Geometry & Topology Volume 29, issue 3 (2025)

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

Author index

To appear

Other MSP journals

The motivic lambda algebra and motivic Hopf invariant one problem

William Balderrama, Dominic Leon Culver and J D Quigley

Geometry & Topology 29 (2025) 1489–1570

DOI: 10.2140/gt.2025.29.1489

Bibliography

1	J F Adams, On the structure and applications of the Steenrod algebra, Comment. Math. Helv. 32 (1958) 180 MR0096219
2	J F Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. 72 (1960) 20 MR0141119
3	J F Adams, Vector fields on spheres, Ann. of Math. 75 (1962) 603 MR0139178
4	M Andrews, H Miller, Inverting the Hopf map, J. Topol. 10 (2017) 1145 MR3743072
5	A Asok, B Doran, J Fasel, Smooth models of motivic spheres and the clutching construction, Int. Math. Res. Not. 2017 (2017) 1890 MR3658186
6	T Bachmann, η-periodic motivic stable homotopy theory over Dedekind domains, J. Topol. 15 (2022) 950 MR4441609
7	T Bachmann, M J Hopkins, η-periodic motivic stable homotopy theory over fields, preprint (2020) arXiv:2005.06778
8	T Bachmann, H J Kong, G Wang, Z Xu, The Chow t-structure on the ∞-category of motivic spectra, Ann. of Math. 195 (2022) 707 MR4387236
9	W Balderrama, Algebraic theories of power operations, J. Topol. 16 (2023) 1543
10	M Behrens, J Shah, C₂-equivariant stable homotopy from real motivic stable homotopy, Ann. K-Theory 5 (2020) 411 MR4132743
11	E Belmont, D C Isaksen, ℝ-motivic stable stems, J. Topol. 15 (2022) 1755 MR4461846
12	E Belmont, B J Guillou, D C Isaksen, C₂-equivariant and ℝ-motivic stable stems, II, Proc. Amer. Math. Soc. 149 (2021) 53 MR4172585
13	A K Bousfield, E B Curtis, D M Kan, D G Quillen, D L Rector, J W Schlesinger, The mod − p lower central series and the Adams spectral sequence, Topology 5 (1966) 331 MR0199862
14	E H Brown Jr., S Gitler, A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra, Topology 12 (1973) 283 MR0391071
15	R R Bruner, The homotopy theory of H_∞ ring spectra, from: "H_∞ ring spectra and their applications" (editors R R Bruner, J P May, J E McClure, M Steinberger), Lecture Notes in Math. 1176, Springer (1986) 88 MR0836132
16	R R Bruner, The Adams spectral sequence of H_∞ ring spectra, from: "H_∞ ring spectra and their applications" (editors R R Bruner, J P May, J E McClure, M Steinberger), Lecture Notes in Math. 1176, Springer (1986) 169 MR0836132
17	R Bruner, An example in the cohomology of augmented algebras, J. Pure Appl. Algebra 55 (1988) 81 MR0968570
18	R Burklund, J Hahn, A Senger, Galois reconstruction of Artin–Tate ℝ-motivic spectra, preprint (2020) arXiv:2010.10325
19	T W Chen, Determination of Ext_𝒜^5,∗(ℤ∕2, ℤ∕2), Topology Appl. 158 (2011) 660 MR2774051
20	R L Cohen, The immersion conjecture for differentiable manifolds, Ann. of Math. 122 (1985) 237 MR0808220
21	D L Culver, J D Quigley, kq-resolutions, I, Trans. Amer. Math. Soc. 374 (2021) 4655 MR4273174
22	E B Curtis, Simplicial homotopy theory, Adv. Math. 6 (1971) 107 MR0279808
23	E B Curtis, P Goerss, M Mahowald, R J Milgram, Calculations of unstable Adams E₂ terms for spheres, from: "Algebraic topology" (editors H R Miller, D C Ravenel), Lecture Notes in Math. 1286, Springer (1987) 208 MR0922929
24	D Dugger, D C Isaksen, The motivic Adams spectral sequence, Geom. Topol. 14 (2010) 967 MR2629898
25	D Dugger, D C Isaksen, Motivic Hopf elements and relations, New York J. Math. 19 (2013) 823 MR3141814
26	D Dugger, D C Isaksen, Low-dimensional Milnor–Witt stems over ℝ, Ann. K-Theory 2 (2017) 175 MR3590344
27	D Dugger, D C Isaksen, ℤ∕2-equivariant and ℝ-motivic stable stems, Proc. Amer. Math. Soc. 145 (2017) 3617 MR3652813
28	B Gheorghe, G Wang, Z Xu, The special fiber of the motivic deformation of the stable homotopy category is algebraic, Acta Math. 226 (2021) 319 MR4281382
29	P G Goerss, Hopf rings, Dieudonné modules, and E_∗Ω²S³, from: "Homotopy invariant algebraic structures" (editors J P Meyer, J Morava, W S Wilson), Contemp. Math. 239, Amer. Math. Soc. (1999) 115 MR1718079
30	P G Goerss, J D S Jones, M E Mahowald, Some generalized Brown–Gitler spectra, Trans. Amer. Math. Soc. 294 (1986) 113 MR0819938
31	J P C Greenlees, Stable maps into free G-spaces, Trans. Amer. Math. Soc. 310 (1988) 199 MR0938918
32	B J Guillou, D C Isaksen, The η-local motivic sphere, J. Pure Appl. Algebra 219 (2015) 4728 MR3346515
33	B J Guillou, D C Isaksen, The η-inverted ℝ-motivic sphere, Algebr. Geom. Topol. 16 (2016) 3005 MR3572357
34	B J Guillou, D C Isaksen, The Bredon–Landweber region in C₂-equivariant stable homotopy groups, Doc. Math. 25 (2020) 1865 MR4184454
35	B J Guillou, M A Hill, D C Isaksen, D C Ravenel, The cohomology of C₂-equivariant 𝒜(1) and the homotopy of ko_C₂, Tunis. J. Math. 2 (2020) 567 MR4041284
36	M A Hill, Ext and the motivic Steenrod algebra over ℝ, J. Pure Appl. Algebra 215 (2011) 715 MR2747214
37	M Hoyois, S Kelly, P A Østvær, The motivic Steenrod algebra in positive characteristic, J. Eur. Math. Soc. 19 (2017) 3813 MR3730515
38	P Hu, I Kriz, Real-oriented homotopy theory and an analogue of the Adams–Novikov spectral sequence, Topology 40 (2001) 317 MR1808224
39	P Hu, I Kriz, K Ormsby, Convergence of the motivic Adams spectral sequence, J. K-Theory 7 (2011) 573 MR2811716
40	P Hu, I Kriz, K Ormsby, Remarks on motivic homotopy theory over algebraically closed fields, J. K-Theory 7 (2011) 55 MR2774158
41	D J Hunter, N J Kuhn, Mahowaldean families of elements in stable homotopy groups revisited, Math. Proc. Cambridge Philos. Soc. 127 (1999) 237 MR1705457
42	D C Isaksen, Stable stems, 1269, Amer. Math. Soc. (2019) MR4046815
43	D C Isaksen, P A Østvær, Motivic stable homotopy groups, from: "Handbook of homotopy theory" (editor H Miller), CRC Press (2020) 757 MR4197998
44	D C Isaksen, G Wang, Z Xu, Stable homotopy groups of spheres : from dimension 0 to 90, Publ. Math. Inst. Hautes Études Sci. 137 (2023) 107 MR4588596
45	I M James, On the suspension sequence, Ann. of Math. 65 (1957) 74 MR0083124
46	I M James, Cross-sections of Stiefel manifolds, Proc. Lond. Math. Soc. 8 (1958) 536 MR0100840
47	I M James, Spaces associated with Stiefel manifolds, Proc. Lond. Math. Soc. 9 (1959) 115 MR0102810
48	J I Kylling, Hermitian K-theory of finite fields via the motivic Adams spectral sequence, master’s thesis, University of Oslo (2015)
49	J I Kylling, G M Wilson, Strong convergence in the motivic Adams spectral sequence, preprint (2019) arXiv:1901.03399
50	W H Lin, On conjectures of Mahowald, Segal and Sullivan, Math. Proc. Cambridge Philos. Soc. 87 (1980) 449 MR0556925
51	W H Lin, Ext_A^4,∗(ℤ∕2, ℤ∕2) and Ext_A^5,∗(ℤ∕2, ℤ∕2), Topology Appl. 155 (2008) 459 MR2380930
52	J Lurie, Derived algebraic geometry, XIII : Rational and p-adic homotopy theory, preprint (2011)
53	M Mahowald, A new infinite family in ₂π_∗^s, Topology 16 (1977) 249 MR0445498
54	M Mahowald, bo-resolutions, Pacific J. Math. 92 (1981) 365 MR0618072
55	M A Mandell, E_∞ algebras and p-adic homotopy theory, Topology 40 (2001) 43 MR1791268
56	W S Massey, F P Peterson, The cohomology structure of certain fibre spaces, I, Topology 4 (1965) 47 MR0189032
57	J Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1970) 318 MR0260844
58	F Morel, Suite spectrale d’Adams et invariants cohomologiques des formes quadratiques, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 963 MR1696188
59	F Morel, An introduction to 𝔸¹-homotopy theory, from: "Contemporary developments in algebraic K-theory" (editors M Karoubi, A O Kuku, C Pedrini), ICTP Lect. Notes 15, Abdus Salam Int. Cent. Theoret. Phys. (2004) 357 MR2175638
60	F Morel, V Voevodsky, 𝔸¹-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999) 45 MR1813224
61	K M Ormsby, Motivic invariants of p-adic fields, J. K-Theory 7 (2011) 597 MR2811717
62	K M Ormsby, P A Østvær, Motivic Brown–Peterson invariants of the rationals, Geom. Topol. 17 (2013) 1671 MR3073932
63	K Ormsby, O Röndigs, The homotopy groups of the η-periodic motivic sphere spectrum, Pacific J. Math. 306 (2020) 679 MR4129416
64	J H Palmieri, The lambda algebra and Sq⁰, from: "Proceedings of the School and Conference in Algebraic Topology" (editors J Hubbuck, N H V Hung, L Schwartz), Geom. Topol. Monogr. 11, Geom. Topol. Publ. (2007) 201 MR2402807
65	A Polishchuk, L Positselski, Quadratic algebras, 37, Amer. Math. Soc. (2005) MR2177131
66	S B Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970) 39 MR0265437
67	C Rezk, Rings of power operations for Morava E-theories are Koszul, preprint (2012) arXiv:1204.4831
68	J Riou, Opérations de Steenrod motiviques, preprint (2012) arXiv:1207.3121
69	O Röndigs, P A Østvær, Slices of hermitian K-theory and Milnor’s conjecture on quadratic forms, Geom. Topol. 20 (2016) 1157 MR3493102
70	O Röndigs, M Spitzweck, P A Østvær, The first stable homotopy groups of motivic spheres, Ann. of Math. 189 (2019) 1 MR3898173
71	O Röndigs, M Spitzweck, P A Østvær, The second stable homotopy groups of motivic spheres, (2021) arXiv:2103.17116
72	D H Shimamoto, An integral version of the Brown–Gitler spectrum, Trans. Amer. Math. Soc. 283 (1984) 383 MR0737876
73	N E Steenrod, Cohomology operations, 50, Princeton Univ. Press (1962) MR0145525
74	M C Tangora, Computing the homology of the lambda algebra, 337, Amer. Math. Soc. (1985) MR0818916
75	M C Tangora, Some remarks on products in Ext, from: "Algebraic topology" (editor M C Tangora), Contemp. Math. 146, Amer. Math. Soc. (1993) 419 MR1224929
76	M C Tangora, Some Massey products in Ext, from: "Topology and representation theory" (editors E M Friedlander, M E Mahowald), Contemp. Math. 158, Amer. Math. Soc. (1994) 269 MR1263722
77	S Tilson, Squaring operations in the RO(C₂)-graded and real motivic Adams spectral sequences, preprint (2017) arXiv:1702.04632
78	V Voevodsky, Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci. 98 (2003) 1 MR2031198
79	V Voevodsky, On motivic cohomology with ℤ∕l-coefficients, Ann. of Math. 174 (2011) 401 MR2811603
80	J S P Wang, On the cohomology of the mod − 2 Steenrod algebra and the non-existence of elements of Hopf invariant one, Illinois J. Math. 11 (1967) 480 MR0214065
81	G Wang, Z Xu, The algebraic Atiyah–Hirzebruch spectral sequence of real projective spectra, preprint (2016) arXiv:1601.02185
82	G Wang, Z Xu, The triviality of the 61-stem in the stable homotopy groups of spheres, Ann. of Math. 186 (2017) 501 MR3702672
83	G M Wilson, Motivic stable stems over finite fields, PhD thesis, Rutgers University (2016) MR3597836
84	G M Wilson, The eta-inverted sphere over the rationals, Algebr. Geom. Topol. 18 (2018) 1857 MR3784021
85	G M Wilson, P A Østvær, Two-complete stable motivic stems over finite fields, Algebr. Geom. Topol. 17 (2017) 1059 MR3623682