025

Equivariant bordism classification of five-dimensional $(\mathbb{Z}_2)^3$-manifolds with isolated fixed points

Denote by $\mathcal{Z}_5((\mathbb{Z}_2)^3)$ the group, which is also a vector space over $\mathbb{Z}_2$, generated by equivariant unoriented bordism classes of all five-dimensional closed smooth manifolds with effective smooth $(\mathbb{Z}_2)^3$-actions fixing isolated points. We show that $\dim_{\mathbb{Z}_2} \mathcal{Z}_5((\mathbb{Z}_2)^3) = 77$ and determine a basis of $\mathcal{Z}_5((\mathbb{Z}_2)^3)$, each of which is explicitly chosen as the projectivization of a real vector bundle. Thus this gives a complete classification up to equivariant unoriented bordism of all five-dimensional closed smooth manifolds with effective smooth $(\mathbb{Z}_2)^3$-actions with isolated fixed points.

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arXiv2025

Iterated residue, toric forms and Witten genus

We introduce the notion of {\em iterated residue} to study generalized Bott manifolds. When applying the iterated residues to compute the Borisov-Gunnells toric form and the Witten genus of certain toric varieties as well as complete intersections, we obtain interesting vanishing results and some theta function identities, one of which is a twisted version of a classical Rogers-Ramanujan type formula.

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arXiv2022

Equivariant Bordism of 2-Torus Manifolds and Unitary Toric Manifolds

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 $\mathcal{Z}_n(\mathbb{Z}_2^n)$ 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 $\mathcal{Z}_n^U(T^n)$ of unitary torus manifolds, and prove that $\mathcal{Z}_n^U(T^n)$ can be generated by quasitoric manifolds with omniorientations.

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arXiv2021

Cohomology ring of manifold arrangements

We study the cohomology ring of the complement $\mathcal{M}(\mathcal{A})$ of a manifold arrangement $\mathcal{A}$ in a smooth manifold $M$ without boundary. We first give the concept of monoidal cosheaf on a locally geometric poset $\mathfrak{L}$, and then define the generalized Orlik--Solomon algebra $A^*(\mathfrak{L}, \mathcal{C})$ over a commutative ring with unit, which is built by the classical Orlik--Solomon algebra and a monoidal cosheaf $\mathcal{C}$ as coefficients. Furthermore, we construct a monoidal cosheaf $\hat{\mathcal{C}}(\mathcal{A})$ associated with $\mathcal{A}$, so that the generalized Orlik--Solomon algebra $A^*(\mathfrak{L}, \hat{\mathcal{C}}(\mathcal{A}))$ becomes a double complex with suitable multiplication structure and the associated total complex $Tot(A^*(\mathfrak{L}, \hat{\mathcal{C}}(\mathcal{A})))$ is a differential algebra. Our main result is that $H^*(Tot(A^*(\mathfrak{L}, \hat{\mathcal{C}}(\mathcal{A}))))$ is isomorphic to $H^*(\mathcal{M}(\mathcal{A}))$ as algebras. Our argument is of topological with the use of a spectral sequence induced by a geometric filtration associated with $\mathcal{A}$. In particular, we also discuss the mixed Hodge complex structure on our model if $M$ and all elements in $\mathcal{A}$ are complex smooth varieties, and show that it induces the canonical mixed Hodge structure of $\mathcal{M}(\mathcal{A})$. As an application, we calculate the cohomology of chromatic configuration spaces, which agrees with many known results in some special cases. In addition, some explicit formulas with respect to Poincaré polynomial and chromatic polynomial are also given.

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arXiv2021

A proof of Kosniowski conjecture

Let $M$ be a unitary $S^1$-manifold with only isolated fixed points such that $M$ is not a boundary. We show that $4χ(M)>\dim M$, where $χ(M)$ is the Euler characteristic of $M$. This gives an affirmative answer of Kosniowski conjecture.

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arXiv2021

Twisted Milnor Hypersurface I

In this paper, we study {\bf twisted Milnor hypersurfaces} and compute their $\hat A$-genus and Atiyah-Singer-Milnor $α$-invariant. Our tool to compute the $α$-invariant is Zhang's analytic Rokhlin congruence formula. We also give some applications about group actions and metrics of positive scalar curvature on twisted Milnor hypersurfaces.

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arXiv2020

Topology and geometry of flagness and beltness of simple orbifolds

We consider a class of right-angled Coxeter orbifolds, named as simple orbifolds, which are a generalization of simple polytopes. Similarly to manifolds over simple polytopes, the topology and geometry of manifolds over simple orbifolds are closely related to the combinatorics and orbifold structure of simple orbifolds. We generalize the notions of flag and belt in the setting of simple polytopes into the setting of simple orbifolds. To describe the topology and geometry of a simple orbifold in terms of its combinatorics, we focus on {\em simple handlebodies} (that is, simple orbifolds which can be obtained from simple polytopes by gluing some disjoint specific codimension-one faces). We prove the following two main results in terms of combinatorics, which can be understood as "Combinatorial Sphere Theorem" and "Combinatorial Flat Torus Theorem" on simple handlebodies: (A) A simple handlebody is orbifold-aspherical if and only if it is flag. (B) There exists a rank-two free abelian subgroup in $π_1^{orb}(Q)$ of an orbifold-aspherical simple handlebody $Q$ if and only if it contains an $\square$-belt. Furthermore, based on such two results and some results of geometry, it is shown that the existence of some curvatures on a certain manifold cover (manifold double) over a simple handlebody $Q$ can be characterized in terms of the combinatorics of $Q$. In 3-dimensional case, together with the theory of hyperbolic 3-manifolds, we can induce a pure combinatorial equivalent description for a simple $3$-handlebody to admit a right-angled hyperbolic structure, which is a natural generalization of Pogorelov Theorem.

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arXiv2020

Crossing-changeable braids from chromatic configuration spaces

Motivated by the work in [15], this paper deals with the theory of the braids from chromatic configuration spaces. This kind of braids possess the property that some strings of each braid may intersect together and can also be untangled, so they are quite different from the ordinary braids in the sense of Artin. This enriches and extends the theory of ordinary braids.

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arXiv2019

A theory of orbit braids

This paper upbuilds the theoretical framework of orbit braids in $M\times I$ by making use of the orbit configuration space $F_G(M,n)$, which enriches the theory of ordinary braids, where $M$ is a connected topological manifold of dimension at least 2 with an effective action of a finite group $G$ and the action of $G$ on $I$ is trivial. Main points of our work include as follows. We introduce the orbit braid group $\mathcal{B}_n^{orb}(M,G)$, and show that it is isomorphic to a group with an additional endowed operation (called the extended fundamental group of $F_G(M,n)$), formed by the homotopy classes of some paths (not necessarily closed paths) in $F_G(M,n)$, which is an essential extension for fundamental groups. The orbit braid group $\mathcal{B}_n^{orb}(M,G)$ is large enough to contain the fundamental group of $F_G(M,n)$ and other various braid groups as its subgroups. Around the central position of $\mathcal{B}_n^{orb}(M,G)$, we obtain five short exact sequences weaved in a commutative diagram. We also analyze the essential relations among various braid groups associated to those configuration spaces $F_G(M,n), F(M/G,n)$, and $F(M,n)$. We finally consider how to give the presentations of orbit braid groups in terms of orbit braids as generators. We carry out our work by choosing $M=\mathbb{C}$ with typical actions of $\mathbb{Z}_p$ and $(\mathbb{Z}_2)^2$. We obtain the presentations of the corresponding orbit braid groups, from which we see that the generalized braid group $Br(B_n)$ actually agrees with an orbit braid group and $Br(D_n)$ is a subgroup of another orbit braid group. In addition, the notion of extended fundamental groups is also defined in a general way in the category of topology and some characteristics extracted from the discussions of orbit braids are given.

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arXiv2019

Equivariant cohomology Chern numbers determine equivariant unitary bordism for torus groups

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 bordism and cobordism. Our approach heavily exploits Quillen's geometric interpretation of homotopic unitary cobordism theory. Moreover, this method can also be applied to the study of $({\Bbb Z}_2)^k$-equivariant unoriented bordism and can still derive the classical result of tom Dieck.

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arXiv2018

Torus orbifolds with two fixed points

The main objects of this paper are torus orbifolds that have exactly two fixed points. We study the equivariant topological type of these orbifolds and consider when we can use the results of the paper [DKS] (arXiv:1809.03678) to compute its integral equivariant cohomology, in terms of generators and relations, coming from the corresponding orbifold torus graph.

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