Abe Homotopy Framework

• Abe homotopy framework is a generalization of classical homotopy theory that uses semi-direct products to classify interactions between line defects and higher excitations.
• It captures the nontrivial effects of the π1-action on higher homotopy classes, leading to unique charge transformations and symmetry reductions.
• The framework unifies continuous and discrete settings, offering tools like spectral sequences and graph-theoretic A-homotopy to analyze topological defects.

The Abe homotopy framework generalizes the classification of topological excitations in both continuous and discrete structures by extending the traditional homotopy-theoretic approach. It replaces the exclusive use of the $n$th homotopy group $\pi_n$ with a more refined structure capable of capturing the nontrivial interaction between fundamental group elements (vortices) and higher homotopy classes. In its modern development, the framework unifies perspectives from algebraic topology (A-homotopy and A-homology), topological defect theory in condensed matter and field theories, and recent extensions to discrete settings such as graph theory. The resulting structures—including the Abe homotopy group $\kappa_n$ and the A-homotopy groups $\pi_A^n$—encode not just the classification of individual excitations, but the global influence of line defects and nontrivial connectivity, and provide a setting for the development of generalized (co)homological tools and spectral sequences (Kobayashi et al., 2011, Ottina, 2011, Yamagata, 2024).

1. Abe Homotopy Groups and Their Classification Principle

Let $\mathcal{M}$ denote the order parameter manifold (often a homogeneous space $G/H$). Classical homotopy groups capture the types of topological defect via $\pi_1(\mathcal{M})$ (vortices or line defects) and $\pi_n(\mathcal{M})$ ($n$-dimensional defects). In the presence of both, an $n$th homotopy class may no longer be invariant under parallel transport around a vortex due to the nontrivial action of $\pi_1$ on $\pi_n$.

Abe and Fox established that the correct classification object is the $n$th Abe homotopy group, defined as

$$\kappa_n(\mathcal{M}, \phi_0) \cong \pi_1(\mathcal{M},\phi_0)\ltimes \pi_n(\mathcal{M},\phi_0)$$

where the operation is a semi-direct product with group law

$$(\gamma_1,\alpha_1) * (\gamma_2,\alpha_2) = (\gamma_1\gamma_2,\, \gamma_2(\alpha_1)\alpha_2)$$

and the action $\gamma(\alpha)$ is the monodromy: $\gamma(\alpha) = \gamma^{-1}*\alpha*\gamma$ (Kobayashi et al., 2011). This group captures both the independent classification of vortices and "higher" excitations, and crucially, the entanglement between them.

The physical charge of an excitation is then identified not with a single group element, but with its conjugacy class in $\kappa_n$, reflecting the fact that only gauge-invariant, physically observable configurations are meaningful.

2. The $\pi_1$-Action and Noncommutativity

When a higher-dimensional excitation is transported adiabatically around a vortex (loop $\gamma\in\pi_1$), its topological charge (originally labeled by $\alpha\in\pi_n$) may be altered. The transformation is given by

$$\gamma: \pi_n(\mathcal{M}) \rightarrow \pi_n(\mathcal{M}), \qquad \alpha \mapsto \gamma(\alpha) = \gamma^{-1}*\alpha*\gamma$$

If all actions are trivial (i.e., $\gamma(\alpha)=\alpha$), the order-parameter manifold is $n$-simple and the Abe group reduces to a direct product $\pi_1\times\pi_n$. Otherwise, the group is a proper semi-direct product, reflecting the nontrivial influence (noncommutativity) between line and point (or higher) defects (Kobayashi et al., 2011).

This formalism accounts for processes such as vortex–antivortex pair creation or annihilation. For instance, such a process can change a monopole charge $(1,\alpha_1)$ to $(1,\gamma^{-1}(\alpha_1))$ by virtual encirclement, directly corresponding to the group law and its associated conjugacy relations.

3. Explicit Structure for Quotient Manifolds and Physical Consequences

Consider $\mathcal{M}=S^n/K$, with $K$ a discrete subgroup of $SO(n+1)$ acting freely. The main classification result is:

• $\pi_1(S^n/K)\cong K$, $\pi_n(S^n/K)\cong\mathbb{Z}$, $\pi_k=0$ for $1
• The action $\gamma_g(\alpha)$ depends on whether $g\in K$ reverses orientation: for $n$ even and $g\in O(n)\setminus SO(n)$, $\gamma_g(\alpha) = \alpha^{-1}$; otherwise, $\gamma_g(\alpha) = \alpha$.

This can be summarized as:

Case	$\gamma(\alpha)$	Structure of $\kappa_n$
$n$ odd or $K$ has no reflection	$\alpha$	$\pi_1\times\pi_n$
$n$ even, $K$ contains a reflection	$\alpha^{-1}$	Nontrivial semi-direct product

The practical effect is a reduction of possible physical charges: e.g., a monopole charge classified by $\mathbb{Z}$ is reduced to $\mathbb{Z}_2$ when orientation-reverse action is present. Examples include uniaxial nematics, spinor Bose–Einstein condensates, and spin-2 nematic order parameters (Kobayashi et al., 2011).

4. A-Homotopy and A-Homology: Homotopy-theoretic Generalization

Ottina formalized the A-homotopy group as

$\pi^n_A(X) := [\Sigma^nA, X]_*$

where $A$ is a pointed CW-complex. This defines based homotopy classes of maps from suspensions $\Sigma^nA$ to $X$. Corresponding A-shaped homology groups $H^A_n(X)$ are defined using the Dold–Thom construction: $H^n_A(X) := \pi^n_A(SP(X)) = [\Sigma^nA, SP(X)]_*$ with $SP(X)$ the infinite symmetric product, providing an Eilenberg–Steenrod homology theory. This generalization extends Whitehead and Hurewicz theorems and enables the computation of these invariants via spectral sequences, including a relative Federer spectral sequence with

$$E^2_{p,q} \cong H^{-p}(A;\,\pi^q(Y,B)),\quad p+q\geq2,\ p\leq-1$$

and convergence to $A$-homotopy groups $\pi_A^{p+q}(Y,B)$ (Ottina, 2011). The A-Whitehead theorem establishes that homotopy equivalences can be detected via A-homotopy and A-homology isomorphisms when $A$ is sufficiently connected.

5. Discrete (Naive) A-Homotopy: Graph-Theoretic Perspective

The discrete A-homotopy framework realizes these principles in the context of combinatorial structures, specifically graphs. Here, the $n$th A-homotopy group $A_n(X,x_0)$ classifies based graph maps from an $n$-cube to a pointed graph $(X,x_0)$, modulo discrete (A-)homotopy. This classification aligns with the cubical realization $|X|_\square$ by

$A_n(X,x_0) \cong \pi_n(|X|_\square, x_0)$

for all $n\geq0$ (Yamagata, 2024).

The principal constructions are:

• Mapping fiber $Mf(f)$ of a graph map $f:X\to Y$;
• Loop graph $\Omega Y$, as pointed graph-maps from a path graph $I_m$ to $Y$, modulo stabilization;
• Reduced suspension $\Sigma X$, formed by collapsing both cylinder ends and the side over the basepoint in $X\otimes I_\ell$.

Important results include the existence of a "Puppe-type" sequence—$$\Omega X \rightarrow \Omega Y \rightarrow Mf(f) \rightarrow X \rightarrow Y$$—with the second iterated fiber $Mf_2$ homotopy equivalent to $\Omega Y$. However, there is typically no well-behaved mapping cone or cofiber sequence since the naive "endpoint" projection from $Mf(f)$ to $Y$ fails to be a graph map except for the trivial map. The adjointness between suspension and loop persists: $$[\Sigma_\ell X, Y]_{hGraph_*} \cong [X, \Omega_\ell Y]_{hGraph_*}$$ (Yamagata, 2024).

6. Applications and Examples

Representative cases illustrating the effect of the Abe framework include:

• Nematic liquid crystal ($\mathbb{RP}^2 \simeq S^2/\mathbb{Z}_2$): $\kappa_2=\mathbb{Z}_2\ltimes\mathbb{Z}$, and physical monopole charges collapse to $\mathbb{Z}_2$ due to the orientation-reversing action.
• Spin-1 polar and ferromagnetic BECs: For the polar case, $n=2$ even and $K=\mathbb{Z}_2$, again giving reduction $\mathbb{Z}\to\mathbb{Z}_2$. For the ferromagnetic case, $n=3$ odd, so the defect classes remain $\mathbb{Z}_2\times\mathbb{Z}$.
• Spin-2 nematic order, $n=4$: Nontrivial orientation reversal again reduces the instanton charge $\mathbb{Z}\to\mathbb{Z}_2$.

In the discrete setting, these constructions enable A-homotopy theoretic invariants in networks, with mapping fiber and loop graph constructions yielding discrete analogs of the exact fiber sequence and adjoint functor pairs, respectively.

7. Significance, Mathematical Context, and Further Developments

The Abe homotopy framework systematically codifies the impact of topological constraints and environmental structures—such as the presence of line defects—on the classification of topological excitations. By integrating semi-direct product constructions, generalized (A-)homotopy theory, and spectral sequence computations, it provides a unified methodology valid for both continuous and combinatorial geometric objects.

A significant consequence is the recognition that classical homotopy invariants may be insufficient in the presence of nontrivial fundamental group actions. The requirement to classify excitations by conjugacy classes in $\kappa_n$ is both mathematically inevitable and physically necessary in systems with non-Abelian topological defects.

Ongoing research extends these ideas in both directions: deeper exploration of spectral sequences and generalized (co)homology associated to A-homotopy; and further combinatorial generalizations, including the mapping fiber sequence and defect sequences in graphs and metric spaces, as outlined in discrete A-homotopy theory (Yamagata, 2024). This suggests potential for broad applicability in algebraic topology, mathematical physics, and combinatorial geometry.

References: (Kobayashi et al., 2011, Ottina, 2011, Yamagata, 2024)