Changing the diagram category preserves stable motivic homotopy theory: the comparison cofibres become arbitrarily connective.
Garkusha’s Comparison Theorem recovers stable motivic homotopy theory from GL-, SL-, Sp-, SO- and O-spectra, under the field hypotheses stated below. The diagram morphism objects are induced Thom spaces. After untwisting, the comparison maps are induced by . Their pointed cofibres use , whose connectivity increases for fixed n as q grows. This eventual connectivity makes the comparison cofibre spectra vanish. The experiments show the quotient towers, a compact real model of the bundle twist, and the geometry of two fibre sequences.
Symbolic: the elementary-piece recipes, dimension sums, connectivity bounds and mod-2 Poincaré polynomials. Numerical: rotation reconstruction, tangent decompositions and winding indices at isolated zeros. Geometric models: real and complex compact homotopy types; these do not display the full motivic spaces or their weight grading. Not shown: semistability (§4), model-category constructions, and §7 beyond its definitions and proof map.
Selection
Computation
Verification
How to operate the experiments
The 3D view
- Drag to rotate, scroll or pinch to zoom, shift-drag (or two-finger drag) to pan. Double-click or Reset view restores the camera. With the view focused: arrow keys rotate, + and − zoom, 0 resets.
- Click a bar, a zero of a vector field, or the identity in the rotation ball to read its definition. Expand enlarges the view; Esc closes it.
- Each 3D page has a Play button; the sliders stop any running animation.
The four pages
- Map of the proof: choose a result to see what it uses (blue) and what uses it (amber).
- Stiefel landscape: choose a family of groups; sweep the homotopy degree ; click a bar to see the tower of and its classical check.
- Twisting: move the point on the sphere; change the vector field and watch the indices of its zeros.
- Fibre towers: the rotation ball over , and the first-column map of .
The mathematics behind the picture
Notation follows G. Garkusha, Group schemes and motivic spectra, Israel J. Math. (2024). Theorem numbers follow the order of the source file supplied: Comparison Theorem 6.1, Proposition 6.3 and Corollary 6.4 (connectivity), Theorem 6.5 (the maps ), Proposition 6.7, and §7 (Definition 7.1, Theorems 7.2 and 7.3). Section 4 (semistability) is not treated here.
A. Mathematical construction
A.1 Diagram spectra
Let be the category of pointed motivic spaces over a field , and a symmetric monoidal -category with a commutative ring object in . A -spectrum is a -space with maps ; by Theorem 2.5 (Mandell–May–Schwede–Shipley) these are the enriched functors on the category with
A.2 Group schemes and the size convention
Let for GL and for SL, Sp, SO and O. The charts use and . Thus, for example, and for Sp. This reindexes the paper’s even-sized objects by ordinary integers.
The unbased group-scheme diagram category has endomorphism space and no morphisms between unequal objects. Adjoining disjoint basepoints gives the pointed enrichment: endomorphism objects , and the one-point object off the diagonal. Block sums and the prescribed shuffle matrices define the symmetric monoidal structure. The sphere ring object sends to . In this reindexed convention, for ,
For the morphism object is the basepoint. The ordinary diagram category contributes ; its inclusion is the fibre over the identity coset. In the original size convention, the first formula is evaluated on objects .
A.3 Stable model structure and the statements
The stable model structure on is the Bousfield localisation of the pointwise structure at the maps (Definition 5.4), and the left Kan extension and the forgetful functor form a Quillen pair (Corollary 5.6).
Scope of §7. The framed-motive application is stated under the source’s assumption that is an (infinite) perfect field, with for SO and O. Definition 7.1 uses all original indices for GL; for the other families it uses even and the colimit over even . This is separate from the any-field scope of the GL, SL and Sp comparison theorem.
A.4 The Stiefel quotients and their towers
The proof of Theorem 6.5 reduces everything to the smooth varieties . For the homogeneous-space chains used here, with the local-triviality and motivic fibre-sequence results cited in the proof, there is a fibre sequence , so is built from the elementary quotients, which the paper identifies as follows (the first-column maps and the quadrics , ):
The motivic sphere has bidegree and is --connected.
The elementary formulas use for GL and Sp, for SL, and for the two SO formulas (with characteristic different from 2). The groups are trivial. The O diagonal includes the additional component piece . Here (p,w) denotes , corresponding to standard bidegree (p+w,w).
B. Geometric interpretation
The visualisation map. Motivic spaces have no canonical picture. I use two exact devices. (i) Arithmetic of towers: each is replaced by the list of its elementary quotients with their bidegrees ; the bar height is the guaranteed connectivity plus one. (ii) Real and complex shadows: on real points , so ; on complex points , so it goes to . The groups realise as , , , , and on complex points as . These are classical facts, not from the paper.
| Graphical element | Mathematical object | Status |
|---|---|---|
| Bar at in the landscape | the quotient ; height , a guaranteed vanishing threshold in simplicial degree p=a−b | exact arithmetic from the paper’s isomorphisms; a lower bound for the connectivity |
| Horizontal plane, warm bars | homotopy degree ; warm means the quotient can have | exact |
| White line through row | the layers of the comparison cofibre | exact |
| Green dot on a bar | connectivity of the compact real Stiefel model, checked against its mod-2 Poincaré polynomial; green means it equals the bar | classical check, outside the paper |
| Sphere , tangent plane, normal line | the base of , the bundle and the tautological line | real shadow |
| Arrows, zeros (gold , blue ) | a section of and its zeros with their indices | exact up to numerical winding numbers |
| Rotation ball | as the ball of radius with antipodal boundary identified; pink curves are fibres of | real shadow, exact |
| Vertical lines over the punctured plane | the fibres of | real points; real-point trivialisation using J; no global algebraic section over the whole punctured plane |
In the rotation model, the displayed representative is , where . The choice of reference frame is local: it has a gauge discontinuity at . No global continuous section of the circle bundle is asserted. The whole fibre is independent of this choice. Its continuous lift has antipodal endpoints, including when the fibre lies entirely in the cut.
C. Mathematical mechanism
C.1 The induced Thom space and its actual base
Put . Over the Stiefel variety , the associated bundle
is the complement of the canonical trivial frame of rank . Its Thom space is . On this base there is an algebraic direct-sum trivialisation , induced by the ambient representation. Equivalently, for a -space extending the given -action, the shearing map gives . Applied to , it gives
The Twisting page shows the compact real GL model with : over . The map trivialises . On , isolated-zero fields have index sum . Repeated eigenvalues of the quadratic field produce a zero circle or the whole sphere, shown in purple; the ordinary isolated-zero sum is then not applied.
The punctured affine base is different. On the bundle satisfies in , but is not algebraically stably trivial for . Indeed, for , the Čech calculation gives . Stable triviality would make this module projective, split the defining sequence, and give , impossible at the origin. The pullback to the affine Stiefel variety has the complementary-bundle trivialisation above; the real compact restriction also has the displayed topological stable trivialisation. Neither supplies a global algebraic trivialisation of Q on U.
C.2 The normalised comparison cofibre and its indexing
Since reflects stable equivalences, it suffices to prove that the comparison maps are stable equivalences. They are induced by : the basepoint goes to the disjoint basepoint, and the non-basepoint goes to the identity coset. The pointed cofibre is .
After the harmless suspension and reindexing used in the proof, one comparison cofibre has the tail
For GL this is . For the other families, and the last smash factor is . The skeletal approximation agrees with through level i and continues thereafter by smashing with K. The natural map from the ordinary free spectrum to is a stable equivalence, giving
C.3 The connectivity of grows without bound
The bottom elementary quotient of is the one with the smallest index. Reading off from A.4, the guaranteed -connectivity of and the connectivity bound for the pointed quotient defining (where ) are
| Family | () | quotient bound , | bottom piece |
|---|---|---|---|
| GL | |||
| SL (size ) | |||
| Sp (size ) | |||
| SO, O (size ) | and |
By Proposition 6.3 and Corollary 6.4 (which use Morel’s stable connectivity theorem) an --connected space has a suspension spectrum in , so .
C.4 Why the filtered homotopy colimit is zero
The bounds in the table are affine functions of with positive slope. Hence ; the exceptional initial layer has no effect on this conclusion. The cofibrant skeletal filtration computes a filtered homotopy colimit. Equivalently, use its derived version if the original presentation is not cofibrant. Homotopy sheaves commute with this filtered homotopy colimit.
We use the standard bigrading . The degree plane displays . Since , for each fixed the sheaf is zero once . Therefore
The same reasoning applies to both comparison cofibres, proving that are stable equivalences. The argument uses eventual vanishing in each degree; it does not assert that any individual finite Stiefel quotient is contractible.
C.5 From to the Quillen equivalence
Since the generators of the localisation are already stable motivic equivalences of the underlying ordinary spectra, a map of -spectra is a stable equivalence if and only if its underlying map is one (Proposition 6.7). For a cofibrant the comparison squares built from show that the unit is a stable equivalence, and reflects stable equivalences between fibrant objects, so is a Quillen equivalence (Theorem 6.1).
C.6 What the pages show about the families
All four families have connectivity tending to , but at different rates in the -index : slope for and , slope for and , and slope for in the -index. For the symmetric groups is a finite set of points: nothing grows, and the argument has no analogue (the comparison of ordinary and symmetric spectra is quoted from the literature, not reproved).
D. Connection with the theory
| Page | Statement in the paper | What is computed |
|---|---|---|
| Map of the proof | §§2–7 | a dependency diagram read off from the proofs; no mathematics is computed |
| Stiefel landscape | Theorem 6.5 (the connectivity of and the vanishing of ); Proposition 6.3, Corollary 6.4; the proof of Theorem 6.1; the varieties of Definition 7.1 | the tower of each quotient; its connectivity; the layers; the real and complex checks by exact Poincaré polynomials; dimension additivity |
| Twisting | the untwisting step of Theorem 6.5, | the explicit isomorphism ; Poincaré–Hopf index sums |
| Fibre towers | the fibre sequences and the first-column equivalence, in the proof of Theorem 6.5 | the rotation geometry of ; the class of the fibre in ; the affine-line fibres |
| Not drawn | §4 (excluded); §5 model categories; the proofs of Proposition 6.7 and Theorems 7.2, 7.3 | nothing |
E. Limitations
Arithmetic and numerical checks. The tower dimensions, bidegrees, lower bounds and Poincaré coefficients are computed by integer arithmetic. The compact-model Poincaré polynomials use coefficients; connectivity is also supported by the known Stiefel fibre sequences, rather than inferred from cohomology for arbitrary spaces. Winding indices and rotations are numerical. The rotation panel checks reconstruction of R from the displayed coordinate and checks the fibre class using a continuous quaternion lift.
Lower bounds. The connectivity from fibre sequences is a lower bound, as in the paper. The real and complex checks show it is attained for the four families; this is not claimed for the motivic spaces themselves.
Shadows. Real and complex points of the Stiefel varieties are not the motivic spaces; they discard the weight grading. In particular the real picture of uses a trivialisation by which does not exist algebraically: a section would give polynomials with , impossible at .
Initial groups and components. is a point, so the even-quadric formula is used only for . On the diagonal , the selectors show SO and O separately: as varieties (after choosing a reflection), and the additional component piece is included. Their complex shadows have respectively two and one connected components. The split real forms have four and two. For their quotients agree, as used in the eventual-connectivity argument.
Not covered. Semistability (§4), excluded by design; the model-category arguments; the passage from spaces to beyond quoting Morel’s theorem; characteristic for ; Theorems 7.2 and 7.3 beyond the map of the proof.
What the experiments show
- The group action is a Thom space over a Stiefel variety. The morphism spaces of the new diagram category are Thom spaces of the complementary bundle over ; the ordinary spectra see only the fibre over the base point.
- The twist is real and one suspension removes it. On the bundle is not trivial (the indices add up to for every displayed field with isolated zeros), yet is trivial through an explicit isometry.
- Connectivity, not the group, is what matters. The same argument works for four families because the quotients are towers of spheres, and their connectivity grows with at a rate that depends on the family.
- A cross-check that is independent of the paper. The recipe for the connectivity, realised on real and complex points, reproduces the classical compact Stiefel manifolds , , , (and the square for ) in the finite cases checked; O has its additional component factor on the diagonal.
- Only finitely many layers matter in each degree. The degree plane makes the vanishing of the comparison cofibre visible: for each , the layers that can contribute form a finite initial segment.
Questions the experiments suggest
- Is the growth of the connectivity of also necessary for a family of groups to give a Quillen equivalence by this argument? A test family: the monomial groups , for which the quotient is never connected.
- The rates differ by a factor of two between and in the -index. Do the rates give quantitative bounds on the connectivity of the unit in terms of the connectivity of ?
- How can one compare the algebraic obstruction to stable triviality of on with the Euler number of its compact real restriction? How does pulling back to the Stiefel variety remove the former obstruction while leaving a nontrivial complementary bundle before stabilisation?
Extensions
- The unit of the Quillen equivalence as a bundle. Draw as a bundle over the Stiefel base with the unit as the inclusion of a fibre, with the base connectivity providing the comparison estimate after untwisting.
- Higher fibre towers. Draw and the first stunted-projective-space cells of , to see the lowest cell in degree directly.
- Section 7. Draw the spectrum of Definition 7.1 for several side by side, and the comparison as a family of models of the same localisation.