Torsion-Freeness in Borel–Moore Homology

• Torsion-freeness in Borel–Moore homology means that the homology groups are free abelian modules over their coefficient rings, ensuring no elements are annihilated by non-units.
• This property enables canonical bases, integral presentations, and duality isomorphisms with de Rham or compactly supported cohomology, enhancing both computations and structural integrity.
• Its demonstration in hyperplane arrangements, quiver varieties, and cotangent representations illustrates its practical impact on computational topology and geometric representation theory.

Torsion freeness in Borel-Moore homology is a property of substantial structural and computational significance across singularity theory, the topology of algebraic varieties, and geometric representation theory. In Borel-Moore homology, “torsion freeness” asserts that the homology groups are free abelian over their coefficient ring (typically $\mathbb{Z}$ or $H^*_G$), implying the absence of elements annihilated by non-units. This property enables canonical bases, integral presentations (not requiring field extension or denominators), and powerful duality isomorphisms with de Rham or compactly supported cohomology. Torsion-freeness has been explicitly demonstrated in classical arrangements, quiver varieties, Hall algebra contexts, and stacks of algebraic representations, revealing fundamental links between geometry, algebra, and combinatorics.

1. Borel-Moore Homology: Definition and Context

Borel-Moore homology $H_*^{\mathrm{BM}}(X; R)$ is the covariant homology theory for locally compact, possibly non-compact, real or complex algebraic varieties or stacks, dual to compactly supported cohomology. For a complex algebraic (or analytic) variety $X$, $H_i^{\mathrm{BM}}(X; R)=H^{-i}(a_{X,*} R_X)$, where $a_X$ is the structure morphism, aligning with the dual space to $H_c^*(X; R)$ and equipping the homology with a canonical mixed Hodge structure when $R=\mathbb{Q}$. In equivariant contexts, equivariant Borel-Moore homology $H_{G,*}^{\mathrm{BM}}(X)$ is constructed as a direct limit over finite-dimensional approximations of the classifying stack $BG$.

Torsion freeness means that, as a module over $R$, $H_*^{\mathrm{BM}}(X; R)$ is free—there are no nonzero elements $x$ and non-unit $r\in R$ with $r x=0$. In equivariant settings, the freeness is often over $H^*_G(\mathrm{pt})$, the equivariant cohomology ring. This property is essential for computations, homological dualities, and the integrity of algebraic structures such as Hall algebras and shuffle algebra embeddings (Gubarevich, 31 Dec 2025, Davison, 2016).

2. Torsion-Freeness in Hyperplane Arrangement Complements

For real hyperplane arrangements, the complement

$$M(\mathcal{A}) = \mathbb{C}^\ell \setminus \bigcup_{i=1}^n H_i^\mathbb{C}$$

admits a canonical, torsion-free Borel-Moore homology structure. By constructing a semi-algebraic, real-analytic partition indexed by chambers $C$ in the real arrangement and a sufficiently generic flag, Ito–Yoshinaga established a cell decomposition into contractible, pairwise disjoint pieces $S(C)$ such that ((Ito et al., 2011), Thms. 3.10, 4.4):

• The closures $\overline{S(C)}$ are smooth, oriented real submanifolds, providing explicit cycles.
• For each degree, the classes $[\overline{S(C)}]$ (for $C$ running over a partition of the chambers) form a $\mathbb{Z}$-basis of $H_*^{\mathrm{BM}}(M(\mathcal{A});\mathbb{Z})$, making the group free abelian.
• The intersection matrix with canonical cohomology generators is triangular with units on the diagonal, ensuring linear independence and spanning.
• All Borel-Moore homology groups are torsion-free—no nontrivial torsion elements arise in any step or intersection computation.

This explicit basis enables a concrete duality with de Rham cohomology, given by Poincaré–Alexander duality, and a “combinatorial Morse theory” adapted to hyperplane complements ((Ito et al., 2011), §5).

3. Torsion-Freeness in Stacks of Representations and Hall Algebras

In the context of preprojective algebras $\Pi_Q$ of quivers, the Borel–Moore homology of the stack of representations $\mathfrak{M}_{\Pi_Q,d}=\mu^{-1}(0)/G_d$ and its equivariant version exhibit freeness as modules over the coefficient ring ((Davison, 2016), Main Theorem):

• The CoHA (cohomological Hall algebra) $$\mathcal{H}_{T,\Pi_Q} = \bigoplus_d H^{\mathrm{BM}}_{T\times G_d}(\mu^{-1}(0),\mathbb{Q})\otimes \mathcal{L}^{-(d,d)}$$ is a free $H^*_T$-module for any torus $T$ acting compatibly (preserving preprojective relations).
• The localization map to the fraction field of $H^*_T$ is injective, confirming torsion-freeness at both the equivariant and nonequivariant level.
• There exists an integral, denominator-free embedding into a shuffle algebra of symmetric polynomials, confirming the absence of “hidden $q$-torsion” in the Hall algebra realization.

The proof relies on dimensional reduction to vanishing-cycle cohomology on the 3-Calabi–Yau completion, the PBW theorem for vanishing-cycle complexes, and the purity of BPS perverse sheaves, ensuring that Leray spectral sequences degenerate and no torsion is introduced ((Davison, 2016), §§1–3).

4. Torsion-Freeness for Cotangent Representations and Hall Induction

For cotangent stacks of the form $T^*(V/G)\cong \mu_V^{-1}(0)/G$ for $V$ a finite-dimensional $G$-representation, torsion-freeness in Borel–Moore homology is established under mild assumptions on extra torus actions ((Gubarevich, 31 Dec 2025), Theorem):

• With $T_s$ a torus acting on $T^*V\times \mathfrak{g}$, commuting with $G$ and preserving the null-fiber of the moment map, if $T_s$ contains two specified one-parameter subgroups, then $H^{\mathrm{BM}}_{G\times T_s}(\mu_V^{-1}(0))$ is a torsion-free $H^*_{G\times T_s}(\mathrm{pt})$-module.
• Equivalently, the restriction homomorphism to the equivariant cohomology ring of the point is injective.
• The proof utilizes the Atiyah–Bott localization theorem, reduction to homology over subtori, spectral sequence arguments, and the purity of Borel–Moore homology for nilpotent orbits.

Stratification by nilpotent orbits of $\mathfrak{g}$ and injectivity on stabilizer homology ensures the absence of torsion classes. Concrete examples (e.g., $G = GL_2$, $V = \mathrm{End}(\mathbb{C}^2)$) confirm these results without torsion appearing outside explicitly characterized loci ((Gubarevich, 31 Dec 2025), §5).

5. Conceptual Mechanisms and Structural Implications

The proofs of torsion-freeness across the above settings share key mechanisms:

• Explicit cell decompositions: semi-algebraic partitions or CW-complex models provide cycle classes with canonical bases, revealing freeness (Ito et al., 2011).
• Filtration and stratification: stratified spaces (e.g., nilpotent cones, quiver representation spaces) are filtered so that associated graded pieces are themselves known to be free.
• Degeneration of spectral sequences: Purity of cohomology (often informed by mixed Hodge module theory) forces degeneration, thus precluding torsion extension classes (Davison, 2016).
• Integrality in Hall algebra structures: The shuffle algebra embeddings are integral—no denominators are required—mirroring torsion-freeness at the level of algebraic presentations (Davison, 2016).
• Equivariant methods and localizations: Applying localization theorems (Atiyah–Bott/Borel–Weil–Bott) reduces proofs to computations over polynomial rings and their localizations, allowing exact control over possible sources of torsion (Gubarevich, 31 Dec 2025).

The upshot is that torsion-freeness implies well-behaved Poincaré and Alexander dualities, canonical combinatorial or geometric bases, and computational tractability in both algebraic and topological settings.

6. Connections to Positivity, Combinatorics, and Representation Theory

Torsion-freeness in Borel–Moore homology underlies and implies further positivity and combinatorial phenomena:

• For preprojective algebras, torsion-freeness and purity translate into the positivity of restricted Kac polynomials and the factorial plethystic factorization of generating series (Davison, 2016).
• For hyperplane arrangements, the combinatorial nature of the basis links to matroid structures, and a plausible implication is that similar torsion-free bases may exist for a broader class of oriented matroid complements (Ito et al., 2011).
• In geometric representation theory, torsion-freeness ensures that weight spaces, character formulas, and convolution algebras are supported on integral, canonical generators, not requiring denominators or field extensions for their presentations (Gubarevich, 31 Dec 2025).

7. Generalizations and Open Directions

The results indicate, and in special cases confirm, that semi-algebraic decompositions yielding torsion-free Borel–Moore homology should extend to complements of more general real algebraic hypersurfaces and to further classes of moduli stacks, particularly those admitting cellular decompositions or deep filtrations (Ito et al., 2011, Davison, 2016). The explicit absence of torsion, both in homological and Hall-algebraic incarnations, points to broader patterns in the topology and representation theory of algebraic stacks, and suggests potential for further applications in cohomological wall-crossing, positivity results, and categorified representation theory.

Key References:

"Ito–Yoshinaga, Semi-algebraic partition and basis of Borel–Moore homology of hyperplane arrangements" (Ito et al., 2011) "Hall induction for cotangent representations and wheel conditions" (Gubarevich, 31 Dec 2025) "The integrality conjecture and the cohomology of preprojective stacks" (Davison, 2016)