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Feifei Fan

Publications and source records attributed to Feifei Fan.

18 recordsLinked to original sources
arXiv2025

On the integral cohomology of real toric manifolds

Real toric manifolds are the real loci of nonsingular complete toric varieties. We compute the integral cohomology groups of real toric manifolds from the combinatorial data encoded by the underlying simplicial fans, generalizing a formula due to Cai and Choi. Furthermore, we determine the image of the integral cohomology in the $\mathbb{Z}/2$-cohomology under the mod $2$ reduction map. As an application, a combinatorial-algebraic criterion is established for when a real toric manifold is $\mathrm{spin}^c$. We also describe the product structure of their integral cohomology rings. In particular, we establish a simple combinatorial formula for the quotient of the cohomology ring by the ideal consisting of elements annihilated by $2$.

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arXiv2025

A formula for the mod $p$ cohomology of $BPU(p)$

We study the mod $p$ cohomology ring of the classifying space $BPU(p)$ of the projective unitary group $PU(p)$, when $p$ is an odd prime. We prove a mod $p$ formula analogous to a formula of Vistoli for the integral cohomology ring of $BPU(p)$. As an application, we give a simple topological proof of Vistoli's formula.

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arXiv2025

A counterexample to a conjecture of Adams

A conjecture due to J. F. Adams says that, for any odd prime $p$, the mod $p$ cohomology ring of the classifying space of a connected compact Lie group is detected by its elementary abelian $p$-subgroups. In this paper, we show that the mod $3$ cohomology ring of the classifying space of the projective unitary group $PU(9)$ is not detected by its elementary abelian $3$-subgroups, providing a counterexample to this conjecture. We also obtain many algebraic results as byproducts.

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arXiv2024

Operators on symmetric polynomials and applications in computing the cohomology of $BPU_n$

This paper studies the integral cohomology ring of the classifying space $BPU_n$ of the projective unitary group $PU_n$. By calculating a Serre spectral sequence, we determine the ring stucture of $H^*(BPU_n;\mathbb{Z})$ in dimensions $\leq 11$. For any odd prime $p$, we also determine the $p$-primary subgroups of $H^i(BPU_n;\mathbb{Z})$ in the range $i\leq 2p+13$ for $i$ odd and $i\leq 4p+8$ for $i$ even. The main technique used in the calculation is applying the theory of Young diagrams and Schur polynomials to certain linear operators on symmetric polynomials.

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arXiv2024

Cohomology of $BPU_n$ and rings of invariants of Weyl groups

Let $PU_n$ denote the projective unitary group of rank $n$ and $BPU_n$ be its classifying space, for $n>1$. Using the Serre spectral sequence associated to the fibration $BU_n\to BPU_n\to K(\mathbb{Z},3)$, we compute the integral cohomology group of $BPU_n$ in dimensions $\leq 14$. In addition, we determine the ring structure of $H^*(BPU_n;\mathbb{Z})$ up to dimension $13$ by computing the ring of invariants $H^*(BT_{PU_n})^W$ of the Weyl group action in dimensions $\leq 12$.

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arXiv2024

The generic anisotropy of strongly edge decomposable spheres

The generic anisotropy is an important property in the study of Stanley-Reisner rings of homology spheres, which was introduced by Papadakis and Petrotou. This property can be used to prove the strong Lefschetz property as well as McMullen's $g$-conjecture for homology spheres. It is conjectured that for an arbitrary field $\mathbb{F}$, any $\mathbb{F}$-homology sphere is generically anisotropic over $\mathbb{F}$. In this paper, we prove this conjecture for all strongly edge decomposable spheres.

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arXiv2022

On the anisotropy and Lefschetz property for PL-spheres

A simplicial sphere $Δ$ is said to be generically anisotropic over a field $\mathbb{F}$ if, for a certain purely transcendental field extension $\mathbf{k}$ of $\mathbb{F}$, a certain Artinian reduction $A$ of the face ring $\mathbf{k}[Δ]$ has the following property: For every nonzero homogeneous element $α\in A$ of degree at most $(\dimΔ+1)/2$, its square $α^2$ is also nonzero. The importance of this property is that the hard Lefschetz property for simplicial spheres can be derived from it. A recent result of Papadakis and Petrotou shows that every simplicial sphere is generically anisotropic over any field of characteristic $2$. In this paper, we give an equivalent condition of being generically anisotropic, and use it to present a simplified proof of Papadakis-Petrotou theorem for PL-spheres. We also prove that the simplicial spheres of dimension $2$ are generically anisotropic over any field $\mathbb{F}$.

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arXiv2021

Weak Lefschetz property of PL-spheres

A recent result of Papadakis-Petrotou shows that every simplicial sphere has the weak Lefschetz property in characteristic $2$. In this paper, we give a simpler proof of this result for PL-spheres by showing that the weak Lefschetz property in characteristic $2$ is preserved by bistellar moves. Several applications are given.

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arXiv2020

Toric spaces and face enumeration on simplicial manifolds

In this paper, we study the well-know $g$-conjecture for rational homology spheres in a topological way. To do this, we construct a class of topological spaces with torus actions, which can be viewed as topological generalizations of toric varieties. Along this way we prove that after doing stellar subdivision operations at some middle dimensional faces of an arbitrary rational homology sphere, the $g$-conjecture is valid. Furthermore, we give topological proofs of several fundamental algebraic results about Buchsbaum complexes and simplicial manifolds. In this process, we also get a few interesting results in toric topology.

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arXiv2020

Cohomological rigidity of manifolds with torus actions: I

We study the cohomological rigidity problem of two families of manifolds with torus actions: the so-called moment-angle manifolds, whose study is linked with combinatorial geometry and combinatorial commutative algebra; and topological toric manifolds, which are topological generalizations of toric varieties. In this paper we prove that when a simplicial sphere satisfies certain combinatorial conditions, the corresponding moment-angle manifold and topological toric manifolds are cohomologically rigid, i.e. their homeomorphism classes in their own families are determined by their cohomology rings. In the case of toric varieties, cohomology even determine the isomorphism classes of varieties. Our main strategy is to show that the combinatorial types of these simplicial spheres are determined by the $\mathrm{Tor}$-algebras of their face rings. This turns out to be a solution to a known problem in combinatorial commutative algebra for a class of spheres.

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arXiv2016

The topology of the moment-angle manifolds--On a conjecture of S. Gitler ans S. Lopez

Let $P$ be a simple polytope of dimension $n$ with $m$ facets and $P_{v}$ be a polytope obtained from $P$ by cutting off one vertex $v$. Let $Z=Z(P)$ and $Z_{v}=Z(P_{v})$ be the corresponding moment-angle manifolds. In \cite{[GL]} S.Gitler and S.López conjectured that: $Z_{v}$ is diffeomorphic to $\partial[(Z-int(D^{n+m}))\times D^{2}]\sharp \mathop{\sharp} \limits_{j=1}^{m-n} \binom{m-n}{j} (S^{j+2}\times S^{m+n-j-1})$, and they have proved the conjecture in the case $m<3n$. In this paper we prove the conjecture in general case.

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arXiv2016

On the cohomology of moment-angle complexes associated to Gorenstein* complexes

The main goal of this article is to study the cohomology rings and their applications of moment-angle complexes associated to Gorenstein* complexes, especially, the applications in combinatorial commutative algebra and combinatorics. First, we give a topological characterization of Gorenstein* complexes in terms of Alexander duality (as an application we give a topological proof of Stanley's Theorem). Next we give some cohomological transformation formulae of $\mathcal {Z}_{K}$, which are induced by some combinatorial operations on the Gorenstein* complex $K$, such as the connected sum operation and stellar subdivisions. We also prove that $\mathcal {Z}_{K}$ is a prime manifold whenever $K$ is a flag $2$-sphere by proving the indecomposability of their cohomology rings. Then we use these results to give the unique decomposition of the cohomology rings of moment-angle manifolds associated to simplicial $2$-spheres, and explain how to use it to detect the cohomological rigidity problem of these moment-angle manifolds.

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arXiv2015

$B$-Rigidity of flag $2$-spheres without $4$-belt

Associated to every finite simplicial complex $K$, there is a moment-angle complex $\mathcal {Z}_{K}$. In this paper, we use some algebraic invariants to solve the $B$-rigidity problem for some special simplicial compelexes.

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arXiv2015

Homology groups of simplicial complements: A new proof of Hochster theorem

In this paper, we consider homology groups induced by the exterior algebra generated by a simplicial compliment of a simplicial complex $K$. These homology groups are isomorphic to the Tor-groups $\mathrm{Tor}_{i, J}^{\mathbf{k}[m]}(\mathbf{k}(K),\mathbf{k})$ of the face ring $\mathbf{k}(K)$, which is very useful and much studied in toric topology. By using $\check{C}ech$ homology theory and Alexander duality theorem, we prove that these homology groups have dualities with the simplicial cohomology groups of the full subcomplexes of $K$. Then we give a new proof of Hochster's theorem.

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arXiv2014

Simplicial (co)homeology groups: New P.L. homeomorphism invariants of polyhedra

In this paper, we define (reduced) homeology groups and (reduced) cohomeology groups on finite simpicial complexes and prove that these groups are PL homeomorphsm invariants of polyhedra, while they are not homotopy invariants. So these groups can reflect some information that (co)homology groups can not tell. We also define homeotopy type of polyhedra which is finer than homotopy type but coarser than homeomorphism class, and prove that (co)homeology groups are actually homeotopy invariants. In the last section of this paper, we give a geometric description of some special (co)homeology groups.

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arXiv2014

Moment-angle manifolds and connected sums of sphere products

This paper investigates the moment-angle manifolds whose cohomology ring is isomorphic to that of a connected sum of sphere products. We first give a example of moment-angle manifolds corresponding to a 4 dimentional simplicial polytope. It has the property that its cohomology ring is isomorphic to that of a connected sum of sphere products with one produt of thress spheres. Finally, we give some general properties of this kind of moment-angle manifolds.

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arXiv2011

Simplicial Homeology and Homeotopy

In this paper, we define homeology group, reduced homeology group, cohomeology group and reduced cohomeology group on finite simpicial complexes and prove that these groups are homeomorphism invariants of polyhedra. We also define homeotopy type of polyhedra which is finer than homotopy type but coarser than homeomorphism class.

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