## Dimensions of multi-fan duality algebras.(English)Zbl 1453.13066

The paper under review considers the dimension vector of the graded Poincare duality algebra associated to a complete simplicial multi-fan.
Therefore, one starts with a multi-fan $$\Delta$$ in an oriented space $$V\cong \mathbb{R}^n$$, whose cones are simplicial and may overlap. Then one considers the pair $$(\omega, \lambda)$$, where $\omega=\sum_{I\subset [m],|I|=n}\omega(I)I\in Z_{n-1}(\Delta_{[m]}^{(n-1)};\mathbb{R})$ is a simplicial cycle on $$m$$ vertices, and $$\lambda:[m]\to V$$ is a function such that $$\{\lambda(i)\mid i\in I\}$$ is a basis of $$V$$ if $$|I|=n$$ and $$\omega(I)\ne 0$$.
With respect to this $$\Delta$$, one also need to consider the simple multi-polytope $$P=(\Delta,\{H_1,\dots,H_m\})$$ with the support parameters $$c_1,\dots,c_m\in \mathbb{R}$$. Here, $$H_i=\{x\in V^*\mid \langle x,\lambda(i)\rangle =c_i\}$$ is a hyperplane in $$V^*$$ perpendicular to $$\lambda(i)\in V$$. This multi-polytope has a well-defined volume depending on the support parameters, which leads to the volume polynomial $$V_\Delta \in \mathbb{R}[c_1,\dots,c_m]$$.
Now the graded Poincare duality algebra under consideration is simply $$\mathcal{A}^*(\Delta):=\mathcal{D}/ \{D\in \mathcal{D}\mid D V_\Delta=0\}$$. Here, $$\mathcal{D}=\mathbb{R}[\partial_1,\dots,\partial_m]$$, where $$\partial_i$$ is the partial derivative operator $$\partial/\partial c_i$$.
The paper under review first considers the question: whether the dimensions of graded components of $$\mathcal{A}^*(\Delta)$$ depend only on $$\omega$$, but not on $$\lambda$$? As found out by this paper, the dimensions do not depend on the values of $$\lambda$$ at the non-singular vertices, but may depend crucially on the values of $$\lambda$$ at the singular vertices.
The next question is then: what can be said when $$\omega$$ is the fundamental cycle of an $$(n-1)$$-dimensional oriented simplicial pseudomanifold $$K$$, i.e., when the multi-fan is supported on $$K$$? Let $$r(K)$$ be the number of distinct dimension vectors of multi-fans on $$K$$. Then, this paper shows that $$r(K)=1$$ when $$K$$ is a homology manifold, or a $$3$$-dimensional pseudomanifold with isolated singularities. But $$r(K)$$ might be nontrivial in general.
Another class of examples studied in this paper is by considering a link (namely a collection of knots in $$S^3$$) $$\ell:\bigsqcup_\alpha S_\alpha^1 \hookrightarrow S^3$$ and then collapsing each of its components to a point. The examples computed by the author lead to the interesting question: is it true that in this case $$r(K)=1$$ if and only if the components in $$\ell$$ are pairwise unlinked?

### MSC:

 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 05E45 Combinatorial aspects of simplicial complexes 05E40 Combinatorial aspects of commutative algebra 28A75 Length, area, volume, other geometric measure theory 52B70 Polyhedral manifolds 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry) 57N65 Algebraic topology of manifolds 13A02 Graded rings 13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes

Zbl 1255.13014

### Software:

GAP; MultiFans; GitHub; simpcomp
Full Text:

### References:

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