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Chromatic defect, Wood's theorem and higher real K-theories

Christian Carrick

Geometry & Topology 30 (2026) 1051–1108
Bibliography
1 J F Adams, H R Margolis, Modules over the Steenrod algebra, Topology 10 (1971) 271 MR294450
2 G Angelini-Knoll, C Ausoni, J Rognes, Algebraic K-theory of real topological K-theory, preprint (2023) arXiv:2309.11463
3 G Angelini-Knoll, J D Quigley, The Segal conjecture for topological Hochschild homology of Ravenel spectra, J. Homotopy Relat. Struct. 16 (2021) 41 MR4225506
4 M F Atiyah, K-theory and reality, Quart. J. Math. Oxford Ser. (2) 17 (1966) 367 MR206940
5 A Baker, A Lazarev, On the Adams spectral sequence for R-modules, Algebr. Geom. Topol. 1 (2001) 173 MR1823498
6 T Bauer, Computation of the homotopy of the spectrum tmf, from: "Groups, homotopy and configuration spaces" (editors N Iwase, T Kohno, R Levi, D Tamaki, J Wu), Geom. Topol. Monogr. 13, Geom. Topol. Publ. (2008) 11 MR2508200
7 J Beardsley, A theorem on multiplicative cell attachments with an application to Ravenel’s X(n) spectra, J. Homotopy Relat. Struct. 14 (2019) 611 MR3987551
8 J Beardsley, T Lawson, Skeleta and categories of algebras, Adv. Math. 457 (2024) 109944 MR4796061
9 A Beaudry, M A Hill, X D Shi, M Zeng, Models of Lubin–Tate spectra via real bordism theory, Adv. Math. 392 (2021) 108020 MR4313964
10 M Behrens, Topological modular and automorphic forms, from: "Handbook of homotopy theory" (editor H Miller), CRC (2020) 221 MR4197986
11 P Bhattacharya, H Chatham, On the EO-orientability of vector bundles, J. Topol. 15 (2022) 2017 MR4584584
12 R R Bruner, J Rognes, The Adams spectral sequence for topological modular forms, 253, Amer. Math. Soc. (2021) MR4284897
13 C D Carrick, Stacks and real-oriented homotopy theory, PhD thesis, University of California, Los Angeles (2022)
14 C Carrick, M A Hill, On higher real K-theories and finite spectra, preprint (2025) arXiv:2507.07051
15 C Carrick, M A Hill, D C Ravenel, The homological slice spectral sequence in motivic and real bordism, Adv. Math. 458 (2024) 109955 MR4798807
16 D Clausen, A Mathew, N Naumann, J Noel, Descent in algebraic K-theory and a conjecture of Ausoni–Rognes, J. Eur. Math. Soc. 22 (2020) 1149 MR4071324
17 D M Davis, The cohomology of the spectrum bJ, Bol. Soc. Mat. Mexicana (2) 20 (1975) 6 MR467749
18 E S Devinatz, M J Hopkins, The action of the Morava stabilizer group on the Lubin–Tate moduli space of lifts, Amer. J. Math. 117 (1995) 669 MR1333942
19 E S Devinatz, M J Hopkins, J H Smith, Nilpotence and stable homotopy theory, I, Ann. of Math. (2) 128 (1988) 207 MR960945
20 C L Douglas, J Francis, A G Henriques, M A Hill, editors, Topological modular forms, 201, Amer. Math. Soc. (2014) MR3223024
21 Z Duan, G Li, X D Shi, Vanishing lines in chromatic homotopy theory, Geom. Topol. 29 (2025) 903 MR4900032
22 P G Goerss, M J Hopkins, Moduli spaces of commutative ring spectra, from: "Structured ring spectra" (editors A Baker, B Richter), London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 151 MR2125040
23 J Hahn, A Raksit, D Wilson, A motivic filtration on the topological cyclic homology of commutative ring spectra, preprint (2022) arXiv:2206.11208
24 M A Hill, M J Hopkins, D C Ravenel, The homotopy of EO2(p1), unpublished note (2008)
25 M A Hill, M J Hopkins, D C Ravenel, On the nonexistence of elements of Kervaire invariant one, Ann. of Math. (2) 184 (2016) 1 MR3505179
26 M A Hill, L Meier, The C2-spectrum Tmf1(3) and its invertible modules, Algebr. Geom. Topol. 17 (2017) 1953 MR3685599
27 M J Hopkins, Stable decompositions of certain loop spaces, PhD thesis, Northwestern University (1984)
28 M J Hopkins, Complex-oriented cohomology theories and the language of stacks, course notes (1999)
29 M J Hopkins, M Mahowald, H Sadofsky, Constructions of elements in Picard groups, from: "Topology and representation theory" (editors E M Friedlander, M E Mahowald), Contemp. Math. 158, Amer. Math. Soc. (1994) 89 MR1263713
30 P Hu, I Kriz, Real-oriented homotopy theory and an analogue of the Adams–Novikov spectral sequence, Topology 40 (2001) 317 MR1808224
31 N M Katz, B Mazur, Arithmetic moduli of elliptic curves, 108, Princeton Univ. Press (1985) MR772569
32 N Kitchloo, V Lorman, W S Wilson, Multiplicative structure on real Johnson–Wilson theory, from: "New directions in homotopy theory", Contemp. Math. 707, Amer. Math. Soc. (2018) 31 MR3807740
33 N Kitchloo, W S Wilson, The second real Johnson–Wilson theory and nonimmersions of RPn, Homology Homotopy Appl. 10 (2008) 223 MR2475624
34 N Kitchloo, W S Wilson, The second real Johnson–Wilson theory and nonimmersions of RPn, II, Homology Homotopy Appl. 10 (2008) 269 MR2475625
35 T Lawson, En-spectra and Dyer–Lashof operations, from: "Handbook of homotopy theory" (editor H Miller), CRC (2020) 793 MR4197999
36 J Lurie, Chromatic homotopy theory, lecture notes (2010)
37 M Mahowald, C Rezk, Brown–Comenetz duality and the Adams spectral sequence, Amer. J. Math. 121 (1999) 1153 MR1719751
38 A Mathew, Torus actions on stable module categories, Picard groups, and localizing subcategories, preprint (2015) arXiv:1512.01716
39 A Mathew, The homology of tmf, Homology Homotopy Appl. 18 (2016) 1 MR3515195
40 A Mathew, L Meier, Affineness and chromatic homotopy theory, J. Topol. 8 (2015) 476 MR3356769
41 A Mathew, N Naumann, J Noel, Nilpotence and descent in equivariant stable homotopy theory, Adv. Math. 305 (2017) 994 MR3570153
42 L Meier, United elliptic homology, PhD thesis, Rheinische Friedrich-Wilhelms-Universität Bonn (2012)
43 L Meier, X D Shi, M Zeng, A stratification of the equivariant slice filtration, preprint (2023) arXiv:2310.12105
44 H R Miller, On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space, J. Pure Appl. Algebra 20 (1981) 287 MR604321
45 H Miller, C Wilkerson, Vanishing lines for modules over the Steenrod algebra, J. Pure Appl. Algebra 22 (1981) 293 MR629336
46 S A Mitchell, Finite complexes with A(n)-free cohomology, Topology 24 (1985) 227 MR793186
47 O Nakamura, On the squaring operations in the May spectral sequence, Mem. Fac. Sci. Kyushu Univ. Ser. A 26 (1972) 293 MR413103
48 P Pstrągowski, Synthetic spectra and the cellular motivic category, Invent. Math. 232 (2023) 553 MR4574661
49 D Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969) 1293 MR253350
50 D C Ravenel, The non-existence of odd primary Arf invariant elements in stable homotopy, Math. Proc. Cambridge Philos. Soc. 83 (1978) 429 MR474291
51 D C Ravenel, Complex cobordism and stable homotopy groups of spheres, 121, Academic (1986) MR860042
52 D C Ravenel, Nilpotence and periodicity in stable homotopy theory, 128, Princeton Univ. Press (1992) MR1192553
53 C Rezk, Notes on the Hopkins–Miller theorem, from: "Homotopy theory via algebraic geometry and group representations" (editors M Mahowald, S Priddy), Contemp. Math. 220, Amer. Math. Soc. (1998) 313 MR1642902
54 J P Serre, Local fields, 67, Springer (1979) MR554237
55 , The Stacks project, electronic reference (2005–)
56 R Wood, Banach algebras and Bott periodicity, Topology 4 (1965/66) 371 MR185598