[Paper Review] The topology of the space of matrices of Barvinok rank two

Axel Hultman, Jakob Jönsson|ArXiv.org|Feb 19, 2009

Advanced Combinatorial Mathematics12 references3 citations

TL;DR

This paper establishes that the space $ B_{d,n} $ of $ d \times n $ real matrices of Barvinok rank two, modulo rescaling and translation, is homeomorphic to the quotient $ (S^{d-2} \times S^{n-2}) / \mathbb{Z}_2 $, proving it is a manifold and computing its integral homology via discrete Morse theory. For $ d \geq n $, the homology is isomorphic to that of $ S^{d-2} \times \mathbb{RP}^{n-2} $ when $ d $ is odd, with torsion arising in specific degrees.

ABSTRACT

The Barvinok rank of a $d imes n$ matrix is the minimum number of points in $\mathbb{R}^d$ such that the tropical convex hull of the points contains all columns of the matrix. The concept originated in work by Barvinok and others on the travelling salesman problem. Our object of study is the space of real $d imes n$ matrices of Barvinok rank two. Let $B_{d,n}$ denote this space modulo rescaling and translation. We show that $B_{d,n}$ is a manifold, thereby settling a conjecture due to Develin. In fact, $B_{d,n}$ is homeomorphic to the quotient of the product of spheres $S^{d-2} imes S^{n-2}$ under the involution which sends each point to its antipode simultaneously in both components. In addition, using discrete Morse theory, we compute the integral homology of $B_{d,n}$. Assuming $d \ge n$, for odd $d$ the homology turns out to be isomorphic to that of $S^{d-2} imes \mathbb{RP}^{n-2}$. This is true also for even $d$ up to degree $d-3$, but the two cases differ from degree $d-2$ and up. The homology computation straightforwardly extends to more general complexes of the form $(S^{d-2} imes X)/\mathbb{Z}_2$, where $X$ is a finite cell complex of dimension at most $d-2$ admitting a free $\mathbb{Z}_2$-action.

Motivation & Objective

• To resolve Develin's conjecture that $ B_{d,n} $, the space of $ d \times n $ matrices of Barvinok rank two modulo rescaling and translation, is a manifold.
• To compute the integral homology of $ B_{d,n} $, particularly identifying torsion and free parts.
• To establish a homology isomorphism between $ (S^{d-2} \times X)/\mathbb{Z}_2 $ and $ S^{d-2} \times (X/\mathbb{Z}_2) $ for finite cell complexes $ X $ with free $ \mathbb{Z}_2 $-action and $ \dim X \leq d-2 $.
• To clarify the difference in homology between even and odd $ d $, especially in higher degrees.

Proposed method

• The authors define $ B_{d,n} $ as the quotient of the space of $ d \times n $ matrices of Barvinok rank two by rescaling and translation, and construct a simplicial decomposition using trees.
• They prove that $ B_{d,n} $ is homeomorphic to $ (S^{d-2} \times S^{n-2}) / \mathbb{Z}_2 $, where the $ \mathbb{Z}_2 $-action is the simultaneous antipodal map on both spheres.
• Discrete Morse theory is applied to the chain complex of the space, using a $ \mathbb{Z}_2 $-equivariant decomposition of the complex into $ \hat{\mathsf{U}}^{(0)} $ and $ \hat{\mathsf{U}}^{(D)} $ subcomplexes.
• A key construction involves the map $ \varphi(w{\bf b}_{0,k}) = \sum_{i=0}^D q^i(w){\bf b}_{i,k-i} $, which relates boundary maps across the decomposition.
• The homology is computed by relating $ H_*(\left(\mathsf{V} \otimes \mathsf{W}\right)^+) $ to $ H_*(\mathsf{W}^+) $ and $ H_*(\mathsf{W}^-) $, depending on the parity of $ D $.
• The final homology computation uses the standard hemispherical cell decomposition of $ S^{n-2} $ as the complex $ \mathsf{W} $, and applies Theorem 4.12 to derive the torsion and free parts.

Experimental results

Research questions

• RQ1 Is the space $ B_{d,n} $ of matrices of Barvinok rank two a topological manifold?
• RQ2 What is the integral homology of $ B_{d,n} $, particularly its torsion and free parts?
• RQ3 How does the homology of $ B_{d,n} $ compare to that of $ S^{d-2} \times \mathbb{RP}^{n-2} $, especially for odd and even $ d $?
• RQ4 Under what conditions does $ (S^{d-2} \times X)/\mathbb{Z}_2 $ have the same homology as $ S^{d-2} \times (X/\mathbb{Z}_2) $ for a $ \mathbb{Z}_2 $-complex $ X $?
• RQ5 What is the role of discrete Morse theory in computing the homology of $ B_{d,n} $, and how does it simplify the chain complex?

Key findings

• The space $ B_{d,n} $ is homeomorphic to $ (S^{d-2} \times S^{n-2}) / \mathbb{Z}_2 $, confirming Develin's conjecture that it is a manifold.
• For $ d \geq n $, the reduced integral homology of $ B_{d,n} $ is isomorphic to that of $ S^{d-2} \times \mathbb{RP}^{n-2} $ when $ d $ is odd.
• When $ d $ is even, the homology of $ B_{d,n} $ differs from that of $ S^{d-2} \times \mathbb{RP}^{n-2} $ in degrees $ d-2 $ and above.
• The free part of the homology is $ \mathbb{Z} $ in degrees $ i $ satisfying specific conditions: $ i+2 = d = n $ and $ i $ odd, or $ i+2 = n \neq d $, or $ i+2 = d \neq n $, or $ i = d+n-4 $ and $ i $ even.
• The torsion part is $ \mathbb{Z}_2 $ in degrees $ i $ such that $ 1 \leq i \leq n-3 $ and $ i $ odd, or $ d-2 \leq i \leq d+n-5 $ and $ i $ even.
• The homology computation generalizes to $ (S^{d-2} \times X)/\mathbb{Z}_2 $ for any finite cell complex $ X $ with free $ \mathbb{Z}_2 $-action and $ \dim X \leq d-2 $, with isomorphism to $ S^{d-2} \times (X/\mathbb{Z}_2) $ holding for odd $ d $.