Equivariant Tubular Neighborhoods

• Equivariant tubular neighborhoods are G-invariant constructions created via the exponential map on the normal bundle, extending classical tubular neighborhoods by incorporating group symmetries.
• They play a crucial role in equivariant topology and gauge theory by ensuring local charts and deformation retractions are compatible with smooth group actions.
• These structures facilitate advanced applications such as equivariant Morse theory, stratified space analysis, and stable homotopy splittings in configuration spaces.

An equivariant tubular neighborhood is a generalization of the classical tubular neighborhood in differential geometry, constructed so as to respect the symmetry given by a group action. Specifically, let $G$ be a group acting smoothly and isometrically on a (possibly infinite-dimensional) Riemannian manifold $M$, and let $N\subset M$ be a $G$-invariant submanifold whose normal bundle is locally trivial. An equivariant tubular neighborhood of $N$ is an open $G$-invariant neighborhood of $N$ in $M$ together with a $G$-equivariant homeomorphism from a corresponding neighborhood in the normal bundle $\nu(N)$ onto this open set, restricting to the identity on $N$. These objects underpin central constructions in equivariant topology, gauge theory, controlled topology, and equivariant homotopy theory, with applications extending from the analysis of moduli spaces to the topology of configuration spaces and the computation of equivariant invariants.

1. Foundational Definitions

Given a Riemannian manifold $(M,g)$—possibly infinite-dimensional and modeled on a (real or complex) Hilbert space—with a group $G$ acting smoothly by isometries, a $G$-invariant submanifold $N\subset M$ possesses a normal bundle $\nu(N)\to N$ which is a $G$-equivariant vector bundle. The exponential map $\exp\colon TM\to M$ for the $G$-invariant metric sends the zero section into $N$ and restricts to a tubular embedding from an open subset $U\subset \nu(N)$ onto a neighborhood $\tau(N)\subset M$ containing $N$. If both $U$ and $\tau(N)$ are $G$-invariant and the map is $G$-equivariant, this structure is an equivariant tubular neighborhood.

In the context of a $G$-stratified space (Whitney regular), each stratum admits a $G$-invariant tubular neighborhood, a $G$-equivariant projection (retraction) to the stratum, and a $G$-invariant “radial” distance function compatible across nested strata, forming a system of $G$-equivariant control data (Pflaum et al., 2017).

For proper actions of discrete groups, the existence of $G$-invariant tubular neighborhoods relies on the ability to average the Riemannian metric to $G$-invariance. The normal bundle $\nu(F)$ to a $G$-invariant closed submanifold $F$ then admits a canonical $G$-action induced by differentials, and the exponential map provides an equivariant diffeomorphism from a neighborhood of the zero section in $\nu(F)$ onto a $G$-invariant open neighborhood in $M$ (Mishchenko et al., 2011).

2. Main Existence and Uniqueness Theorems

The key structural result is that if $G$ acts by isometries on a strongly Riemannian (possibly infinite-dimensional) manifold $M$ and $N\subset M$ is a $G$-invariant submanifold with a locally trivial (smooth) normal bundle, then a $G$-invariant total tubular neighborhood exists, homeomorphic (and $G$-equivariantly so) to the entire normal bundle $\nu(N)$ (Ramras, 2010). The construction does not require compactness of $G$; strong regularity of the metric and its invariance under $G$ suffice.

In $G$-stratified spaces, every closed union of strata $A$ admits a $G$-equivariant tubular neighborhood in the ambient stratified space $X$, constructed via induction on the depth of strata and compatibility axioms—the existence result also yielding a $G$-equivariant strong deformation retraction onto $A$ (Pflaum et al., 2017).

Uniqueness (up to $G$-equivariant homeomorphism) follows from the homotopy-theoretic nature of the construction, with compression along fiberwise radial geodesics or appropriate $G$-equivariant homotopies.

3. Construction Methods and Local Models

The standard construction uses the exponential map of a $G$-invariant metric. For each point $y\in N$, the exponential map is a local diffeomorphism from a ball in the normal fiber $\nu_y(N)$. By building $G$-invariant families of such balls and gluing local charts via $G$-translates, one obtains a $G$-invariant open subset $Z\subset \nu(N)$ where $\exp$ is injective and $G$-equivariant (Ramras, 2010). To extend this to the full normal bundle (the "total tube" property), one compresses the fibers outside $Z$ into $Z$ via a $G$-equivariant homotopy.

For $G$-stratified spaces, one equips each stratum $S$ with a $G$-invariant tubular neighborhood $T_S$, a $G$-equivariant projection $\pi_S:T_S\to S$, and a $G$-invariant radius function $\rho_S$ (measuring transverse distance to $S$), all constructed using $G$-invariant Riemannian metrics and local charts provided by the equivariant submersion theorem. Compatibility on overlaps of neighborhoods between strata is enforced by the normal form in equivariant tubular neighborhoods and submersions (Pflaum et al., 2017).

When $G$ is discrete and acting properly (stabilizers finite), the $G$-invariant structure arises by averaging metrics and using the group action to lift to the normal bundle. The exponential map, being $G$-equivariant due to metric invariance, then produces the tubular neighborhood (Mishchenko et al., 2011).

4. Applications in Gauge Theory and Stratified Topology

Equivariant tubular neighborhoods are fundamental in Morse theory on infinite-dimensional Riemannian manifolds, particularly in Yang–Mills theory on surfaces. The gauge group $\G^k(P)$ acts isometrically on the affine Hilbert manifold of Sobolev connections $\A^{k-1}(P)$. The Yang–Mills functional is Morse–Bott, and its critical points decompose into closed submanifolds (Morse strata) with smooth normal bundles. The main existence theorem guarantees that each stratum $C_j$ admits a $\G^k(P)$-invariant total tubular neighborhood, crucial for the excision and Thom isomorphism arguments in gauge-equivariant cohomology (Ramras, 2010). In orientable cases, this recovers the equivariant perfection property (Atiyah–Bott recursion). In non-orientable cases, one obtains equivariant Morse inequalities and anti-perfection.

In Whitney stratified spaces with compact Lie group action, a full system of $G$-equivariant control data leads to equivariant versions of neighborhood deformation retractions and cofibrations, critical for analyzing mapping cylinders, constructing collar neighborhoods, and understanding the topology of singular spaces with group symmetries (Pflaum et al., 2017).

Equivariant tubular neighborhoods feature in the localization of the equivariant signature in the context of proper actions of discrete groups. The equivariant signature reduces to data computable on fixed-point sets (up to tubular neighborhood homeomorphism) and their normal bundles. Thus, the computation of noncommutative signatures is localized to equivariant geometry of these neighborhoods (Mishchenko et al., 2011).

In the study of configuration spaces, spaces of tubular neighborhoods ("tubular configurations") bear a strictly equivariant structure under diffeomorphism groups, leading to robust equivariant scanning maps and stable splittings generalizing Snaith decompositions of loop spaces (Manthorpe et al., 2013). This machinery establishes stable injectivity for stabilization maps and split injections on homology for diffeomorphism groups with marked points, tightly linking the local tubular geometry with global homotopical properties.

5. Algebraic and Topological Structures

The $G$-equivariant normal bundle $\nu(N)$ is intrinsically a $G$-equivariant vector bundle with fiberwise $G$-actions given by the induced differentials of $G$ on $M$. These lift canonically to the total space of $\nu(N)$. In equivariant bordism and K-theory, the classification of such bundles (especially over fixed-point sets) is governed by classifying spaces built from the action of the normalizer $N(H)$ for a subgroup $H<G$, and the equivariant tubular neighborhood enables reduction of global equivariant invariants to such local data (Mishchenko et al., 2011).

The deformation retraction properties associated with equivariant tubular neighborhoods facilitate the identification of equivariant mapping cylinders and cofibrations, permitting induction over stratified filtrations and application of homotopy-theoretic tools for equivariant spaces (Pflaum et al., 2017).

In configuration spaces, the tubular configuration model provides a $\Sigma_k$-equivariant bundle over the configuration space $C_k(M)$ with contractible fibers. Scanning maps and power set splittings, which depend crucially on the equivariant tubular geometry, allow for spectrum-level decompositions and stable equivalences respecting the diffeomorphism symmetries (Manthorpe et al., 2013).

6. Explicit Formulas and Local Models

Representative formulas for key constructions include:

• The $G$-equivariant embedding via the exponential map:

$$\exp\colon \left\{v\in\nu(N)\mid \|v\|<\varepsilon\right\}\longrightarrow M, \quad \exp(g\cdot v)=g\cdot \exp(v),$$

for $v\in\nu_x(N)$, $g\in G$ (Ramras, 2010, Mishchenko et al., 2011).

• Splitting of the tangent bundle over $TX$:

$$0\to\pi^*TX\to T(TX)\xrightarrow{\;\alpha\;}\pi^*TX\to0,$$

with a $G$-invariant symmetric bilinear form from the metric spray providing the splitting (Ramras, 2010).

• In stratified control data:

$$\pi_S: T_S\to S, \quad \pi_S(g\cdot x)=g\cdot\pi_S(x),\quad \rho_S:T_S\to[0,\infty),\quad \rho_S(g\cdot x)=\rho_S(x).$$

These satisfy compatibility axioms across overlapping tubes (Pflaum et al., 2017).

• In the stable splitting of tubular configuration spaces:

$$\Sigma^\infty_+\;\Gamma_c(W\setminus M_0;\mathcal{E}) \simeq \bigvee_{k\ge1}\Sigma^\infty_+\,D_k(M;T),$$

for suitable group actions on $M$ and $T$, yielding a $G$-equivariant splitting at the spectrum level (Manthorpe et al., 2013).

7. Significance and Advanced Applications

Equivariant tubular neighborhoods are instrumental in the analysis of equivariant cohomology, equivariant Morse theory, the computation of equivariant signatures, and the homotopy theory of configuration spaces, particularly in settings with intricate group symmetries or infinite-dimensional manifolds. These constructions lay the foundation for excision and localization statements, deformation retraction arguments, and stable splitting theorems that respect symmetries, serving as cornerstones in both geometric analysis and algebraic topology (Ramras, 2010, Pflaum et al., 2017, Mishchenko et al., 2011, Manthorpe et al., 2013).