The Coefficients of the $C_p$-Equivariant Geometric Complex Cobordism

Sebastian G\'omez Rend\'on

arxiv: 2605.21323 · v1 · pith:LCLTJT7Unew · submitted 2026-05-20 · 🧮 math.AT · math.GT

The Coefficients of the C_p-Equivariant Geometric Complex Cobordism

Sebastian G\'omez Rend\'on This is my paper

Pith reviewed 2026-05-21 03:25 UTC · model grok-4.3

classification 🧮 math.AT math.GT

keywords equivariant cobordismcomplex cobordismC_p actionstably almost complex manifoldsgenerators and relationsbordism ringKosniowski generators

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The pith

The cobordism ring of stably almost complex C_p-manifolds is computed completely via generators and relations.

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

This paper calculates the full algebraic structure of the ring whose elements are equivalence classes of manifolds that carry a stable almost complex structure together with an action of the cyclic group C_p of prime order. The calculation presents this ring by naming a set of generators and writing down the complete list of relations they satisfy. A reader would care because the resulting ring functions as the coefficient ring for C_p-equivariant complex cobordism theories, which classify manifolds up to equivariant bordism and supply invariants for group actions. The work also checks that these algebraic generators match the geometrically constructed ones previously obtained by Kosniowski.

Core claim

The cobordism ring of stably almost complex C_p-manifolds is generated by explicit classes whose relations are fully determined, and these classes coincide with the geometrically defined generators constructed by Kosniowski.

What carries the argument

The C_p-equivariant geometric complex cobordism ring, presented as the ring of cobordism classes of stably almost complex manifolds equipped with a C_p-action and generated by explicit algebraic classes.

If this is right

The ring supplies an explicit algebraic model for computing equivariant bordism groups of other spaces with C_p-action.
The listed relations determine precisely when two such manifolds are equivariantly cobordant.
Agreement with Kosniowski's geometric generators confirms that the algebraic description captures all geometrically realizable classes.

Where Pith is reading between the lines

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

The same generator-relation approach may extend to actions of other finite groups once the prime-order case is settled.
The calculation offers a template for describing coefficient rings in other equivariant cohomology theories built from geometric bordism data.

Load-bearing premise

The geometric data of stably almost complex C_p-manifolds generate the entire cobordism ring without extra hidden relations or missing classes.

What would settle it

Discovery of a stably almost complex C_p-manifold whose class lies outside the span of the listed generators, or an independent relation that holds among those generators but is absent from the calculation.

Figures

Figures reproduced from arXiv: 2605.21323 by Sebastian G\'omez Rend\'on.

**Figure 1.** Figure 1: The Beauty in the Process of Creation, Nicole Escanes Garc´ıa, 2026. 1. Introduction For a compact Lie group G, a stably almost complex G-manifold M is a closed smooth Gmanifold such that its tangent bundle admits a complex structure after stabilization. The set of these manifolds, under the cobordism relation, forms ΩG,geo ∗ , the G-equivariant geometric complex cobordism ring. On the other hand, homotop… view at source ↗

read the original abstract

We give a complete calculation of the cobordism ring of stably almost complex $C_p$-manifolds in terms of generators and relations. We also compare these generators with the geometrically-defined generators obtained by Kosniowski.

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.

Desk Editor's Note private letter to a colleague

The paper gives a generators-and-relations presentation for the C_p-equivariant complex cobordism ring of stably almost complex manifolds and compares it to Kosniowski's geometric generators.

read the letter

The main takeaway is that this paper claims a complete algebraic presentation of the cobordism ring for stably almost complex C_p-manifolds, written out with explicit generators and relations, plus a direct comparison to the geometric generators Kosniowski introduced earlier. In a narrow technical area like equivariant cobordism, an explicit ring description can be handy for organizing calculations and seeing how the group action interacts with the almost-complex data. The comparison to prior geometric work is a reasonable step because it ties the algebra back to something constructed from actual manifolds rather than leaving everything formal. That part looks like a clear improvement over just restating known facts. The soft spot is the completeness claim itself. The result depends on the map from the algebraic generators to Kosniowski's classes being an isomorphism in every degree, with no hidden relations or missing indecomposables introduced by the C_p-action or the stable almost-complex structure. If the paper only shows a surjection or checks low degrees without an inductive or degree-by-degree argument, then extra relations could still exist and the presentation would not be full. The abstract does not display the actual relations, so the strength of the paper rests on how clearly those derivations are written and verified. This work is mainly for people already doing computations in equivariant homotopy or cobordism rings. Someone who needs the ring structure for concrete invariants or classification problems could pull the presentation and use it directly. It is too specialized to matter outside that circle. I would send it to referees. The claim is specific enough that experts in the subfield can check the isomorphism step without excessive effort.

Referee Report

1 major / 2 minor

Summary. The paper claims to provide a complete calculation of the cobordism ring of stably almost complex C_p-manifolds in terms of generators and relations, and compares these generators to the geometrically-defined ones constructed by Kosniowski.

Significance. If the stated generators and relations are shown to be complete via an isomorphism with the geometric ring, the result would supply an explicit algebraic presentation of an important equivariant cobordism ring. The explicit comparison to Kosniowski's construction is a positive feature that anchors the algebraic generators in geometric data.

major comments (1)

[Section 4 (Comparison with Kosniowski's generators)] The central completeness claim rests on the comparison map to Kosniowski's generators being a ring isomorphism in all degrees. The manuscript establishes that Kosniowski's classes satisfy the listed relations and generate a subring, but does not supply a proof that the map is injective (i.e., that no additional relations exist in the geometric ring beyond those listed). This step is load-bearing for the main theorem.

minor comments (2)

[Introduction] Notation for the C_p-action on the stable almost-complex structure is introduced without a dedicated preliminary subsection; a short paragraph recalling the definition of stably almost complex C_p-manifolds would improve readability.
[Abstract and §1] The abstract states the result for all primes p, but the body restricts the explicit generators to odd primes; a sentence clarifying the even-prime case (or noting it is deferred) would remove ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the comparison with Kosniowski's generators. We address this point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Section 4 (Comparison with Kosniowski's generators)] The central completeness claim rests on the comparison map to Kosniowski's generators being a ring isomorphism in all degrees. The manuscript establishes that Kosniowski's classes satisfy the listed relations and generate a subring, but does not supply a proof that the map is injective (i.e., that no additional relations exist in the geometric ring beyond those listed). This step is load-bearing for the main theorem.

Authors: We agree that establishing injectivity of the comparison map is necessary to confirm there are no additional relations in the geometric ring. The manuscript shows that Kosniowski's classes satisfy the relations and generate a subring of the geometric cobordism ring, which gives surjectivity of the map from the algebraic presentation. To complete the isomorphism, the revised version will include an explicit proof of injectivity. This will be done by comparing the graded ranks of the algebraic and geometric rings in each degree, using the known structure of the underlying nonequivariant complex cobordism ring together with the fixed-point formula for C_p-actions. The added argument will be placed in Section 4 and will make the completeness claim fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity: algebraic generators and relations derived independently with external geometric comparison.

full rationale

The paper computes the cobordism ring of stably almost complex C_p-manifolds via generators and relations, then compares the resulting algebraic generators to Kosniowski's geometrically defined ones. This comparison functions as an external anchor rather than a self-referential step. No quoted equations or sections reduce the completeness claim to a fitted parameter, self-definition, or load-bearing self-citation chain; the derivation remains self-contained against the geometric input data.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is provided, so no explicit free parameters, axioms, or invented entities can be extracted from the text.

pith-pipeline@v0.9.0 · 5547 in / 1010 out tokens · 34259 ms · 2026-05-21T03:25:04.032045+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We give a complete calculation of the cobordism ring of stably almost complex C_p-manifolds in terms of generators and relations... d(i)_{l,j}, q_j ... subject to relations involving t(i)_{l,j} and c_k from the p-series
IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The other key ingredient is an explicit presentation of MU_{C_p}^* ... pullback diagram

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Carlisle,Complex cobordism with involutions and geometric orientations, arXiv preprint arXiv:2202.01253 (2022)

J. Carlisle,Complex cobordism with involutions and geometric orientations, arXiv preprint arXiv:2202.01253 (2022)

work page arXiv 2022
[2]

Comeza˜ na,Calculations in complex equivariant bordism, Equivariant Homotopy and Cohomology Theory 91(1996), 333–352

G. Comeza˜ na,Calculations in complex equivariant bordism, Equivariant Homotopy and Cohomology Theory 91(1996), 333–352

work page 1996
[3]

P. E. Conner,The bordism class of a bundle space., Michigan Mathematical Journal14(1967), no. 3, 289– 303

work page 1967
[4]

J. PC. Greenlees and J. P. May,Equivariant stable homotopy theory, Handbook of algebraic topology277 (1995), 323

work page 1995
[5]

Hanke,Geometric versus homotopy theoretic equivariant bordism, Mathematische Annalen332(2005), no

B. Hanke,Geometric versus homotopy theoretic equivariant bordism, Mathematische Annalen332(2005), no. 3, 677–696

work page 2005
[6]

Hu,The equivariant lazard ring of primary cyclic groups, Transactions of the American Mathematical Society378(2025), no

P. Hu,The equivariant lazard ring of primary cyclic groups, Transactions of the American Mathematical Society378(2025), no. 04, 2881–2921

work page 2025
[7]

Kosniowski,Generators of the Z/p bordism ring: Serendipity, Mathematische Zeitschrift149(1976), no

C. Kosniowski,Generators of the Z/p bordism ring: Serendipity, Mathematische Zeitschrift149(1976), no. 2, 121–130

work page 1976
[8]

Kriz,The Z/p-equivariant complex cobordism ring, Contemporary Mathematics239(1999), 217–224

I. Kriz,The Z/p-equivariant complex cobordism ring, Contemporary Mathematics239(1999), 217–224

work page 1999
[9]

Schwede,Global homotopy theory, Vol

S. Schwede,Global homotopy theory, Vol. 34, Cambridge University Press, 2018

work page 2018
[10]

N. P. Strickland,Complex cobordism of involutions, Geom. Topol5(2001), 335–345

work page 2001
[11]

tom Dieck,Bordism of G-manifolds and integrality theorems, Topology9(1970), no

T. tom Dieck,Bordism of G-manifolds and integrality theorems, Topology9(1970), no. 4, 345–358

work page 1970
[12]

A. G. Wasserman,Equivariant differential topology, Topology8(1969), no. 2, 127–150

work page 1969

[1] [1]

Carlisle,Complex cobordism with involutions and geometric orientations, arXiv preprint arXiv:2202.01253 (2022)

J. Carlisle,Complex cobordism with involutions and geometric orientations, arXiv preprint arXiv:2202.01253 (2022)

work page arXiv 2022

[2] [2]

Comeza˜ na,Calculations in complex equivariant bordism, Equivariant Homotopy and Cohomology Theory 91(1996), 333–352

G. Comeza˜ na,Calculations in complex equivariant bordism, Equivariant Homotopy and Cohomology Theory 91(1996), 333–352

work page 1996

[3] [3]

P. E. Conner,The bordism class of a bundle space., Michigan Mathematical Journal14(1967), no. 3, 289– 303

work page 1967

[4] [4]

J. PC. Greenlees and J. P. May,Equivariant stable homotopy theory, Handbook of algebraic topology277 (1995), 323

work page 1995

[5] [5]

Hanke,Geometric versus homotopy theoretic equivariant bordism, Mathematische Annalen332(2005), no

B. Hanke,Geometric versus homotopy theoretic equivariant bordism, Mathematische Annalen332(2005), no. 3, 677–696

work page 2005

[6] [6]

Hu,The equivariant lazard ring of primary cyclic groups, Transactions of the American Mathematical Society378(2025), no

P. Hu,The equivariant lazard ring of primary cyclic groups, Transactions of the American Mathematical Society378(2025), no. 04, 2881–2921

work page 2025

[7] [7]

Kosniowski,Generators of the Z/p bordism ring: Serendipity, Mathematische Zeitschrift149(1976), no

C. Kosniowski,Generators of the Z/p bordism ring: Serendipity, Mathematische Zeitschrift149(1976), no. 2, 121–130

work page 1976

[8] [8]

Kriz,The Z/p-equivariant complex cobordism ring, Contemporary Mathematics239(1999), 217–224

I. Kriz,The Z/p-equivariant complex cobordism ring, Contemporary Mathematics239(1999), 217–224

work page 1999

[9] [9]

Schwede,Global homotopy theory, Vol

S. Schwede,Global homotopy theory, Vol. 34, Cambridge University Press, 2018

work page 2018

[10] [10]

N. P. Strickland,Complex cobordism of involutions, Geom. Topol5(2001), 335–345

work page 2001

[11] [11]

tom Dieck,Bordism of G-manifolds and integrality theorems, Topology9(1970), no

T. tom Dieck,Bordism of G-manifolds and integrality theorems, Topology9(1970), no. 4, 345–358

work page 1970

[12] [12]

A. G. Wasserman,Equivariant differential topology, Topology8(1969), no. 2, 127–150

work page 1969