Equivariant complex cobordism. So we may consider its complex cobordism class Abstract.

Equivariant complex cobordism. 6. Feb 1, 1970 · We calculate the cobordism ring of stably almost complex manifolds with involution, and investigate the equivariant spectrum which represents it. The -equivariant complex cobordism ring for a compact Lie group can be Computing the coefficient ring of homotopical equivariant complex cobordism for general groups G turns out to be a difficult problem. mp4 (61M) We give a combinatorial description of the integral equivariant cohomology ring of certain simplicial GKM orbifold complexes. Introduction The calculation of the non-equivariant cobordism ring due to Milnor and Quillen [13, 14] was one of the great successes of algebraic topol-ogy. An old conjecture, which was established over 30 years ago for abelian groups, is that the graded coeffi-cient rings are concentrated in even degrees. Abstract. Jan 5, 2024 · Tom Dieck’s homotopical equivariant bordism MUG [17], defined with the help of equivariant Thom spaces, strives to be the legitimate equivariant refinement of complex cobordism, for compact Lie groups G. By the Pontryagin-Thom theorem this is equivalently framed cobordism cohomology theory. 02348 [math. We compute the integral equivariant cohomology ring, equivariant K -theory ring and equivariant cobordism ring of divisive simplicial GKM orbifold complexes. Appendices: Signs and normalization conventions Proper actions of Lie groups Equivariant cohomology Stable complex and Spin The Equivariant Complex Cobordism Ring of a Finite - Mathematics 28. They were subsequently developed into a complete theory of equivariant cobordism for all k-schemes in [19]. ⊕ R −→ ⊕ · · · ⊕ This endows M with the canonical equivariant stably complex struc-ture. The paper [2] of the author and Kriz provides the corresponding alge-braic computation of the Aug 18, 2025 · cobordism group, cobordism ring Thom spectrum, Thom's theorem cobordism category (infinity,n)-category of cobordisms, cobordism hypothesis cobordism cohomology theory extended cobordism B-bordism orbifold cobordism Lagrangian cobordism algebraic cobordism Sullivan chord diagram link cobordism braid cobordism 4. Other examples of geometrically oriented spectra include The cobordism rings we have in mind here are homotopy rings of the corresponding equivariant Thom spectra, as defined in [10]. Kriz [21] computed the equivariant complex cobordism ring of the p-primary cyclic group, and Strickland [31] gave a description of generators and relations for the equivariant complex cobordism ring of Z=2. Our proof method generalizes well to other settings, such as equivariant complex cobordism, and affords a Coefficients of equivariant complex cobordism Yunze Lu University of Michigan August, 2019 Complex manifolds: Compact smooth manifolds, with a tangential stable almost complex structure. Jun 22, 2023 · Orientation Theory in Mackey Cohomology and Disproving the Evenness Conjecture for Homotopical Equivariant Complex Cobordism Watch video | Download video: 202306191600-Kriz. More precisely, Z/2 acts by conjugation on BU(n) and the Thom space BU(n)γn of the universal complex n-bundle, and we have a map 2 Equivariant complex cobordism spectra 2. G/–graded equivariant cohomology theory. We also make some remarks on general methods of In this paper, we calculate the coefficient ring of equivariant Thom complex cobordism for the symmetric group on three elements. This is an equivariant re nement of the complex cobordism spectrum and the theory of complex orientations. Cobordism: Hamiltonian cobordism Abstract moment maps The linearization theorem Reduction and applications Part 2. 01926 Coefficients of the Σ3-equivariant complex cobordism ring,. The G-spectrum MUG that represents stabilized equivariant complex cobordism is an algebra over the equivariant sphere spectrum SG, and there is an MUG-module MG whose underlying MU-module is M. We compute the G-equivariant formal group law Abstract The article gives an introduction to equivariant formal group laws, and explains its relevance to complex oriented cohomol-ogy theories in general and to complex cobordism in particular. The Segal conjecture Chapter XXI. We introduce the notion of geometrically oriented spectra, which extends the notion of complex oriented spectra [more] Apr 16, 2025 · We give a combinatorial description of the integral equivariant cohomology ring of certain simplicial GKM orbifold complexes. 2 As algebraic K-theory over 1. Because of transversality issues, it has been known since the 60’s that these homotopical cobordism rings are diferent from geometric cobordism rings, which are actual cobordism rings of weakly stably complex G-manifolds. There is a G-equivariant analogue of this s Algebraic cobordism is a bigraded cohomology theory of smooth schemes represented by the Thom spectrum MGL in the stable motivic homotopy category. We prove the Thom iso 11. Oct 3, 2023 · We explore several examples of this structure on familiar T -equivariant commutative ring spectra including the periodic T -equivariant complex cobordism spectrum MUPT and a new (connective) equivariant version of the complex cobordism spectrum MU. Feb 4, 2022 · the geometric complex cobordism ring Ω∗. This ring is known to coincide with the coefficient ring of the equivariant stable complex cobordism spectrum, which I compute by the method of isotropy separation, using a “staircase diagram. Costenoble) Chapter XVI. In particular, we use methods from equivariant stable homotopy theory to cal-culate the unoriented C2-equivariant bordism ring ΩC2 ∗ , which was originally calculated by Alexander using other methods. The coefficients of MUG for G Z/p with = p prime were calculated in [21]. Abstract We calculate the cobordism ring of stably almost complex manifolds with involution, and investigate the equivariant spectrum which represents it. In this talk, I will discuss the Real bordism spectrum and the theory of Real orien-tations. In this paper, we calculate the coefficient ring of equi-variant Thom complex cobordism for the symmetric group on three elements. Thm 3 (Davis–Januszkiewicz). the geometric complex cobordism ring Ω∗. I will talk about their solution and explain how the Real We begin with a development of equivariant stable homotopy theory relevant to our work, including a new result on shift desuspension of suspension spectra. References The notion originates with Another key ingredient in our computation is the concept of geometric fixed points of an RO. 1 Complex cobordism and formal groups De nition 2. Tom Dieck’s homotopical equivariant bordism MU [17], defined with the help of equivariant Thom spaces, strives to be the legitimate equivariant refinement of complex cobordism, for compact Lie groups G. [14]) that because of fail-ure of equivariant transversality We begin with a development of equivariant stable homotopy theory relevant to our work, including a new result on shift desuspension of suspension spectra. 13). By recent Nov 25, 2020 · Equivariant complex cobordism For an abelian compact Lie group G G, equivariant complex cobordism theory MU G MU_G is an equivariant complex oriented cohomology theory (Greenlees 01, Sec. The G-equivariant complex cobordism ring for G a compact Lie group can be de ned analogously to the non-equivariant case. ” Lecture notes of a course taught by Daniel Quillen Transcription (and any typos) by Marco Mendez Jun 8, 2023 · cobordism theory = manifolds and cobordisms + stable homotopy theory / higher category theory equivariant cobordism theory Concepts of cobordism theory manifold, differentiable manifold, smooth manifold tangential structure cobordism, cobordism class cobordism ring submanifold, normal bundle Pontrjagin's theorem (equivariant, twisted): ↔ Sep 27, 2021 · As a matter of fact, because of failure of transversality, geometric cobordism of G -equivariant compact smooth manifolds is not represented by a complex-oriented equivariant spectrum. Examples include ordinary cohomology, complex topological K-theory, elliptic cohomology and cobordism cohomology. The reason is that the equivariant Thom spectra MUG, which represent them, are proving to play the same key role as complex cobordism plays in non-equivariant homotopy theory. However, a complex-oriented G-equivariant analog MUG of the complex Thom spectrum can be formed, and represents a certain stabilization of equivariant complex cobordism, corresponding precisely to “fixing” the transversality problem. 8, pp. Elements of Ω∗ are represented by stably almost complex manifolds, and we declare [M] = 0 in Ω∗ if there is a stably almos complex manifold W with boundary ∂W = M. Other examples of geometrically oriented spectra include the Coefficients of equivariant complex cobordism Yunze Lu University of Michigan August, 2019 1 / 24 Complex cobordism Complex manifolds: Compact smooth manifolds, with a tangential stable almost complex structure. Jul 8, 2024 · Idea 0. The -equivariant complex cobordism ring for a compact Lie group can be Abstract: This paper calculates the equivariant Lazard ring for primary cyclic groups, in terms of explicit generators and defining relations. The -equivariant complex cobordism ring for a compact Lie group can be 6 Precyclotomic bases, equivariant factorization homology, and global commutative ring spectra with multiplicative deflations laws, and this is the proposed equivariant analog to Quillen's Theorem. We survey some results in the field of equivariant cobordism. On this way we get an explicit formula for the cobordism classes of such spaces in terms of the weights at the fixed point 𝑒 𝐻 eH. Quantization: Geometric quantization The quantum version of the linearization theorem Quantization commutes with reduction Part 3. Methods of isotropy separation via Tate diagrams are heavily employed, and the key computational tool is the Isotropy Jun 6, 2025 · We give counterexamples to the evenness conjecture for homotopical equivariant cobordism. We will write MU for the global Thom ring spectrum defined in [21, Example 6. The Real bordism spectrum and its norms are crucial in Hill{Hopkins{Ravenel's solution of the Kervaire invariant one problem in 2009. Some remarks on Mackey functors (arXiv: 2205. We discuss how to compute the weighted structure constants for the integral equivariant cohomology ring of a divisive weighted Grassmann orbifold. There is a G-equivariant analogue of this s Tangentially stable almost complex G–structures are also called G–equivariant unitary structures, and the equivalence classes of manifolds under the bordism relation in the realm of G–equivariant unitary structures is called the G–equivariant unitary bordism group. Relative cyclotomic structures and equivariant complex cobordism arXiv:2310. We compute the G-equivariant formal group law We begin with a development of equivariant stable homotopy theory relevant to our work, including a new result on shift desuspension of suspension spectra. R. A cohomology theory is called complex oriented if it is multiplica-tive and it satis es Thom isomorphism for (almost) complex vector bundles. Jun 22, 2025 · G. The G-equivariant complex cobordism ring for G a compact Lie group can be defined analogously to the non-equivariant case. We also make some remarks on general methods of calculating certain… Expand [PDF] We also study the relation between the coeficient ring of equivariant complex cobordism and the universal Lazard ring of equivariant formal group laws for finite abelian groups, and prove a result generalizing classical Quillen’s Theorem. On completion and the evenness conjecture for homotopical equivariant cobordism , (Peking Mathematical Journal (2025)) 7. Mandell. To this end, we prove a completion theorem for certain complex cobordism modules which does not involve higher derived functors. Features: Idea 0. Equivalently, it is the cohomological dual concept to stable homotopy homology theory. INTRODUCTION The goal of the present paper is to provide an algebraic description of the equivariant formal group law corresponding to the equivariant complex cobordism ring MUG for a finite abelian group G, which is given as Theorem 4. However, a complex-oriented G-equivariant ana-log MUG of the complex Th Finally, the book gives an introduction to “brave new algebra”, the study of point-set level algebraic structures on spectra and its equivariant applications. The paper [2] of the author and Kriz provides the corresponding alge-braic computation of the See full list on deepblue. 15346) 8. It was noticed almost immediately however (e. Two closed manifolds are cobordant, if their disjoint union is the boundary of a third manifold. Chapter XV. This paper investigates the bigraded motivic homotopy groups ˇ Sep 2, 2021 · In this paper, we calculate the coefficient ring of equivariant Thom complex cobordism for the symmetric group on three elements. Methods of isotropy separation via Tate diagrams are heavily employed, and the key computational tool is the Isotropy Sep 28, 2015 · Since the equivariant complex cobordism spectrum has a canonical complex orientation, there is a corresponding equivariant formal group law. We introduce the notion of geometrically oriented spectra, which extends the notion of complex oriented spectra, and of which the geometric cobordism spectrum is the universal example. A first idea for defining cobordism in algebraic ge-ometry is to naively follow the well-known presentation of complex cobordism. umich. We also make some remarks on general methods of cal-culating certain pullbacks of rings which typically occur in calcu-lations of equivariant cobordism. References The notion originates with M orbifold complex is equivariantly formal, and a divisive simplicial GKM orbifold complex is integrally equivariantly formal We give a combinatorial description of the integral equivariant cohomology ring of certain simplicial GKM orbifold complexes. Methods of isotropy separation via Tate diagrams are heavily employed, and the key computational tool is the Isotropy Introduction Part 1. So we may consider its complex cobordism class Abstract. g. Sep 28, 2015 · We give a simple and explicit presentation of the Z/2-equivariant complex cobordism ring. Let X be a variety over a field k. Algebraic cobordism was introduced by Vo- evodsky as an analogue to complex cobordism to help solve the Milnor conjecture [Voe98]. Jan 1, 2022 · We calculate the cobordism ring of stably almost complex manifolds with involution, and investigate the equivariant spectrum which represents it. Jack Carlisle Abstract. Introduction The calculation of the non-equivariant cobordism ring due to Milnor and Quillen [13, 14] was one of the great successes of algebraic topology. The paper [2] of the author and Kriz provides the corresponding alge-braic computation of the Finally, the book gives an introduction to ``brave new algebra", the study of point-set level algebraic structures on spectra and its equivariant applications. T-Algebras and the Vector Delannoy Category, Stability in Topology, Arithmetic, and Representation Theory, Purdue University, July 2023 13. This formula also shows that the cobordism class for 𝐺 𝐻 G/H related to an invariant almost complex structure can be computed without information about cohomology for 𝐺 𝐻 G/H. Dec 1, 2012 · The equivariant algebraic cobordism groups for smooth varieties were defined by Deshpande in [8]. Methods of isotropy separation via Tate diagrams are heavily employed, and the key computational tool is the We calculate the cobordism ring of stably almost complex manifolds with involution, and investigate the equivariant spectrum which represents it. The paper [2] of the author and Kriz provides the corresponding alge-braic computation of the f complex lines through the origin in U. 127-150, 1969 (pdf) Aug 18, 2025 · cobordism group, cobordism ring Thom spectrum, Thom's theorem cobordism category (infinity,n)-category of cobordisms, cobordism hypothesis cobordism cohomology theory extended cobordism B-bordism orbifold cobordism Lagrangian cobordism algebraic cobordism Sullivan chord diagram link cobordism braid cobordism 4. We also make some remarks on general methods of calculating certain pullbacks of rings which typically occur in calculations of equivariant cobordism. Thom’s method was then used by other More generally, the ring of $n$-fold cooperations of equivariant complex bordism is shown to be universal among functors equipped with a strict $n$-tuple of coordinates. 2014 • Views Embed 1. AT] Andrew J. 1, and is adapted from the author’s doctoral thesis [1]. Mandell, and Allen Yuan. Hill, and Michael A. Some Recent Results on Homotopical and Geometrical Equivariant Complex Cobordism, Equivariant Bordism Theory and Applications, Banff International Research Station, CMO, Oaxaca, Mexico, June 2023, 12. We prove the Thom iso Discussion of the stable cohomotopy (framed cobordism cohomology theory) in the equivariant cohomology -version of cohomotopy (equivariant cohomotopy): Arthur Wasserman, section 3 of Equivariant differential topology, Topology Vol. The theory MU is the universal equivariantly complex oriented theory, and for abelian compact Lie groups, the coefficient ring MU∗ carries the universal G-equivariant formal group law [7 Sep 2, 2021 · Abstract In this paper, we calculate the coefficient ring of equivariant Thom complex cobordism for the symmetric group on three elements. 1. 11404 [math. 0722) Abstract. We give counterexamples to the evenness conjecture for homotopical equivariant cobordism. Volume 3, Number 2 (2001), ii-451 (EUCLID) William Abram, On the equivariant formal group law of the equivariant complex cobordism ring, (arXiv:1309. We shall explain a still more recent result which states that a similar analysis works to give the same kind of localization and completion theorems for the spectrum MUG that represents a stabilized version of equivariant complex cobordism and for all module spectra over MUG. The Burnside ring Chapter XVIII. On the structure of equivariant formal group laws, joint with Igor Kriz, accepted by Homotopy, Homology and Applications, available upon request On the RO (G)-graded coefficients of Q8-equivariant cohomology, accepted by Topology and its Applications, arXiv:2111. We compute the integral equivariant cohomology ring, equivari-ant K-theory ring and equivariant cobordism ring of divisive simplicial GKM orbifold complexes. We introduce the notion of geometrically oriented Abstract. Let E be a complex oriented cohomology theory, then The notion of cobordism goes back to Poincar ́e, but the method that allows us to successfully study it is more recent: Pontrjagin noticed that the study of cobordism for framed manifolds is related to the study of certain homotopy groups; Thom generalized this approach and successfully completed the compu-tation of the unoriented cobordism groups. Emphasis is placed on equivariant complex cobordism, and related results on that topic are presented in detail. Transfer maps in equivariant bundle theory Chapter XIX. This work gives explicit computations of the coefficients of homotopical complex equivariant cobordism theory MU G, when G is abelian, and defines a family of operations which essentially divide by Euler classes and use these operations to define the authors' generating sets. This theory is based on the analogous construction of the equivariant Chow groups by Totaro [31]and Edidin–Graham [9]. Other examples of geometrically oriented spectra include We begin with a development of equivariant stable homotopy theory relevant to our work, including a new result on shift desuspension of suspension spectra. Two closed manifolds are We begin with a development of equivariant stable homotopy theory relevant to our work, including a new result on shift desuspension of suspension spectra. Sep 25, 2013 · Abstract We begin with a development of equivariant stable homotopy theory relevant to our work, including a new result on shift desuspension of suspension spectra. Stable homotopy and Mackey functors Chapter XX. An introduction to equivariant cobordism (by S. There is an isomorphism of real vector bundles τ M 2(m n) ∼= − ρ1 ρm. Blumberg, Michael A. 2. Norms for compact Lie groups in equivariant stable homotopy theory arXiv:2212. Methods of isotropy separation via Tate diagrams are heavily employed, and the key computational tool is the Isotropy ABSTRACT We calculate the cobordism ring of stably almost complex manifolds with involution, and investigate the equivariant spectrum which represents it. If proved, Greenlees' Conjecture will allow similar algebraic methods to those developed in the 1. Proposition 2. 1 The generalized cohomology theory which is represented by the sphere spectrum is also called stable cohomotopy, as it is the stable homotopy theory version of cohomotopy. Spectra and G -spectra; change of groups; duality Chapter XVII. 04. 1. of the spectrum KG that represents equivariant K-theory. With the modern development of equivariant stable homotopy theory it is possible to carry this classical approach to group actions much further than in its first development. Comezana and Peter May, A completion theorem in complex cobordism, in Equivariant Homotopy and Cohomology Theory, CBMS Regional conference series in Mathematics, American Mathematical Society Publications, Volume 91, Providence, 1996. The idea of using equivariant cobordism theory to study questions about the existence and structure of group actions on manifolds dates back to work of Conner and Floyd, soon after cobordism theory was invented. edu Properties Universal orientation For an abelian compact Lie group G G, equivariant complex cobordism theory MU G MU_G is an equivariant complex oriented cohomology theory (Greenlees 01, Sec. While geometric com-plex cobordism rings are of geometric interest, homotopy cobordism rings are of much more fundamental homotopy-theoretical interest. 53] which is a model for the homotopical equivariant bordism MUG introduced by tom Dieck in [7] for every compact Lie group G. We introduce the notion of geometrically oriented In this paper, we calculate the coefficient ring of equivariant Thom complex cobordism for the symmetric group on three elements. Properties 0. Introduction The calculation of the non-equivariant cobordism ring due to Milnor and Quillen [9, 10] was one of the great successes of algebraic topol-ogy. lib. By work of Pontrj ∼= nd the algebraic geometry of formal groups. We calculate the equivariant cohomology ring, equivariant K–theory ring and equivariant cobordism ring of a divisive weighted Grassmann orbifold with integer coeficients. A key step in the proof is provided by a certain new relation between Mackey and Borel cohomology. This concept was introduced (using a different terminology) by tom Dieck[6], who calculated the geometric fixed points of equivariant complex cobordism. We then build on existing techniques of Kriz to compute the equivariant complex cobordism ring of a finite abelian group. M orbifold complex is equivariantly formal, and a divisive simplicial GKM orbifold complex is integrally equivariantly formal We give a combinatorial description of the integral equivariant cohomology ring of certain simplicial GKM orbifold complexes. On the local cohomology of L-shaped integral FI-modules ( Journal of Algebra 611 (2022) 149-174) 9. We begin with a development of equivariant stable homotopy theory relevant to our work, including a new result on shift desuspension of suspension spectra. Methods of isotropy separation via Tate diagrams are heavily employed, and the key computational tool is the Isotropy 6 Precyclotomic bases, equivariant factorization homology, and global commutative ring spectra with multiplicative deflations laws, and this is the proposed equivariant analog to Quillen's Theorem. 12192. 1 A complex oriented cohomology theory is a Whitehead-generalized cohomology theory which is oriented on all complex vector bundles. We calculate the cobordism ring of stably almost complex manifolds with involution, and investigate the equivariant spectrum which represents it. 5 Global complex cobordism Since our work concerns the complex bordism ring, we recollect here some facts about this theory. Jan 19, 2025 · John Greenlees, Equivariant formal group laws and complex oriented cohomology theories, Homology Homotopy Appl. 2. [18]) that because of failure of equivariant transversality Real cobordism is defined by considering the action of complex conjugation on complex cobordism. As a matter of fact, be-cause of failure of transversality, geometric cobordism of G-equivariant compact smooth manifolds is not represented by a complex-oriented equivariant spectrum. ipnb diyrmkel ubu as jz17t vk2rum ey1 8dn du li6d