A slice model turns a neighborhood of an orbit into a quotient built from a compact stabiliser acting on a normal space. The previous article used that model to reconstruct local quotient structure. The next question is sharper: when do two different-looking slice models describe the same local object?
Coordinate changes, different complements and conjugate stabilisers can produce presentations that are plainly equivalent. A homeomorphism between quotient spaces is weaker. It may preserve the visible cone while forgetting which group acted, how isotropy changes across strata, or which smooth invariant functions came from the representation.
The Gold Standard is to state the category before claiming uniqueness. Compare equivariant germs with equivariant maps, stratified quotient germs with stratum-preserving maps, and bare topological quotient germs only when the lost structure is genuinely irrelevant.
A practical orientation
Use this guide after identifying a stabiliser and normal slice. The worked examples compare finite-group representations near the origin and track what survives as an equivariant germ, a differentiable stratified quotient germ and a bare topological quotient.
Choose a reading route
Fix the comparison object
Read Chapters 1–3 to define germs, normalise stabilisers and distinguish three levels of equivalence.
Compare slice data
Use Chapters 4–6 for equivariant linear tests, invariant functions and quotient stratification.
Work the examples
Read Chapters 7–9 for equivalent and non-equivalent models plus changed-context transfer.
Stress uniqueness
Use Chapters 10–11 for limits, misconceptions and a practical comparison checklist.
Continue the verified route
Use Chapter 12 for FAQs and links to the subject root, parent, hub and preceding specialist articles.
Chapter contents
- 1. Decide which germ is being compared
- 2. Normalise stabilisers before comparing representations
- 3. Separate three levels of sameness
- 4. Use the derivative to test an alleged equivariant equivalence
- 5. Compare invariant functions when quotient coordinates are deceptive
- 6. Track orbit types rather than only the underlying cone
- 7. Worked comparison: two coordinate presentations, one equivariant germ
- 8. Counterexample pattern: the quotient forgets the presenting group
- 9. Transfer check: add a trivial summand, restrict a subgroup, change the base point
- 10. Limits and misconceptions
- 11. Practical comparison checklist
- 12. FAQs and the next reading route
1. Decide which germ is being compared
A germ records behavior in some sufficiently small neighborhood of a chosen point, with two representatives identified when they agree after restriction. For a slice representation H acting on V near 0, the equivariant germ retains the group action and maps that intertwine it. The quotient germ retains a neighborhood of [0] in V/H.
There are several quotient categories. A bare topological germ remembers open sets and continuity. A stratified germ also remembers orbit-type pieces. A differentiable quotient germ remembers the sheaf of smooth functions induced from H-invariant smooth functions on V. Claims of sameness grow weaker as structure is forgotten.
Write the comparison target first. “These quotients look like a half-line” can be enough for one topological question and completely inadequate for an orbifold or equivariant question. The category is part of the statement, not a technical footnote.
- Equivariant germ: group action and intertwining maps.
- Stratified quotient germ: orbit-type pieces retained.
- Differentiable quotient germ: invariant smooth-function structure retained.
- Topological quotient germ: only neighborhood topology retained.
2. Normalise stabilisers before comparing representations
Slice representations at points on the same orbit have conjugate stabilisers. Choose an explicit group isomorphism before comparing vector spaces. Without it, the equation relating two actions is not even well typed.
A representation equivalence is a linear isomorphism A: V → W together with a group isomorphism φ: H → K such that A(h·v)=φ(h)·A(v). For orthogonal slice representations, A need not be the original coordinate choice; any local equivariant diffeomorphism with invertible derivative gives a linear intertwiner at the fixed point.
Different invariant complements to the orbit tangent can yield different concrete slices while the resulting germs remain equivalent. Comparison should therefore target the stabiliser representation up to the declared isomorphism, not the literal coordinates used to draw the slice.
| Input difference | Usually harmless when | Required record |
|---|---|---|
| Conjugate stabiliser | Conjugating isomorphism is specified | Group isomorphism |
| Different basis | Linear map intertwines actions | Change-of-basis matrix |
| Different invariant complement | Local equivariant diffeomorphism exists | Slice maps and derivative |
| Different group | Only quotient homeomorphism observed | Do not claim equivariant equivalence |
3. Separate three levels of sameness
Level one is equivariant equivalence of slice germs. It is the strongest practical comparison here and carries stabiliser, representation and orbit-type information. It induces equivalence of quotient germs.
Level two is equivalence of differentiable stratified quotient germs. It preserves the quotient’s smooth invariant functions and orbit-type organization but may not identify a unique acting group presentation without additional hypotheses.
Level three is a homeomorphism of bare quotient germs. It preserves local topology but may merge models with different isotropy or smooth structure. The implications generally run downward: equivariant equivalence gives a structured quotient equivalence, which gives a topological equivalence. Reverse implications require proof, not intuition.
| Level | Retained structure | Safe conclusion |
|---|---|---|
| Equivariant germ | Group, action, local map | Same slice model in declared category |
| Differentiable stratified quotient | Invariant smooth functions and strata | Same structured quotient germ |
| Bare quotient topology | Neighborhood topology | Same local topological shape only |
4. Use the derivative to test an alleged equivariant equivalence
Suppose F is an equivariant local diffeomorphism between slice neighborhoods, sends 0 to 0 and is equivariant with respect to φ. Differentiating the identity F(h·v)=φ(h)·F(v) at 0 gives an intertwining linear map dF₀. Because F is a local diffeomorphism, dF₀ is invertible.
This derivative test is fast: compare dimensions, fixed subspaces, character data for finite or compact groups, and multiplicities of irreducible summands. A mismatch proves that no such smooth equivariant germ equivalence exists. A match suggests a candidate intertwiner but does not by itself construct the nonlinear germ when extra structure is present.
For purely linear slice models, an invertible intertwiner already supplies the equivalence. Check the chosen group isomorphism and the full representation, not only the stabiliser order or dimension.
- Compare vector-space dimension.
- Compare fixed-subspace dimension.
- Compare irreducible multiplicities or characters.
- Construct an explicit intertwiner.
- Verify compatibility with the chosen group isomorphism.
5. Compare invariant functions when quotient coordinates are deceptive
A quotient can be embedded using generators of invariant polynomials or described by smooth invariant functions. Relations among generators encode more than a sketch of the orbit space. They can reveal singular strata and how quotient coordinates meet.
For the sign action of C₂ on R, the invariant coordinate is u=x² with u≥0. The quotient is a half-line, and the boundary point corresponds to the fixed point. Scaling x by a nonzero constant rescales u but preserves the germ and its stratification.
Do not assume that any homeomorphism of half-lines preserves the differentiable quotient structure. A map such as u↦u³ is a homeomorphism near zero, yet its inverse is not smooth in the ordinary coordinate. The permitted map depends on the chosen differentiable-space definition.
| Model | Invariant description | Visible quotient feature |
|---|---|---|
| C₂ on R by x↦−x | u=x², u≥0 | Boundary stratum at 0 |
| Trivial action on R | u=x | No isotropy change |
| C₂ on R² by central inversion | Quadratic invariants with one relation | Cone-like singular quotient |
6. Track orbit types rather than only the underlying cone
The quotient stratification records which points have conjugate stabilisers. Near the slice origin, the fixed point belongs to the deepest local stratum, while nearby directions may have smaller stabiliser. The link of the quotient singularity is built from the unit sphere in V modulo H.
Compare the dimensions and incidence relations of strata, not just the overall dimension of V/H. A topological cone can carry different stratified data. If a proposed equivalence sends a singular stratum into a regular one, it is not a stratified equivalence even when it is a homeomorphism.
The link is a useful changed-scale view. Removing the cone point and intersecting with a small sphere separates radial behavior from angular orbit structure. Equivalent linear slice representations yield compatible quotient links; different links disprove local quotient equivalence in any stronger category.
- List stabiliser conjugacy classes.
- Record stratum dimensions and incidences.
- Compare the quotient link S(V)/H.
- Check whether the cone point and singular sets are preserved.
7. Worked comparison: two coordinate presentations, one equivariant germ
Let H=C₂ act on V=R by x↦−x. Let K=C₂ act on W=R by y↦−y. Choose the group isomorphism sending the nontrivial element to the nontrivial element and the linear map A(x)=3x. Then A(−x)=−A(x), so A is an invertible intertwiner.
The quotient coordinates u=x² and v=y² satisfy v=9u under A. Near zero, u↦9u is a diffeomorphism of half-line quotient germs, preserves the boundary stratum and carries invariant smooth functions by composition. The differing scale is presentation, not topology.
Now replace A by the coordinate drawing y=x³. It remains equivariant and induces v=u³, a homeomorphism of half-lines, but its derivative at zero is zero. It is not a local diffeomorphism and therefore does not prove equivalence of smooth equivariant germs. The same picture supports different conclusions in different categories.
| Map | Equivariant? | Derivative at 0 | Valid conclusion |
|---|---|---|---|
| A(x)=3x | Yes | 3, invertible | Equivariant linear germ equivalence |
| F(x)=x³ | Yes | 0 | Topological quotient homeomorphism only |
| G(x)=x+ x³ | Yes | 1, invertible | Local equivariant diffeomorphism near 0 |
8. Counterexample pattern: the quotient forgets the presenting group
Consider a trivial action of C₂ on R and a trivial action of C₃ on R. Their bare quotient spaces are both R, with identical local topology and no orbit-type change inside each presentation. Yet the equivariant germs are not equivalent under a group isomorphism because C₂ and C₃ are not isomorphic.
If group labels are retained as isotropy data, the models also differ as orbifold-style objects even though the underlying topological and ordinary smooth spaces agree. This is the simplest warning that a quotient map can forget automorphisms.
Change the context by declaring that only the underlying topological neighborhood matters. The two models then become equivalent for that limited question. This is not a contradiction; it is a change of category. The proof obligation is to say which information is intentionally discarded.
| Comparison | Same? | Reason |
|---|---|---|
| Bare topological quotient germ | Yes | Both are neighborhoods in R |
| Ordinary smooth quotient space | Yes | Both have ordinary smooth coordinate |
| Equivariant germ | No | Presenting groups are non-isomorphic |
| Isotropy-labelled object | No | Local automorphism groups differ |
9. Transfer check: add a trivial summand, restrict a subgroup, change the base point
First add the same trivial one-dimensional summand to both representations. An existing intertwiner extends by the identity, and the quotient gains a product direction. If a claimed invariant cannot survive this stabilisation, it was tied to coordinates rather than the representation germ.
Second restrict both actions along a subgroup. Equivalence should restrict to equivalence. Failure can reveal that the original comparison matched quotient shapes without respecting the chosen group isomorphism.
Third move from the fixed point into a neighboring orbit-type stratum. The stabiliser and normal representation change. A local comparison at the origin does not automatically classify every nearby base point; repeat the slice construction at the new orbit and compare the induced models.
| Changed context | Expected behavior under true equivariant equivalence | What failure reveals |
|---|---|---|
| Add equal trivial summand | Intertwiner extends by identity | Coordinate-dependent criterion |
| Restrict subgroup | Representations remain equivalent | Group map was not respected |
| Move base point | New slice comparison required | Origin germ was overextended |
10. Limits and misconceptions
The slice theorem supplies a local model under hypotheses such as properness; it is not a global quotient classification. Two actions can have equivalent germs near selected orbits and very different global orbit spaces.
Matching stabiliser groups is not enough. The same compact group can act through non-equivalent representations with different fixed subspaces and quotient strata. Conversely, a bare quotient may fail to remember the group at all.
An invariant-ring or smooth-function comparison depends on the chosen category and regularity. Polynomial, smooth and continuous invariants carry different information. State which algebra or sheaf is being used before invoking reconstruction or uniqueness.
- Misconception: same stabiliser means same slice.
- Misconception: homeomorphic quotients imply equivalent actions.
- Misconception: a local slice classifies the global quotient.
- Limit: polynomial, smooth and continuous quotient structures differ.
11. Practical comparison checklist
Record the acting groups, base points, stabilisers, group isomorphism, slice spaces and regularity category. Verify properness or the hypotheses under which the slice model is being used. State whether the object is equivariant, stratified, differentiable or merely topological.
For an equivariant claim, compute dimensions, fixed spaces and irreducible data; then exhibit an intertwiner or local equivariant diffeomorphism. For a quotient claim, compare invariant functions, strata, links and the behavior at the distinguished orbit.
Finally, test one changed context and one potential counterexample. Write down the strongest conclusion actually proved and the structure forgotten in passing to the quotient. A good comparison ends with a boundary, not only an isomorphism symbol.
- Groups, stabilisers and explicit group isomorphism.
- Slice spaces and representation data.
- Chosen equivalence category and regularity.
- Intertwiner or structured quotient map.
- Strata, link and distinguished point.
- Changed-context test and stated uniqueness limit.
12. FAQs and the next reading route
Slice-model comparison turns a local reconstruction into a disciplined uniqueness question. It prevents a convenient quotient picture from being mistaken for recovery of the acting group or representation.
Read the Topology root for the broader route. The stabiliser-stratification article introduces automorphism-sensitive classification, and the isotropy-representation article constructs the normal slice model used here. This article then compares those models and states what the quotient can forget.
The parent Gold Standard and How X Works hub connect the method to a general learning habit: name the object, choose the comparison category, test a changed context and state the information lost under abstraction.
Does an equivariant linear isomorphism give the same quotient germ?
Yes. It induces a compatible quotient map and preserves the representation data under the chosen group isomorphism.
Does a homeomorphism of quotient germs recover the stabiliser?
Not in general. The quotient topology can forget the presenting group and isotropy labels; additional structured data and hypotheses are needed.
Why compare links?
The quotient link isolates angular orbit structure around the cone point and can disprove local equivalence when overall dimensions or drawings look similar.
Continue through the verified ecosystem
Sources and scope note
- Orbispaces as differentiable stratified spaces — used for local quotient models of compact-group representations and differentiable stratified structure.
- Slice theorem and orbit type stratification in infinite dimensions — used for the relationship between slices and orbit-type stratification, with its stated hypotheses.
The finite-group examples and comparison tables are original teaching examples. The sources support slice, quotient and stratification concepts; the article does not claim a general reconstruction theorem beyond the categories and limits stated.