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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

Citations:

Zbl 1255.13014
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References:

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