On the Rational Type 0f Moment Angle Complexes

The homology of simpicial complements and the cohomology of moment-angle complexes

We use the Taylor resolution of a monomial ideal to compute the Tor algebra of the Stanley-Reisner ring of a simplicial complement of a simplicial complex.

与旗复形相关的力矩角复形的环路空间分解

Loop space decompositions of moment-angle complexes associated to flag complexes

本文证明了与旗复形 $k$-骨架相关的力矩角复形的环路空间属于类 $\mathcal{P}$，并通过证明 $\mathcal{P}$ 在收缩下封闭来完成。

Rational approximations of sectional category and Poincaré duality

Félix, Halperin, and Lemaire have shown that the rational module category Mcat and the rational Toomer invariant $e_0$ coincide for simply connected Poincaré duality complexes. We establish an analogue of this result for the sectional category of a fibration.

Toral rank conjecture for moment-angle complexes

We consider an operation K \to L(K) on the set of simplicial complexes, which we call the "doubling operation". This combinatorial operation has been recently brought into toric topology by the work of Bahri, Bendersky, Cohen and Gitler on generalised moment-angle complexes (also known as K-powers). The crucial property of the doubling operation is that the moment-angle complex Z_K can be identified with the real moment-angle complex RZ_L(K) for the double L(K). As an application we prove the toral rank conjecture for Z_K by estimating the lower bound of the cohomology rank (with rational coef…

Rational formality of mapping spaces

Let X and Y be finite nilpotent CW complexes with dimension of X less than the connectivity of Y. Generalizing results of Vigué-Poirrier and Yamaguchi, we prove that the mapping space Map(X,Y) is rationally formal if and only if Y has the rational homotopy type of a finite product of odd dimensional spheres.

Two-loop part of the rational homotopy of spaces of long embeddings

Arone and Turchin defined graph-complexes computing the rational homotopy of the spaces of long embeddings. The graph-complexes split into a direct sum by the number of loops in graphs. In this paper we compute the homology of its two-loop part.

矩角复形的微分分次代数模型

Dga models for moment-angle complexes

本文证明矩角复形的整奇异上链微分分次代数模型可表示为Stanley-Reisner环与外代数的扭曲张量积，并将该结果推广到实矩角复形，同时比较了若干上同调环描述。

Genus of The Hypercube Graph And Real Moment-Angle Complexes

In this paper we demonstrate a calculation to find the genus of the hypercube graph $Q_n$ using real moment-angle complex $\mathcal{Z}_\mathcal{K}(D^1,S^0)$ where $\mathcal{K}$ is the boundary of an $n$-gon. We also calculate an upper bound for the genus of the quotient graph $Q_n/C_n$, where $C_n$ represents the cyclic group with $n$ elements.

The homology of simplicial complement and the cohomology of the moment-angle complexes

A simplicial complement P is a sequence of subsets of [m] and the simplicial complement P corresponds to a unique simplicial complex K with vertices in [m]. In this paper, we defined the homology of a simplicial complement $H_{i,σ}(Λ^{*,*}[P], d)$ over a principle ideal domain k and proved that $H_{*,*}(Λ[P], d)$ is isomorphic to the Tor of the corresponding face ring k(K) by the Taylor resolutions. As applications, we give methods to compute the ring structure of Tor_{*,*}^{k[x]}(k(K), k)$, $link_{K}σ$, $star_{K}σ$ and the cohomology of the generalized moment-angle complexes.

Iterated Higher Whitehead products in topology of moment-angle complexes

In this paper we study the topological structure of moment-angle complexes $\mathcal{Z_K}$. We consider two classes of simplicial complexes. The first class $B_Δ$ consists of simplicial complexes $\mathcal{K}$ for which $\mathcal{Z_K}$ is homotopy equivalent to a wedge spheres. The second class $W_Δ$ consists of $\mathcal{K}\in B_Δ$ such that all spheres in the wedge are realized by iterated higher Whitehead products. Buchstaber and Panov asked if it is true that $B_Δ = W_Δ$. In this paper we show that this is not the case. Namely, we give an example of a simplicial complex whose corresponding…

关于三元组上Kneser图独立复形的有理同调

The rational homology of independence complexes of Kneser graphs on triples

该研究针对三元组上的Kneser图独立复形，在有理数域上肯定回答了Feng和Wang提出的同调非平凡维度问题，确定了其非零Betti数，通过多种同调工具完成了证明。

Families of minimally non-Golod complexes and their polyhedral products

We consider families of simple polytopes $P$ and simplicial complexes $K$ well-known in polytope theory and convex geometry, and show that their moment-angle complexes have some remarkable homotopy properties which depend on combinatorics of the underlying complexes and algebraic properties of their Stanley--Reisner rings. We introduce infinite series of Golod and minimally non-Golod simplicial complexes $K$ with moment-angle complexes $\mathcal Z_K$ having free integral cohomology but not homotopy equivalent to a wedge of spheres or a connected sum of products of spheres respectively. We then…

m-structures determine integral homotopy type

This paper proves that the functor $C(*)$ that sends pointed, simply-connected CW-complexes to their chain-complexes equipped with diagonals and iterated higher diagonals, determines their integral homotopy type --- even inducing an equivalence of categories between the category of CW-complexes up to homotopy equivalence and a certain category of chain-complexes equipped with higher diagonals. Consequently, $C(*)$ is an algebraic model for integral homotopy types similar to Quillen's model of rational homotopy types. For finite CW complexes, our model is finitely generated. Our result implies …

On Davis-Januszkiewicz homotopy types I; formality and rationalisation

For an arbitrary simplicial complex K, Davis and Januszkiewicz have defined a family of homotopy equivalent CW-complexes whose integral cohomology rings are isomorphic to the Stanley-Reisner algebra of K. Subsequently, Buchstaber and Panov gave an alternative construction (here called c(K)), which they showed to be homotopy equivalent to Davis and Januszkiewicz's examples. It is therefore natural to investigate the extent to which the homotopy type of a space is determined by having such a cohomology ring. We begin this study here, in the context of model category theory. In particular, we ext…

Equivariant cohomology of moment-angle complexes with respect to coordinate subtori

We compute the equivariant cohomology $H^*_{T_I}(\mathcal Z_K)$ of moment-angle complexes $\mathcal Z_K$ with respect to the action of coordinate subtori $T_I \subset T^m$. We give a criterion for the equivariant formality of $\mathcal Z_K$ and obtain specifications for the cases of flag complexes and graphs.

Realizing spaces as classifying spaces

Which spaces occur as a classifying space for fibrations with a given fibre? We address this question in the context of rational homotopy theory. We construct an infinite family of finite complexes realized (up to rational homotopy) as classifying spaces. We also give several non-realization results, including the following: the rational homotopy types of $\mathbb{C}P^2$ and $S^4$ are not realized as the classifying space of any simply connected, rational space with finite-dimensional homotopy groups.

On the cohomology of quotients of moment-angle complexes

We describe the cohomology of the quotient Z_K/H of a moment-angle complex Z_K by a freely acting subtorus H in T^m by establishing a ring isomorphism of H*(Z_K/H,R) with an appropriate Tor-algebra of the face ring R[K], with coefficients in an arbitrary commutative ring R with unit. This result was stated in [BP02, 7.37] for a field R, but the argument was not sufficiently detailed in the case of nontrivial H and finite characteristic. We prove the collapse of the corresponding Eilenberg-Moore spectral sequence using the extended functoriality of Tor with respect to `strongly homotopy multipl…

The rational homotopy type of the space of self-equivalences of a fibration

Let Aut(p) denote the topological monoid of self-fibre-homotopy equivalences of a fibration p:E\to B. We make a general study of this monoid, especially in rational homotopy theory. When E and B are simply connected CW complexes with E finite, we identify the rational Samelson Lie algebra of the identity component of Aut(p) as the homology of a certain DG Lie algebra of derivations arising from the Koszul-Sullivan model of p. We obtain related identifications for the rational homotopy groups of fibrewise mapping spaces and for the rationalization of a natural nilpotent subgroup of Aut(p).

Lowest-degree triple Massey products in moment-angle complexes

We give a combinatorial classification of non-trivial triple Massey products of three dimensional classes in the cohomology of a moment-angle complex. This work improves on a result by Denham and Suciu (2007) by considering triple Massey products with non-trivial indeterminacy.

$\frac{n}{3}$-Neighbourly Moment-Angle Complexes and their Unstable Splittings

Given an $\frac{n}{3}$-neighbourly simplicial complex $K$ on vertex set $[n]$, we show that the moment-angle complex $\mathcal{Z}_K$ is a $co$-$H$-space if and only if $K$ satisfies a homotopy analogue of the Golod property.

Rational Homotopy Type of the Classifying Space for Fibrewise Self-Equivalences

Let p be a fibration of simply connected CW complexes with finite base B and fibre F. Let aut_1(p) denote the identity component of the space of all fibre-homotopy self-equivalences of p and Baut_1(p) the classifying space for this topological monoid. We give a differential graded Lie algebra model for Baut_1(p). We use this model to give classification results for the rational homotopy types represented by Baut_1(p) and also to obtain conditions under which the monoid aut_1(p) is a double loop-space after rationalization.

角度复形与紧流形的对偶性

Moment angle complexes and duality for tight manifolds

研究角度复形的同调秩与紧流形三角化的关系，提出新的双同调对偶定理。

关于矩角复形双上同调的秩

On the rank of the double cohomology of moment-angle complexes

该文研究矩角复形双上同调秩在粘贴n-单纯形后的变化规律，正面解答了公开问题——任意偶数r都存在对应单纯复形使双上同调秩为r。

Duality in filtered Floer-Novikov complexes

We prove that a certain bilinear pairing (analagous to the Poincare-Lefschetz intersection pairing) between filtered sub- and quotient complexes of a Floer-type chain complex and of its "opposite complex" is always nondegenerate on homology. This implies a duality relation for the Oh-Schwarz-type spectral invariants of these complexes which (in Hamiltonian Floer theory) was established in the special case that the period map has discrete image by Entov and Polterovich. The duality relation served as a key lemma in Entov and Polterovich's construction of a Calabi quasimorphism on certain ration…

Homotopy types of moment-angle complexes for flag complexes

We study the homotopy types of moment-angle complexes, or equivalently, of complements of coordinate subspace arrangements. The overall aim is to identify the simplicial complexes K for which the corresponding moment-angle complex Z_K has the homotopy type of a wedge of spheres or a connected sum of sphere products. When K is flag, we identify in algebraic and combinatorial terms those K for which Z_K is homotopy equivalent to a wedge of spheres, and give a combinatorial formula for the number of spheres in the wedge. This extends results of Berglund and Joellenbeck on Golod rings and homotopy…

Discrete Morse theory for moment-angle complexes of pairs (D^n,S^{n-1})

For a finite simplicial complex K and a CW-pair (X,A), there is an associated CW-complex Z_K(X,A), known as a polyhedral product. We apply discrete Morse theory to a particular CW-structure on the n-sphere moment-angle complexes Z_K(D^{n}, S^{n-1}). For the class of simplicial complexes with vertex-decomposable duals, we show that the associated n-sphere moment-angle complexes have the homotopy type of wedges of spheres. As a corollary we show that a sufficiently high suspension of any restriction of a simplicial complex with vertex-decomposable dual is homotopy equivalent to a wedge of sphere…

Whitehead products in moment-angle complexes

In toric topology, to a simplicial complex $K$ with $m$ vertices, one associates two spaces, the moment-angle complex $\mathcal{Z}_K$ and the Davis-Januszkiewicz space $DJ_K$. These spaces are connected by a homotopy fibration $\mathcal{Z}_K\to DJ_K\to(\mathbb{C}P^\infty)^m$. In this paper, we show that the map $\mathcal{Z}_K\to DJ_K$ is identified with a wedge of iterated (higher) Whitehead products for a certain class of simplicial complexes $K$ including dual shellable complexes. We will prove the result in a more general setting of polyhedral products.

矩角复形的双上同调

Double cohomology of moment-angle complexes

本文在矩角复形的上同调上定义双上同调结构，给出微分的三个等价定义，并计算包含弦图团复形的单纯复形族的双上同调。

Non-trivial higher Massey products in moment-angle complexes

As part of various obstruction theories, non-trivial Massey products have been studied in symplectic and complex geometry, commutative algebra and topology for a long time. We introduce a general approach to constructing non-trivial Massey products in the cohomology of moment-angle complexes, using homotopy theoretical and combinatorial methods. Our approach sets a unifying way of constructing higher Massey products of arbitrary cohomological classes and generalises all existing examples of non-trivial Massey products in moment-angle complexes. As a result, we obtain explicit constructions of …

Pontryagin algebras of some moment-angle-complexes

We consider the problem of describing the Pontryagin algebra (loop homology) of moment-angle complexes and manifolds. The moment-angle complex Z_K is a cell complex built of products of polydiscs and tori parametrised by simplices in a finite simplicial complex K. It has a natural torus action and plays an important role in toric topology. In the case when K is a triangulation of a sphere, Z_K is a topological manifold, which has interesting geometric structures. Generators of the Pontryagin algebra H_*(ΩZ_K) when K is a flag complex have been described in the work of Grbic, Panov, Theriault a…

Rational toral rank of a map

Let $X$ and $Y$ be simply connected CW complexes with finite rational cohomologies. The rational toral rank $r_0(X)$ of a space $X$ is the largest integer $r$ such that the torus $T^r$ can act continuously on a CW-complex in the rational homotopy type of $X$ with all its isotropy subgroups finite \cite{H}. As a rational homotopical condition to be a toral map preserving almost free toral actions for a map $f:X\to Y$, we define the rational toral rank $r_0(f)$ of $f$, which is a natural invariant with $r_0(id_X)=r_0(X)$ for the identity map $id_X$ of $X$. We will see some properties of it by Su…

Rational homotopy type of mapping spaces via cohomology algebras

In this paper, we show that for finite $CW$-complexes $X$ and two-stage space $Y$ (for example $n$-spheres $S^n$, homogeneous spaces and $F_0$-spaces), the rational homotopy type of $\map(X, Y)$ is determined by the cohomology algebra $H^*(X; \Q)$ and the rational homotopy type of $Y$. From this, we deduce the existence of H-structures on a component of the mapping space $\map(X, Y)$, assuming the cohomology algebras of $X$ and $Y$ are isomorphism. Finally, we will show that $\map(X, Y; f)\simeq\map(X, Y; f')$ if the corresponding \emph{Maurer-Cartan elements} are connected by an algebra autom…

Moment-angle complexes and combinatorics of simplicial manifolds

Let $ρ:(D^2)^m\to I^m$ be the orbit map for the diagonal action of the torus $T^m$ on the unit poly-disk $(D^2)^m$, $I^m=[0,1]^m$ is the unit cube. Let $C$ be a cubical subcomplex in $I^m$. The moment-angle complex $\ma(C)$ is a $T^m$-invariant bigraded cellular decomposition of the subset $ρ^{-1}(C)\subset(D^2)^m$ with cells corresponding to the faces of $C$. Different combinatorial problems concerning cubical complexes and related combinatorial objects can be treated by studying the equivariant topology of corresponding moment-angle complexes. Here we consider moment-angle complexes defined …

Moment-angle manifolds and complexes. Lecture notes KAIST'2010

These are notes of the lectures given during the Toric Topology Workshop at the Korea Advanced Institute of Science and Technology in February 2010. We describe several approaches to moment-angle manifolds and complexes, including the intersections of quadrics, complements of subspace arrangements and level sets of moment maps. We overview the known results on the topology of moment-angle complexes, including the description of their cohomology rings, as well as the homotopy and diffeomorphism types in some particular cases. We also discuss complex-analytic structures on moment-angle manifolds…

Torus equivariant algebraic models and compact realization

Let $T$ be a compact torus. We prove that, up to equivariant rational equivalence, the category of $T$-simply connected, $T$-finite type $T$-spaces with finitely many isotropy types is completely described by certain finite systems of commutative differential graded algebras with consistent choices of degree $2$ cohomology classes. We show that the algebraic systems corresponding to finite $T$-CW-complexes are exactly those which satisfy the necessary condition imposed by the Borel localization theorem along with certain finiteness conditions. We derive an algebraic characterization of when an…

Moment-angle complexes from simplicial posets

We extend the construction of moment-angle complexes to simplicial posets by associating a certain T^m-space Z_S to an arbitrary simplicial poset S on m vertices. Face rings Z[S] of simplicial posets generalise those of simplicial complexes, and give rise to new classes of Gorenstein and Cohen--Macaulay rings. Our primary motivation is to study the face rings Z[S] by topological methods. The space Z_S has many important topological properties of the original moment-angle complex Z_K associated to a simplicial complex K. In particular, we prove that the integral cohomology algebra of Z_S is iso…

Higher Whitehead products in moment-angle complexes and substitution of simplicial complexes

We study the question of realisability of iterated higher Whitehead products with a given form of nested brackets by simplicial complexes, using the notion of the moment-angle complex $Z_K$. Namely, we say that a simplicial complex $K$ realises an iterated higher Whitehead product $w$ if $w$ is a nontrivial element of $π_*(Z_K)$. The combinatorial approach to the question of realisability uses the operation of substitution of simplicial complexes: for any iterated higher Whitehead product $w$ we describe a simplicial complex $\partialΔ_w$ that realises $w$. Furthermore, for a particular form o…

Moment-angle complexes, monomial ideals, and Massey products

Associated to every finite simplicial complex K there is a "moment-angle" finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Massey products of a moment-angle complex, relating these topological invariants to the algebraic combinatorics of the underlying simplicial complex. Applications to the study of non-formal manifolds and subspace arrangements are given.

H-space structure on pointed mapping spaces

We investigate the existence of an H-space structure on the function space, F_*(X,Y,*), of based maps in the component of the trivial map between two pointed connected CW-complexes X and Y. For that, we introduce the notion of H(n)-space and prove that we have an H-space structure on F_*(X,Y,*) if Y is an H(n)-space and X is of Lusternik-Schnirelmann category less than or equal to n. When we consider the rational homotopy type of nilpotent finite type CW-complexes, the existence of an H(n)-space structure can be easily detected on the minimal model and coincides with the differential length co…

光滑环面簇的上同调环与矩角复形的商

The cohomology rings of smooth toric varieties and quotients of moment-angle complexes

本文研究矩角复形部分商（光滑环面簇的拓扑类比）的上同调环，证明Buchstaber-Panov公式仅在2可逆时成立，并通过定义挠积乘法的显式形变加以修正。

On Hochster's formula for a class of quotient spaces of moment-angle complexes

Any finite simplicial complex K and a partition of the vertex set of K determines a canonical quotient space of the moment-angle complex of K. We prove that the cohomology groups of such a space can be computed via some Hochster's type formula, which generalizes the usual Hochster's formula for the cohomology groups of moment-angle complexes. In addition, we show that the stable decomposition of moment-angle complexes can also be extended to such spaces. This type of spaces include all the quasitoric manifolds that are pullback from the linear models. And we prove that the moment-angle complex…

旗情形下循环同调的moment-angle复形

Loop homology of moment-angle complexes in the flag case

本文研究了旗情形下moment-angle复形的循环同调，通过同调方法描述了其生成元和关系，并证明了其在有理数上的coformal性质及同调群的计算方法。

映射到注入词复合体的同调类型

The homotopy type of the moment-angle complex associated to the complex of injective words

本文研究利用多面体积 functor 构造有向图的拓扑空间，并探讨其同调类型，揭示了同调与组合学的紧密联系，通过计算注入词复合体的同调类型，展示了其由 h-向量决定。

楔积、有理完备化与$\mathbb Q$-坏空间

Wedges, Rational completion and $\mathbb Q$-bad spaces

本研究针对特定连通基于CW复形，探究楔积的有理完备化映射在相关对象上诱导的映射的余核维数下界，进而得出特定对象为坏空间的结论。

Cup-products in generalized moment-angle complexes

Given a family of based CW-pairs $(\underline{X},\underline{A})=\{(X;A)\}^m_{i=1}$ together with an abstract simplicial complex $K$ with $m$ vertices, there is an associated based CW-complex $Z(K;(\underline{X},\underline{A}))$ known as a generalized moment-angle complex. The decomposition theorem of \cite{bbcg}, \cite{bbcg2} splits the suspension of $Z(K; (\underline{X}, \underline{A}))$ into a bouquet of spaces determined by the full sub-complexes of $K$. Thatdecomposition theorem is used here to describe the ring structure for the cohomology of Z(K; (\underline{X}, \underline{A})). Explicit…

与几乎线性分解相关的矩角复形的同伦型

Homotopy types of moment-angle complexes associated to almost linear resolutions

本文研究矩角复形及流形的同伦型，证明其Hurewicz像含线性链，并推广至几乎线性分解的Gorenstein理想，引入新单纯复形类，证明相关流形为形式且具有特定有理与整同伦型。

The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces

This article gives a natural decomposition of the suspension of generalized moment-angle complexes or {\it partial product spaces} which arise as {\it polyhedral product functors} described below. In the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the homological decomposition in Goresky-MacPherson \cite{goresky.macpherson}, Hochster\cite{hochster}, Baskakov \cite{baskakov}, Panov \cite{panov}, and Buchstaber-Panov \cite{buchstaber.panov}. Since the splitting is geometric, an analogous homological decomposition for a generalized momen…

Homotopy type of Frobenius complexes

A submonoid A of N^d has a natural order defined by a <= a + b for elements a and b of A. The Frobenius complex is the order complex of an open interval of A with respect to this order. In this paper, the homotopy type of the Frobenius complex of A is determined when A is the submonoid of N generated by two relatively prime integers, or the submonoid of N^2 generated by three elements of which any two are linearly independent. As an application, the multigraded Poincare series of the quotient algebra K[x, y, z] / (x^p y^q - z^r) over a field K is determined and proved to be rational.

Pontryagin algebras and the LS-category of moment-angle complexes in the flag case

For any flag simplicial complex $K$, we describe the multigraded Poincare series, the minimal number of relations and the degrees of these relations in the Pontryagin algebra of the corresponding moment-angle complex $Z_K$. We compute the LS-category of $Z_K$ for flag complexes and give a lower bound in the general case. The key observation is that the Milnor-Moore spectral sequence collapses at the second sheet for flag $K$. We also show that the results of Panov and Ray about the Pontryagin algebras of Davis-Januszkiewicz spaces are valid for arbitrary coefficient rings, and introduce the $(…

矩角复形的上同调运算与Stanley-Reisner环的消解

Cohomology operations for moment-angle complexes and resolutions of Stanley-Reisner rings

本文通过研究矩角复形上标准环面作用诱导的上同调运算，将Stanley-Reisner环的极小自由消解赋予拓扑解释，并利用所得工具完全刻画了矩角复形上环面作用的等变形式性。

关于实时刻-角流形拓扑刚性的评论

Remarks on Topological Rigidity of Real Moment-Angle Manifolds

本文研究实时刻-角流形与旗帜复形的拓扑刚性，通过Davis构造证明其具有CAT(0)度量，并验证fundamental group满足Farrell--Jones猜想，进一步推导出五维以上实时刻-角流形满足Borel猜想，揭示实情况下的刚性现象在复和四元数情形中失效。

Möbius transform, moment-angle complexes and Halperin-Carlsson conjecture

In this paper, we give an algebra-combinatorics formula of the Möbius transform for an abstract simplicial complex $K$ on $[m]=\{1, ..., m\}$ in terms of the Betti numbers of the Stanley-Reisner face ring of $K$. Furthermore, we employ a way of compressing $K$ to estimate the lower bound of the sum of those Betti numbers by using this formula. As an application, associating with the moment-angle complex $\mathcal{Z}_K$ (resp. real moment-angle complex ${\Bbb R}\mathcal{Z}_K$) of $K$, we show that the Halperin-Carlsson conjecture holds for $\mathcal{Z}_K$ (resp. ${\Bbb R}\mathcal{Z}_K$) under t…

A Golod complex with non-suspension moment-angle complex

It could be expected that the moment-angle complex associated with a Golod simplicial complex is homotopy equivalent to a suspension space. In this paper, we provide a counter example to this expectation. We have discovered this complex through the studies of the Golod property of the Alexander dual of a join of simplicial complexes, and that of a union of simplicial complexes.

$LS$-category of moment-angle manifolds and higher order Massey products

Using the combinatorics of the underlying simplicial complex $K$, we give various upper and lower bounds for the Lusternik-Schnirelmann (LS) category of moment-angle complexes $\zk$. We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS category. In particular, we characterise the LS category of moment-angle complexes $\zk$ over triangulated $d$-manifolds $K$ for $d\leq 2$, as well as higher dimension spheres built up via connected sum, join, and vertex doubling operations. %This characterisation is given in terms of the co…