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arxiv: 2512.08304 · v3 · submitted 2025-12-09 · 🧮 math.KT · math.OA· math.QA

Milnor meets Hopf and Toeplitz at the K-theory of quantum projective planes

Francesco D'Andrea , Piotr M. Hajac , Tomasz Maszczyk , Bartosz Zieli\'nski This is my paper

Pith reviewed 2026-05-16 23:57 UTC · model grok-4.3

classification 🧮 math.KT math.OAmath.QA
keywords K-theoryMilnor connecting homomorphismquantum projective planesHopf-Galois theorypiecewise cleft algebrasToeplitz C*-algebranoncommutative Hopf fibrationsprojective modules
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The pith

Explicit Milnor idempotents determine the K-theory classes of projective modules from noncommutative Hopf fibrations over quantum projective planes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies Milnor's connecting homomorphism to compute clutching K1-classes explicitly for finitely generated projective modules associated to piecewise cleft principal comodule algebras. This yields an explicit Milnor idempotent that determines the module from its representation matrix. The construction is used for quantum complex projective planes to find K0-generators via modules attached to noncommutative Hopf fibrations. By rewriting these classes through explicit homotopies as elementary projections in the Toeplitz C*-algebra, the generators are shown to lie in the positive cone of K0, a feature absent from the classical commutative case.

Core claim

For a finitely generated projective module associated to any piecewise cleft principal comodule algebra, the clutching K1-class equals an explicit formula in terms of the representation matrix, so the module corresponds to an explicit Milnor idempotent. Applied to quantum projective planes, this produces K0-generators from noncommutative Hopf fibrations. These classes are then rewritten, via explicit homotopies of unitaries inside the Toeplitz C*-algebra, as elementary projections, placing every generator in the positive cone of K0.

What carries the argument

The Milnor connecting homomorphism from odd to even K-groups, applied to piecewise cleft principal comodule algebras to produce explicit Milnor idempotents from representation matrices.

If this is right

  • The K0-group of quantum projective planes is generated by classes of modules coming from noncommutative Hopf fibrations.
  • All such K0-generators lie in the positive cone of K0.
  • The clutching K1-class of any such module is computed directly from its representation matrix.
  • Explicit homotopies inside the Toeplitz C*-algebra convert the Milnor idempotents into elementary projections.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same explicit formula may compute clutching classes for modules over other noncommutative spaces that admit piecewise cleft comodule algebra structures.
  • Positivity of K0-generators could serve as a computational signature that distinguishes quantum projective planes from their classical limits.
  • The approach may extend to K-theory calculations for higher-dimensional quantum projective spaces.

Load-bearing premise

The principal comodule algebras arising from the noncommutative Hopf fibrations over quantum projective planes are piecewise cleft, and the relevant unitaries admit explicit homotopies inside the Toeplitz C*-algebra.

What would settle it

A projective module over a quantum projective plane whose actual clutching K1-class fails to match the value given by the formula from its representation matrix.

read the original abstract

We explore applications of the celebrated construction of the Milnor connecting homomorphism from the odd to the even K-groups in the context of Hopf--Galois theory. For a finitely generated projective module associated to any piecewise cleft principal comodule algebra, we provide an explicit formula computing the clutching $K_1$-class in terms of the representation matrix defining the module. Thus, the module is determined by an explicit Milnor idempotent. We apply this new tool to the K-theory of quantum complex projective planes to determine their $K_0$-generators in terms of modules associated to noncommutative Hopf fibrations. On the other hand, using explicit homotopy between unitaries, we express the $K_0$-class of the Milnor idempotents in terms of elementary projections in the Toeplitz C*-algebra. This allows us to infer that all our generators are in the positive cone of the $K_0$-group, which is a purely quantum phenomenon absent in the classical case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops an explicit formula for the clutching K1-class of finitely generated projective modules over any piecewise cleft principal comodule algebra, expressed in terms of the representation matrix and yielding an explicit Milnor idempotent via the Milnor connecting homomorphism. It applies this to the K-theory of quantum complex projective planes by associating modules to noncommutative Hopf fibrations, thereby determining the K0-generators. Using explicit homotopies between unitaries inside the Toeplitz C*-algebra, the K0-classes of these idempotents are rewritten in terms of elementary projections, from which the authors conclude that all generators lie in the positive cone of K0—a feature absent in the classical case.

Significance. If the constructions are verified, the work supplies concrete, representation-matrix-based generators for the K0-group of quantum projective planes and isolates a quantum-specific positivity phenomenon. The explicit Milnor idempotent formula and the Toeplitz homotopy technique constitute reusable tools that link Hopf-Galois theory with classical K-theory and C*-algebra methods; these strengths would be valuable for computations in noncommutative geometry.

major comments (2)
  1. [§4.2] §4.2: The principal comodule algebras arising from the noncommutative Hopf fibrations over quantum projective planes are asserted to be piecewise cleft, yet no explicit local cleaving maps or coaction-compatible sections are constructed for these specific extensions; this verification is load-bearing for transferring the clutching K1-class formula to the claimed K0-generators.
  2. [§5.3, Eq. (5.4)] §5.3, Eq. (5.4): The homotopy between unitaries is stated to exist inside the Toeplitz C*-algebra and to produce the positive-cone representatives, but the argument does not detail how the homotopy remains within the algebra once the quantum deformation parameters are inserted; this step directly supports the claim that the generators lie in the positive cone.
minor comments (2)
  1. [§2] The notation for the coaction and the representation matrix is introduced in §2 but used with slight variations in later sections; a single consolidated definition would improve readability.
  2. A brief comparison table of the classical versus quantum K0-generators would clarify the claimed absence of the positive-cone property in the commutative case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying these two points that require clarification. We will revise the manuscript to supply the missing explicit constructions and verifications.

read point-by-point responses

  1. Referee: [§4.2] §4.2: The principal comodule algebras arising from the noncommutative Hopf fibrations over quantum projective planes are asserted to be piecewise cleft, yet no explicit local cleaving maps or coaction-compatible sections are constructed for these specific extensions; this verification is load-bearing for transferring the clutching K1-class formula to the claimed K0-generators.

    Authors: We agree that an explicit verification is necessary. In the revised version we will add, in §4.2, the local cleaving maps for the Hopf-Galois extensions arising from the noncommutative Hopf fibrations. These maps are obtained by restricting the standard coaction on the quantum sphere to the appropriate open sets of the base and composing with the projection onto the coinvariants; the resulting sections are coaction-compatible by construction and satisfy the piecewise cleft condition. revision: yes

  2. Referee: [§5.3, Eq. (5.4)] §5.3, Eq. (5.4): The homotopy between unitaries is stated to exist inside the Toeplitz C*-algebra and to produce the positive-cone representatives, but the argument does not detail how the homotopy remains within the algebra once the quantum deformation parameters are inserted; this step directly supports the claim that the generators lie in the positive cone.

    Authors: We accept that the argument must be made uniform in the deformation parameter. In the revision we will expand §5.3 to exhibit an explicit straight-line homotopy between the deformed unitaries that stays inside the Toeplitz algebra for all admissible values of the quantum parameter. The homotopy is defined by convex combination in the multiplier algebra and is shown to remain norm-continuous and to satisfy the required relations by direct computation with the deformed commutation relations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit general formula applied after assumption check

full rationale

The paper states a general result: for any piecewise cleft principal comodule algebra, an explicit formula computes the clutching K1-class from the representation matrix, yielding a Milnor idempotent. It then specializes to the Hopf-Galois extensions over quantum projective planes by invoking the piecewise cleft property for those specific comodule algebras and constructing explicit unitaries homotopies inside the Toeplitz algebra. These steps consist of new explicit constructions resting on standard K-theory and C*-algebra machinery rather than any reduction of the output K0-generators or positive-cone claim back to the input data by definition, fitting, or self-citation chain. No equation or claim equates a derived quantity to its own defining assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard background from algebraic K-theory, Hopf-Galois theory, and C*-algebras without introducing new free parameters or invented entities; the piecewise-cleft condition is a domain assumption for the applications.

axioms (2)
  • domain assumption Milnor connecting homomorphism extends to the Hopf-Galois setting for piecewise cleft principal comodule algebras
    Invoked to obtain the explicit clutching formula.
  • domain assumption Homotopies between unitaries exist in the Toeplitz C*-algebra that realize the K0-classes
    Used to express Milnor idempotents via elementary projections.

pith-pipeline@v0.9.0 · 5492 in / 1308 out tokens · 55783 ms · 2026-05-16T23:57:45.052241+00:00 · methodology

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Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    Adams, J. F. Vector fields on spheres. Annals of Mathematics 75 , 3 (1962), 603--632

    work page 1962

  2. [2]

    M., and Tobolski, M

    Arici, F., D'Andrea, F., Hajac, P. M., and Tobolski, M. An equivariant pullback structure of trimmable graph C*-algebras. Journal of Noncommutative Geometry 16 , 3 (2022)

    work page 2022

  3. [3]

    Atiyah, M. F. K-theory , Harvard Lecture 1964 (notes by D. W. Anderson). Benjamin, 1967

    work page 1964

  4. [4]

    F., and Todd, J

    Atiyah, M. F., and Todd, J. A. On complex S tiefel manifolds. Mathematical Proceedings of the Cambridge Philosophical Society 56 , 4 (1960), 342–353

    work page 1960

  5. [5]

    F., De Commer, K., and Hajac, P

    Baum, P. F., De Commer, K., and Hajac, P. M. Free actions of compact quantum group on unital C* -algebras. Documenta Mathematica 22\/ (2017), 825--849

    work page 2017

  6. [6]

    F., Dabrowski, L., and Hajac, P

    Baum, P. F., Dabrowski, L., and Hajac, P. M. Noncommutative B orsuk- U lam-type conjectures. Banach Center Publications 106 , 1 (2015), 9--18

    work page 2015

  7. [7]

    F., and Hajac, P

    Baum, P. F., and Hajac, P. M. Local proof of algebraic characterization of free actions. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 10 , 060 (2014), 1--7

    work page 2014

  8. [8]

    F., Hajac, P

    Baum, P. F., Hajac, P. M., Matthes, R., and Szyma \'n ski, W. The K -theory of H eegaard-type quantum 3-spheres. K-Theory 35\/ (2005), 159--186

    work page 2005

  9. [9]

    K-theory for operator algebras , vol

    Blackadar, B. K-theory for operator algebras , vol. 5 of Mathematical Sciences Research Institute Publications . Cambridge University Press, 1998

    work page 1998

  10. [10]

    Covering and gluing of algebras and differential algebras

    Calow, D., and Matthes, R. Covering and gluing of algebras and differential algebras. Journal of Geometry and Physics 32 , 4 (2000), 364 -- 396

    work page 2000

  11. [11]

    Coburn, L. A. The C* -algebra generated by an isometry. Bull. Amer. Math. Soc. 73 , 6 (1967), 722--726

    work page 1967

  12. [12]

    Coburn, L. A. The C* -algebra generated by an isometry. II . Transactions of the American Mathematical Society 137\/ (1969), 211--217

    work page 1969

  13. [13]

    M., Matthes, R., and Wagner, E

    Dąbrowski, L., Hadfield, T., Hajac, P. M., Matthes, R., and Wagner, E. Index pairings for pullbacks of C* -algebras. Banach Center Publications 98\/ (2012), 67--84

    work page 2012

  14. [14]

    Hajac, P. M. Strong connections on quantum principal bundles. Communications in mathematical physics 182 , 3 (1996), 579--617

    work page 1996

  15. [15]

    Hajac, P. M. Bundles over quantum sphere and noncommutative index theorem. K-theory 21 , 2 (2000), 141--150

    work page 2000

  16. [16]

    M., Kaygun, A., and Zieli \'n ski, B

    Hajac, P. M., Kaygun, A., and Zieli \'n ski, B. Quantum projective space from T oeplitz cubes. Journal of Noncommutative Geometry 6 , 3 (2012), 603--621

    work page 2012

  17. [17]

    M., Kr \"a hmer, U., Matthes, R., and Zieli \'n ski, B

    Hajac, P. M., Kr \"a hmer, U., Matthes, R., and Zieli \'n ski, B. Piecewise principal comodule algebras. Journal of Noncommutative Geometry 5 , 4 (2011), 591--614

    work page 2011

  18. [18]

    M., and Maszczyk, T

    Hajac, P. M., and Maszczyk, T. Pullbacks and nontriviality of associated noncommutative vector bundles. Journal of noncommutative geometry 12 , 4 (2018), 1341--1358

    work page 2018

  19. [19]

    M., Matthes, R., and Szyma \'n ski, W

    Hajac, P. M., Matthes, R., and Szyma \'n ski, W. Noncommutative index theory for mirror quantum spheres. Comptes Rendus Mathematique 343 , 11 (2006), 731 -- 736

    work page 2006

  20. [20]

    M., Nest, R., Pask, D., Sims, A., and Zieli \'n ski, B

    Hajac, P. M., Nest, R., Pask, D., Sims, A., and Zieli \'n ski, B. The K -theory of twisted multipullback quantum odd spheres and complex projective spaces. Journal of Noncommutative Geometry 12 , 3 (2018), 823--863

    work page 2018

  21. [21]

    M., Rennie, A., and Zieli \'n ski, B

    Hajac, P. M., Rennie, A., and Zieli \'n ski, B. The K -theory of H eegaard quantum lens spaces. Journal of Noncommutative Geometry 7 , 4 (2013), 1185--1216

    work page 2013

  22. [22]

    M., and Rudnik, J

    Hajac, P. M., and Rudnik, J. Noncommutative bundles over the multi-pullback quantum complex projective plane. New York Journal of Mathematics 23\/ (2017), 295--313

    work page 2017

  23. [23]

    M., and Wagner, E

    Hajac, P. M., and Wagner, E. The pullbacks of principal coactions. Documenta Mathematica 19\/ (2014), 1025--1060

    work page 2014

  24. [24]

    The M ayer-- V ietoris and the P uppe sequences in K -theory for C* -algebras

    Hilgert, J. The M ayer-- V ietoris and the P uppe sequences in K -theory for C* -algebras. Studia Mathematica 83 , 2 (1986), 105--117

    work page 1986

  25. [25]

    H., and Szyma \'n ski, W

    Hong, J. H., and Szyma \'n ski, W. Quantum spheres and projective spaces as graph algebras. Communications in mathematical physics 232 , 1 (2002), 157--188

    work page 2002

  26. [26]

    H., and Szyma \'n ski, W

    Hong, J. H., and Szyma \'n ski, W. Noncommutative balls and mirror quantum spheres. Journal of the London Mathematical Society 77 , 3 (2008), 607--626

    work page 2008

  27. [27]

    a nich, K. Vektorraumb\

    J\" a nich, K. Vektorraumb\" u ndel und der raum der Fredholm-operatoren. Mathematische Annalen 161 , 3 (1965), 129--142

    work page 1965

  28. [28]

    K-theory: An introduction

    Karoubi, M. K-theory: An introduction . Springer, 2009

    work page 2009

  29. [29]

    A two-parameter quantum deformation of the unit disc

    Klimek, S., and Lesniewski, A. A two-parameter quantum deformation of the unit disc. Journal of Functional Analysis 115 , 1 (1993), 1 -- 23

    work page 1993

  30. [30]

    M., Neshveyev, S

    Dabrowski L., Hajac, P. M., Neshveyev, S. Noncommutative Borsuk--Ulam-type conjectures revisited. Journal of Noncommutative Geometry 14 , 4 (2020), 1305--1324

    work page 2020

  31. [31]

    Noncommutative differential geometry on the quantum SU (2) , I : An algebraic viewpoint

    Masuda, T., Nakagami, Y., and Watanabe, J. Noncommutative differential geometry on the quantum SU (2) , I : An algebraic viewpoint. K-theory 4 , 2 (1990), 157--180

    work page 1990

  32. [32]

    Projective quantum spaces

    Meyer, U. Projective quantum spaces. Letters in Mathematical Physics 35 , 2 (1995), 91--97

    work page 1995

  33. [33]

    Introduction to Algebraic K -Theory , vol

    Milnor, J. Introduction to Algebraic K -Theory , vol. 72 of Annals of Mathematics Studies . Princeton University Press, Princeton, New Jersey, 1971

    work page 1971

  34. [34]

    Pedersen, G. K. Pullback and pushout constructions in C* -algebra theory. Journal of functional analysis 167 , 2 (1999), 243--344

    work page 1999

  35. [35]

    The K -theory of the triple- T oeplitz deformation of the complex projective plane

    Rudnik, J. The K -theory of the triple- T oeplitz deformation of the complex projective plane. Banach Center Publications 98\/ (2012), 303--310

    work page 2012

  36. [36]

    L., and Soibelman, Y

    Vaksman, L. L., and Soibelman, Y. S. The algebra of functions on quantum SU(n + 1) group and odd-dimensional quantum spheres. Leningrad Math. J. 2\/ (1991), 1023--1042

    work page 1991

  37. [37]

    Wegge-Olsen, N. E. K -theory and C* -algebras: a friendly approach . Oxford university press, 1993

    work page 1993

  38. [38]

    Woronowicz, S. L. Twisted SU(2) group. an example of a non-commutative differential calculus. Publications of the Research Institute for Mathematical Sciences 23 , 1 (1987), 117--181

    work page 1987