Locally Linear Involutions on S^4

• Locally linear involutions on S^4 are homeomorphisms of order two that act like linear maps in local neighborhoods and are classified by the dimension and topology of their fixed-point sets.
• The classification leverages modified surgery theory and an equivariant Schoenflies theorem to distinguish six distinct types of fixed-point set configurations with practical topological implications.
• Applications include connections to knot concordance and equivariant slicing in B^4, demonstrating the interplay between classical 4-manifold topology and modern equivariant methods.

A locally linear involution on the 4-sphere $S^4$ is a homeomorphism $\tau:S^4\to S^4$ of order two that is locally linear in the sense that each point lies in a $\tau$-invariant neighborhood locally homeomorphic (with involution) to an open set with a linear involution in $\mathbb{R}^4$. The classification, topology, and geometry of such involutions—especially the constraints on their fixed-point sets, the topology of the quotient space, linearizability, and the existence of exotic actions—form a central theme in equivariant 4-manifold topology. Recent results (Chen et al., 2014, Boyle et al., 27 Dec 2025) establish a comprehensive understanding of the possibilities for such involutions, linking deep surgery theory, equivariant Schoenflies theory, and knot concordance.

1. Definitions and Local Models

A locally linear involution on $S^4$ is a homeomorphism $\rho:S^4\to S^4$ with $\rho^2 = \mathrm{Id}$, such that for every point $x\in S^4$ there exists a $\rho$-invariant neighborhood $U$ of $x$ and a $\rho$-equivariant homeomorphism $(U,\rho|_U)\approx (V,\text{linear involution})$ for some open $V\subset\mathbb{R}^4$ equipped with a linear involution. Near each fixed point, the local model is $\mathbb{R}^k\times\mathbb{R}^{4-k}$ with $\rho\equiv\mathrm{Id}$ on $\mathbb{R}^k$ and $\rho(v) = -v$ on $\mathbb{R}^{4-k}$.

An equivariant tubular neighborhood of a component $N\subset\mathrm{Fix}(\rho)$ consists of a closed disk bundle $\nu(N) \subset S^4$ and a bundle involution covering the identity on $N$, compatible with the fiberwise linear involution. The existence and uniqueness of such tubular neighborhoods are nontrivial in the equivariant setting compared to the nonequivariant case (Boyle et al., 27 Dec 2025).

2. Classification of Locally Linear Involutions

Every locally linear involution on $S^4$ is, up to topological conjugacy, a representative of one of six distinct types, differentiated by the dimension and topology of the fixed-point set. The exhaustive classification, as in [(Boyle et al., 27 Dec 2025), Theorem 1.7], is summarized below:

$\dim\,\mathrm{Fix}(\rho)$	Fixed-point set type	Conjugacy classification
$-1$ (none)	free	two classes $\leftrightarrow$ homotopy $\mathbb{R}P^4$
$0$	two points	unique (Kwasik–Schultz)
$1$	circle $S^1$	unique linear class if eq tubular neighborhood exists
$2$	$2$-sphere $S^2$	knotted $2$-spheres with $2$-fold branched cover $S^4$
$3$	$\mathbb{Z}$-homology $3$-sphere	homeomorphism classes of $\mathbb{Z}$-homology $3$-spheres
$4$	$S^4$ (identity)	trivial

For involutions with 1-dimensional fixed set ($S^1$), uniqueness (modulo conjugacy) holds under the existence of an equivariant tubular neighborhood, which is always satisfied if $\rho$ is smooth [(Boyle et al., 27 Dec 2025), Theorem 1.1]. In the 2-dimensional case ($S^2$), linearizability occurs if and only if the $S^2$ is unknotted; conjugacy classes correspond to isotopy classes of such 2-knots in $S^4$ whose 2-fold branched cover is $S^4$ [(Boyle et al., 27 Dec 2025), Theorem 1.6]. All orientation-preserving locally linear involutions are linearizable by conjugation in $SO(5)$ (Chen et al., 2014).

3. Linearizability and the Role of $O(5)$

Any locally linear involution $\tau:S^4 \to S^4$ is, up to topological conjugacy, represented by a diagonal matrix $\mathrm{diag}(\pm 1, \ldots, \pm 1)$ in $O(5)$ (Chen et al., 2014). For orientation-preserving involutions (determinant $+1$), $\tau$ lies in $SO(5)$, and its fixed-point set is determined by the possible dimensions $k=0,2,4$ of the $-1$ eigenspace. The key nontrivial cases are:

• $k=2$: fixed set $S^2$
• $k=4$: fixed set $S^0$ (two points)

For orientation-reversing involutions (determinant $-1$), $k=1,3,5$ are possible for the $-1$ eigenspace:

• $k=1$: fixed set $S^3$
• $k=3$: fixed set $S^1$
• $k=5$: free action (fixed set empty)

No exotic orientation-preserving $C^0$ or locally linear involutions occur: all such involutions are conjugate to some orthogonal model [(Chen et al., 2014), Theorem 1.1]. For orientation-reversing involutions, the classification remains open in greater generality, but no counterexample is known for $G=\mathbb{Z}/2$.

4. Techniques: Surgery Theory and Equivariant Schoenflies Theorems

Proofs of the classification statements, especially for the $S^1$-fixed set case, combine modified surgery theory (Kreck) with an equivariant adaptation of the Schoenflies theorem (Boyle et al., 27 Dec 2025). In the 1-dimensional case, the existence of a suitable equivariant tubular neighborhood reduces the classification problem to the structure of the quotient space $X=(S^4-\mathrm{Int}\,\nu)/\rho$ with boundary $S^1\times\mathbb{R}P^2$ and fundamental group $\mathbb{Z}/2$. Modified surgery obstructions $\Theta(W,\nu)\in\ell_5(\mathbb{Z}[\mathbb{Z}/2],w)$ vanish in the relevant cases, owing to elementary nature, thus ultimately demonstrating topological conjugacy.

The equivariant Schoenflies theorem is developed by adapting Brown’s proof to equivariant contexts: any equivariant embedding of spheres with fixed-point sets differing by one dimension admits complementary regions equivariantly homeomorphic to 4-balls with the standard involution. This underpins the local-global principle in the classification.

5. Equivariant Tubular Neighborhoods and Non-uniqueness

Unlike the nonequivariant case, where a locally flat $S^1\subset S^4$ has a unique (up to ambient isotopy) topological tubular neighborhood [Freedman–Quinn], the equivariant context may admit multiple inequivalent tubular neighborhoods. For a circle fixed by a locally linear involution on $S^4$, precisely two inequivalent choices of equivariant tubular neighborhood exist, distinguished by the value of the Kirby–Siebenmann invariant $ks(X)\in \mathbb{Z}/2$ for the quotient space $X$ [(Boyle et al., 27 Dec 2025), Theorem 1.3]. This reflects a subtle interplay between 4-dimensional topology and the equivariant structure of the normal bundle.

6. Application to Knot Theory and Concordance

A notable application connects involutions on $S^4$ to equivariant knot concordance. Every strongly negative amphichiral knot $K\subset S^3$ with trivial Alexander polynomial ($\Delta_K(t)=1$) is standardly equivariantly topologically slice in $B^4$ with respect to the linear involution $x\mapsto -x$ [(Boyle et al., 27 Dec 2025), Theorem D]. This result leverages the linearizability theorem in the fixed-circle case: a locally linear involution extending the antipodal involution on $S^3$ and fixing a slice disk $D$ can be conjugated globally to the linear involution, yielding an equivariant slicing in $B^4$.

7. Summary and Further Directions

All locally linear involutions $\tau:S^4\to S^4$ are, up to topological conjugacy, orthogonal involutions—diagonal matrices in $O(5)$ with eigenvalues $\pm 1$—with allowable fixed-point set types and dimensions as dictated by the sign of the determinant and the dimension of $-1$ eigenspaces. Involutions on $S^4$ thus display remarkable rigidity in the locally linear and smooth category, with all nontrivial cases related to explicit linear models; no exotic $\mathbb{Z}/2$-actions occur. The classification is robust under stable smoothing, as all such involutions are stably smoothable and preserve the round metric on $S^4$ (Chen et al., 2014).

Open problems include the realization and classification of orientation-reversing actions outside $O(5)$, exploration of equivariant structures in higher dimensions, and further connections to 4-dimensional knot theory via involutive branched covers. The theorems of Chen–Kwasik–Schultz (Chen et al., 2014) and Boyle–Chen–Conway (Boyle et al., 27 Dec 2025) provide the definitive framework for locally linear involutions on $S^4$, integrating topological, geometric, and equivariant techniques.