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Global group laws and equivariant bordism rings. (English) Zbl 07583002

D. Quillen [Bull. Am. Math. Soc. 75, 1293–1298 (1969; Zbl 0199.26705)] proved that the formal group law associated to the complex orientation of complex bordism \(MU\) is the universal one. In particular, the complex bordism ring is isomorphic to the Lazard ring. This result became one of the main tools in understanding the structure of the stable homotopy category.
In view of the importance of Quillen’s theorem much work has been put into obtaining a similar understanding of equivariant bordism rings and their characterization in terms of formal group law data. For an abelian compact Lie group \(A\), the notion of an \(A\)-equivariant formal group law was introduced by M. Cole et al. [Proc. Lond. Math. Soc. (3) 81, No. 2, 355–386 (2000; Zbl 1030.55004)]. As in the non-equivariant case there exists a universal \(A\)-equivariant formal group law over an \(A\)-equivariant Lazard ring \(L_A\) and a map \[ \varphi_A:L_A\rightarrow \pi_*^A(MU_A) \] classifying the Euler class of a product of \(A\)-equivariant complex line bundles. Here \(MU_A\) denotes the homotopical \(A\)-equivariant bordism spectrum.
It has been conjectured by Greenlees that \(\varphi_A\) is an isomorphism. This conjecture has only been proved in case that \(A\) has order two by B. Hanke and M. Wiemeler [Adv. Math. 340, 48–75 (2018; Zbl 1409.55012)]. The paper under review now proves the conjecture in full generality.
It also proves a similar result for homotopy theoretic \(A\)-equivariant unoriented bordism \(MO_A\) in the case that \(A\) is an elementary abelian \(2\)-group. In that case one considers the ring \(L_A^{2\text{-tor}}\) carrying the universal \(A\)-equivariant \(2\)-torsion formal group law. Here the result is:
The map \[ \varphi_A^{2\text{-tor}}:L_A^{2\text{-tor}}\rightarrow \pi_*^A(MO_A) \] classifying the formal group law induced from the real orientation of \(MO_A\) is an isomorphism.
The proof of these results goes as follows. First it is noted that in the complex case it suffices to prove the result for tori \(A\). Then global group laws and global \(2\)-torsion group laws are introduced. These are functors \[ (\text{tori})^{\text{op}}\rightarrow \text{commutative rings} \] and \[ (\text{elementary abelian two-groups})^{\text{op}}\rightarrow \text{commutative rings} \] which have a coordinate, respectively. Examples of such global group laws are given by \(\mathbf{MU}\) and \(\mathbf{MO}\) which assign to a group its equivariant bordism ring. It is shown that in the categories \(GL_{gl}\) and \(GL_{gl}^{2\text{-tor}}\) of these group laws there exist initial objects \(\mathbf{L}\) and \(\mathbf{L}^{2\text{-tor}}\). Moreover functors \[ (\cdot)^\wedge_A:GL_{gl}\rightarrow FGL_A \] and \[ (\cdot)^\wedge_A:GL_{gl}^{2-\text{tor}}\rightarrow FGL_A^{2\text{-tor}} \] for every torus \(A\) and every elementary abelian \(2\)-group \(A\), respectively, are constructed. Here \(FGL_A\) and \(FGL_A^{2\text{-tor}}\) denote the categories of \(A\)-equivariant (2-torsion) formal group laws. It is shown that these functors have right adjoints. In particular,
\[ (\mathbf{L})_A^\wedge=L_A\quad\quad\quad\text{and}\quad\quad\quad(\mathbf{L}^{2\text{-tor}})_A^\wedge=L_A^{2\text{-tor}}. \]
Using the axioms of global group laws it is deduced that the non-trivial Euler classes in \(L_A\) are regular elements. It was previously known by work of Greenlees [The coefficient ring of equivariant homotopical bordism classifies equivariant formal group laws over Noetherian rings, Preprint, 2000] that \(\varphi_A\) and \(\varphi_A^{2\text{-tor}}\) become isomorphisms after inverting all non-trivial Euler classes. Therefore it follows that these maps are injective. Using Greenlees’s result and the properties of global group laws it is then deduced that \(\varphi_A\) and \(\varphi_A^{2\text{-tor}}\) are also surjective. Hence the main results follow.
As a consequence of this line of reasoning it also follows that \(\mathbf{L}=\mathbf{MU}\) and \(\mathbf{L}^{2\text{-tor}}=\mathbf{MO}\).

MSC:

57R85 Equivariant cobordism
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55P91 Equivariant homotopy theory in algebraic topology
14L05 Formal groups, \(p\)-divisible groups
55P42 Stable homotopy theory, spectra
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[1] Abram, William C.; Kriz, Igor, The equivariant complex cobordism ring of a finite abelian group, Math. Res. Lett.. Mathematical Research Letters, 22, 1573-1588 (2015) · Zbl 1373.55005
[2] Ad\'{a}mek, Ji\v{r}\'{\i}; Rosick\'{y}, Ji\v{r}\'{\i}, Locally Presentable and Accessible Categories, London Math. Soc. Lecture Note Ser., 189, xiv+316 pp. (1994) · Zbl 0795.18007
[3] Barthel, Tobias; Greenlees, J. P. C.; Hausmann, Markus, On the {B}almer spectrum for compact {L}ie groups, Compos. Math.. Compositio Mathematica, 156, 39-76 (2020) · Zbl 1431.55012
[4] Br\"{o}cker, Theodor; Hook, Edward C., Stable equivariant bordism, Math. Z.. Mathematische Zeitschrift, 129, 269-277 (1972) · Zbl 0236.57020
[5] Barthel, Tobias; Hausmann, Markus; Naumann, Niko; Nikolaus, Thomas; Noel, Justin; Stapleton, Nathaniel, The {B}almer spectrum of the equivariant homotopy category of a finite abelian group, Invent. Math.. Inventiones Mathematicae, 216, 215-240 (2019) · Zbl 1417.55016
[6] Baker, A.; Morava, J., {\(MSp\)} localized away from \(2\) and odd formal group laws (2014)
[7] Bredon, Glen E., Equivariant stable stems, Bull. Amer. Math. Soc.. Bulletin of the Amer. Math. Soc., 73, 269-273 (1967) · Zbl 0152.21803
[8] Balmer, Paul; Sanders, Beren, The spectrum of the equivariant stable homotopy category of a finite group, Invent. Math.. Inventiones Mathematicae, 208, 283-326 (2017) · Zbl 1373.18016
[9] Cole, Michael; Greenlees, J. P. C.; Kriz, I., Equivariant formal group laws, Proc. London Math. Soc. (3). Proceedings of the London Mathematical Society. Third Series, 81, 355-386 (2000) · Zbl 1030.55004
[10] Cole, Michael; Greenlees, J. P. C.; Kriz, I., The universality of equivariant complex bordism, Math. Z.. Mathematische Zeitschrift, 239, 455-475 (2002) · Zbl 1008.55015
[11] Comeza{\~{n}}a, S., Calculations in complex equivariant bordism. Equivariant Homotopy and Cohomology Theory, CBMS Reg. Conf. Ser. Math., 91, 333-253 (1996)
[12] Degrijse, D.; Hausmann, M.; L{\"u}ck, W.; Patchkoria, I.; Schwede, S., Proper equivariant stable homotopy theory
[13] Firsching, Moritz, Real equivariant bordism for elementary abelian 2-groups, Homology Homotopy Appl.. Homology, Homotopy and Applications, 15, 235-251 (2013) · Zbl 1275.57042
[14] Greenlees, J. P. C.; May, J. P., Localization and completion theorems for {\(M{\rm U}\)}-module spectra, Ann. of Math. (2). Annals of Mathematics. Second Series, 146, 509-544 (1997) · Zbl 0910.55005
[15] Greenlees, J. P. C., The coefficient ring of equivariant homotopical bordism classifies equivariant formal group laws over {N}oetherian rings (2000)
[16] Greenlees, J. P. C., Equivariant formal group laws and complex oriented cohomology theories. Equivariant Stable Homotopy Theory and Related Areas, Homology Homotopy Appl., 3, 225-263 (2001) · Zbl 0973.00048
[17] Hanke, Bernhard, Geometric versus homotopy theoretic equivariant bordism, Math. Ann.. Mathematische Annalen, 332, 677-696 (2005) · Zbl 1073.55006
[18] Hill, M. A.; Hopkins, M. J.; Ravenel, D. C., On the nonexistence of elements of {K}ervaire invariant one, Ann. of Math. (2). Annals of Mathematics. Second Series, 184, 1-262 (2016) · Zbl 1366.55007
[19] Hopkins, Michael J.; Smith, Jeffrey H., Nilpotence and stable homotopy theory. {II}, Ann. of Math. (2). Annals of Mathematics. Second Series, 148, 1-49 (1998) · Zbl 0927.55015
[20] Hanke, Bernhard; Wiemeler, Michael, An equivariant {Q}uillen theorem, Adv. Math.. Advances in Mathematics, 340, 48-75 (2018) · Zbl 1409.55012
[21] Joachim, Michael, Higher coherences for equivariant {\(K\)}-theory. Structured Ring Spectra, London Math. Soc. Lecture Note Ser., 315, 87-114 (2004) · Zbl 1070.19007
[22] Kriz, Igor, The {\({\bf Z}/p\)}-equivariant complex cobordism ring. Homotopy Invariant Algebraic Structures, Contemp. Math., 239, 217-223 (1999) · Zbl 0982.55001
[23] Lazard, Michel, Sur les groupes de {L}ie formels \`a un param\`etre, Bull. Soc. Math. France. Bulletin de la Soci\'{e}t\'{e} Math\'{e}matique de France, 83, 251-274 (1955) · Zbl 0068.25703
[24] L{\"o}ffler, Peter, Equivariant unitary cobordism and classifying spaces. Proceedings of the {I}nternational {S}ymposium on {T}opology and its {A}pplications ({B}udva, 1972), 158-160 (1973)
[25] available on the author’s webpage, Elliptic cohomology {II}: {O}rientations
[26] Novikov, S. P., Methods of algebraic topology from the point of view of cobordism theory, Izv. Akad. Nauk SSSR Ser. Mat.. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 31, 855-951 (1967) · Zbl 0176.52401
[27] Quillen, Daniel, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc.. Bulletin of the Amer. Math. Soc., 75, 1293-1298 (1969) · Zbl 0199.26705
[28] Schwede, Stefan, Global homotopy theory, New Math. Monogr., 34, xviii+828 pp. (2018) · Zbl 1451.55001
[29] Sinha, Dev P., Computations of complex equivariant bordism rings, Amer. J. Math.. American Journal of Mathematics, 123, 577-605 (2001) · Zbl 0997.55008
[30] Sinha, Dev, Real equivariant bordism and stable transversality obstructions for {\( \Bbb Z/2\)}, Proc. Amer. Math. Soc.. Proceedings of the Amer. Math. Soc., 130, 271-281 (2002) · Zbl 0987.57015
[31] Strickland, N. P., Complex cobordism of involutions, Geom. Topol.. Geometry and Topology, 5, 335-345 (2001) · Zbl 1009.55003
[32] Strickland, N. P., Multicurves and equivariant cohomology, Mem. Amer. Math. Soc.. Memoirs of the Amer. Math. Soc., 213, vi+117 pp. (2011) · Zbl 1228.55001
[33] Tate, J. T., {\(p\)}-divisible groups. Proc. {C}onf. {L}ocal {F}ields, 158-183 (1967)
[34] tom Dieck, Tammo, Bordism of {\(G\)}-manifolds and integrality theorems, Topology. Topology. An International Journal of Mathematics, 9, 345-358 (1970) · Zbl 0209.27504
[35] Uribe, Bernardo, The evenness conjecture in equivariant unitary bordism. Proceedings of the {I}nternational {C}ongress of {M}athematicians—{R}io de {J}aneiro 2018. {V}ol. {II}. {I}nvited {L}ectures, 1217-1239 (2018) · Zbl 1450.55001
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