Discrete Invariants of Fibres

Updated 16 December 2025

Discrete invariants of fibres are algebraic, combinatorial, or topological quantities that encode monodromy, singularities, and stability in fibered systems.
They are computed using methods like chart descriptions, group actions, and differential character theories to classify structures such as Lefschetz fibrations and families of flat connections.
These invariants have practical applications in symplectic, algebraic, and tropical geometry, serving as obstructions to fiber equivalence and supporting robust classification techniques.

Discrete invariants of fibres are algebraic, combinatorial, or topological quantities attached to the fibers of fibrations, bundle maps, families of connections, or parameterized systems, typically taking values in finite sets (such as mod 2, orbit sets, tuples of integers, or group data). These invariants distinguish different fiberwise structures, encode monodromy, detect singularities, and classify bundles under equivalence relations. Their explicit computation often depends on subtle group actions, cobordism theory, chart descriptions, rigidity, or intersection data. Discrete invariants play a key role in the classification, stability, and rigidity of fibered objects across symplectic geometry, algebraic geometry, integrable systems, and topological quantum matter.

1. Chart-Based and Group-Theoretic Invariants for Lefschetz Fibrations

Discrete invariants arise naturally in the context of Lefschetz fibrations and their monodromy representations. For hyperelliptic Lefschetz fibrations over closed oriented surfaces, the chart description yields a combinatorial encoding of the monodromy in terms of labeled graphs. In particular, considering the mapping-class group $M_{0,2g+2}$ of the sphere with $2g+2$ marked points, one constructs C $_0$ -charts with labeled black and white vertices, encoding generators and relators including the Dirac braid relator $r_4$ corresponding to the central $\mathbb{Z}_2$ -(Endo et al., 2015).

The discrete invariant $w$ is defined for a C $_0$ -chart $\Gamma$ as

$w(\Gamma) = \text{(number of white vertices of type $r_4 $or$ r_4^{-1} $in$ \Gamma$)} \bmod 2$

which measures the parity of Dirac braid relators present in the monodromy. For genus $g$ odd, this $\mathbb{Z}_2$ -valued invariant is stable under all chart moves (including transition moves), and thus provides a classification obstruction: two hyperelliptic Lefschetz fibrations are stably isomorphic if and only if their singular fiber counts and $w$ values match.

Examples demonstrate the effectiveness of $w$ in distinguishing non-stably isomorphic fibrations with identical fiber type counts, confirming $w$ as a genuine discrete invariant when $g$ is odd. The presence or absence of Dirac braid relators in the chart completely determines stable isomorphism classes modulo fiber summations with a universal fibration.

2. Discrete Invariants in Families of Flat Connections

For smooth vector bundles, discrete invariants can be constructed for families of flat connections via fiber integration of differential characters(Mata, 2019). Given a polynomial invariant $P\in I^p(GL(n,\mathbb{R}))$ and the Cheeger-Simons theory, one associates to each flat connection a class $cs_{(P,u)}(E,\theta)\in H^{2p-1}(B;\mathbb{R})$ , lifting the topological invariant $u(E)$ .

Generalizing to families of relatively flat connections parametrized by simplicial sets, fiber integration along the parameters yields invariants

$\tilde \psi_{p,r}: H_r(\mathcal{D}(E)) \rightarrow H^{2p-r-1}(B; \mathbb{R})$

when $p \ne r, r+1$ . These classes are rigid: they vanish in the low-degree range $p < r$ and are independent of auxiliary choices. In the classical case $(p=2, r=0)$ , one recovers the discrete, rigid Chern-Simons invariants. The construction refines integral characteristic classes to finer discrete classes parametrizing the family, and their vanishing or non-vanishing distinguishes different families up to rigid homotopy.

3. Fiberwise Coincidence, Reidemeister, and Nielsen Invariants

Fiberwise maps between bundles over a common base $B$ admit discrete invariants via the geometry of coincidences(Koschorke, 2013). The Reidemeister set $R_B(f_1, f_2)$ consists of the orbit set of path-components of the path space $E_B(f_1, f_2)$ , which itself arises as a quotient of lifting classes under the fundamental group action:

$R_B(f_1, f_2) \cong \pi_1(B) \backslash \text{LiftClasses}(f_1, f_2)$

Each orbit defines a Nielsen class, and the Nielsen number $N_B(f_1, f_2)$ —the count of essential classes—is a discrete invariant that determines the minimum number of coincidence components or fixed points under fiberwise homotopy.

Explicit computation in the case of torus bundles over $S^1$ reveals that the group $G = \mathbb{Z}^n / L(\mathbb{Z}^m)$ with affine action $\beta(u) = [A_N(u - v)]$ encodes all discrete data; the parity and cardinality of orbits, especially the odd-order orbits, play a critical role in detecting essential coincidences or fixed points. These invariants classify fiberwise maps up to deformation and determine the possibility of fixed-point-free homotopies.

4. Topological and K-Theoretic Discrete Invariants

Discrete invariants also appear as topological indices in the classification of fiber bundles. For Miller–Morita–Mumford characteristic classes(Church et al., 2011), discrete invariance is achieved when the class $x \in C_d \otimes \mathbb{Q}$ is the fiber integral of a near-primitive in $H^*(BSO(d); \mathbb{Q})$ , i.e., it satisfies a cocommutative Hopf algebra condition mimicking additivity modulo lower-degree terms. Only those MMM classes arising from near-primitives yield characteristic numbers independent of the fibering, detecting only the cobordism class of the total space. This dichotomy singles out which discrete characteristic numbers are truly fiber-invariant.

Similarly, in Hamiltonian fibrations with prequantum structures, discrete K-theoretic invariants(Savelyev et al., 2015) are built from the family analytic index of $Spin^c$ -Dirac operators, yielding classes in twisted $K$ -theory

$\text{Ind}(D^k_+) \in K^0(B; \alpha)$

where $\alpha$ is the Dixmier–Douady class. These twisted K-theory classes are stable under homotopy and often capture torsion information invisible to classical characteristic classes, exemplifying the role of discrete index invariants in symplectic topology and quantized system classification.

5. Discrete Invariants in Tropical and Floer Geometry of Singular Fibres

In the tropical and Floer-theoretic study of elliptic K3 surfaces, open Gromov–Witten invariants $\mu(\gamma; u)$ provide discrete counts of holomorphic discs with specified boundary cycles for fibers of various Kodaira types(Lin, 2017). These invariants depend only on the combinatorics of monodromy and the intersection pairing of vanishing cycles; for each singular fibre type (I $_n$ , II, III, IV), the pattern and values of $\mu(\gamma)$ encode discrete wall-crossing data and stipulate the permissible configurations for broken lines in scattering diagrams. The local $\mu(\gamma)$ serve as discrete "initial conditions" for global wall-crossing phenomena, stabilizing the tropical moduli.

Families of polarized varieties admit natural numerical invariants—such as the Cornalba–Harris invariant(Barja et al., 2012)

$e(\mathcal{L}, \mathcal{G}) = r\ell - n d \ell_F$

with $r = \operatorname{rank} \mathcal{G}$ , $\ell = \mathcal{L}^n$ , $d = \deg(\mathcal{G}|_t)$ , $\ell_F = (\mathcal{L}|_F)^{n-1}$ —whose positivity reflects fiberwise and basewise stabilities. These discrete invariants govern inequality relationships (slope inequalities, Severi-type bounds) and are rigidly tied to GIT, slope stability, and linear stability conditions on the fiber and base. Their non-negativity, preserved under geometric invariant theory, slope filtration, or linear stability, yields fine classification results and sharp bounds for algebraic families.

7. Real Lefschetz Chains and Decorated Chain Invariants

In the classification of real Lefschetz fibrations with real critical values, discrete invariants arise as Lefschetz chains(Salepci, 2011)—sequences of isotopy or conjugacy classes of real codes $[c_i, a_i]$ recording the succession of real structures and vanishing cycles across fibers. For genus $g > 1$ , these chains completely classify fibrations up to isomorphism; for $g = 1$ , a binary decoration is required on certain entries to record the ambiguity introduced by nontrivial fundamental group of real structures. The decorated real Lefschetz chain serves as a full discrete invariant for real elliptic Lefschetz fibrations, with the choice of decoration distinguishing non-isomorphic total spaces having the same undeclared sequence. These chain invariants synthesize monodromy, boundary gluing data, and equivariant geometry into a discrete combinatorial object.

In summary, discrete invariants of fibres—arising from group actions, characteristic classes, chart combinatorics, parameterized analytical indices, and stability filtrations—serve as irreplaceable tools for classification, rigidity, and enumeration problems in topological, geometric, and algebraic settings. Their calculations depend on the detailed structure of fiberwise monodromy, intersection theory, symplectic quantization, or flat connection families, and they delineate the boundaries of stable equivalence, deformation classes, and positivity regimes across fibered mathematical objects.