Geometry & Topology Volume 28, issue 9 (2024)

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

On the high-dimensional geography problem

Robert Burklund and Andrew Senger

Geometry & Topology 28 (2024) 4257–4293

DOI: 10.2140/gt.2024.28.4257

Bibliography

1	J F Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. 72 (1960) 20 MR0141119
2	J F Adams, On the groups J(X), IV, Topology 5 (1966) 21 MR0198470
3	J F Adams, A periodicity theorem in homological algebra, Proc. Cambridge Philos. Soc. 62 (1966) 365 MR0194486
4	M Ando, M J Hopkins, C Rezk, Multiplicative orientations of KO–theory and of the spectrum of topological modular forms, preprint (2010)
5	M F Atiyah, R Bott, A Shapiro, Clifford modules, Topology 3 (1964) 3 MR0167985
6	M G Barratt, J D S Jones, M E Mahowald, Relations amongst Toda brackets and the Kervaire invariant in dimension 62, J. Lond. Math. Soc. 30 (1984) 533 MR0810962
7	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
8	M Behrens, The construction of tmf, from: "Topological modular forms" (editors C L Douglas, J Francis, A G Henriques, M A Hill), Math. Surv. Monogr. 201, Amer. Math. Soc. (2014) 131
9	J Bowden, D Crowley, A I Stipsicz, The topology of Stein fillable manifolds in high dimensions, I, Proc. Lond. Math. Soc. 109 (2014) 1363 MR3293153
10	W Browder, The Kervaire invariant of framed manifolds and its generalization, Ann. of Math. 90 (1969) 157 MR0251736
11	W Browder, Surgery on simply-connected manifolds, 65, Springer (1972) MR0358813
12	E H Brown Jr., F P Peterson, The Kervaire invariant of (8k+2)–manifolds, Amer. J. Math. 88 (1966) 815 MR0206970
13	G Brumfiel, On the homotopy groups of BPL and PL∕O, Ann. of Math. 88 (1968) 291 MR0234458
14	R R Bruner, J Rognes, The Adams spectral sequence for topological modular forms, 253, Amer. Math. Soc. (2021) MR4284897
15	R R Bruner, J P May, J E McClure, M Steinberger, H_∞ ring spectra and their applications, 1176, Springer (1986) MR0836132
16	R Burklund, Synthetic cookware, book project (2022)
17	R Burklund, J Hahn, A Senger, On the boundaries of highly connected, almost closed manifolds, Acta Math. 231 (2023) 205 MR4683371
18	D Culver, The Adams spectral sequence for 3–local tmf, J. Homotopy Relat. Struct. 16 (2021) 1 MR4225505
19	D M Davis, M Mahowald, The image of the stable J–homomorphism, Topology 28 (1989) 39 MR0991098
20	P Deligne, Courbes elliptiques : formulaire d’après J Tate, from: "Modular functions of one variable, IV" (editors B J Birch, W Kuyk), Lecture Notes in Math. 476, Springer (1975) 53 MR0387292
21	S K Devalapurkar, The Ando–Hopkins–Rezk orientation is surjective, preprint (2019) arXiv:1911.10534
22	C L Douglas, A G Henriques, Topological modular forms and conformal nets, from: "Mathematical foundations of quantum field theory and perturbative string theory" (editors H Sati, U Schreiber), Proc. Sympos. Pure Math. 83, Amer. Math. Soc. (2011) 341 MR2742433
23	Y Eliashberg editor, Contact topology in higher dimensions: questions and open problems, workshop notes (2012)
24	M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357 MR0679066
25	M A Hill, M J Hopkins, D C Ravenel, On the nonexistence of elements of Kervaire invariant one, Ann. of Math. 184 (2016) 1 MR3505179
26	F Hirzebruch, T Berger, R Jung, Manifolds and modular forms, E20, Vieweg & Sohn (1992) MR1189136
27	M J Hopkins, Algebraic topology and modular forms, from: "Proceedings of the International Congress of Mathematicians, I" (editor T Li), Higher Ed. Press (2002) 291 MR1989190
28	M J Hopkins, H R Miller, Elliptic curves and stable homotopy, I, from: "Topological modular forms" (editors C L Douglas, J Francis, A G Henriques, M A Hill), Math. Surv. Monogr. 201, Amer. Math. Soc. (2014) 209 MR3328535
29	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
30	M A Kervaire, Some nonstable homotopy groups of Lie groups, Illinois J. Math. 4 (1960) 161 MR0113237
31	M A Kervaire, J W Milnor, Groups of homotopy spheres, I, Ann. of Math. 77 (1963) 504 MR0148075
32	J Konter, The homotopy groups of the spectrum Tmf, preprint (2012) arXiv:1212.3656
33	A Kosiński, On the inertia group of π–manifolds, Amer. J. Math. 89 (1967) 227 MR0214085
34	M Krannich, Mapping class groups of highly connected (4k+2)–manifolds, Selecta Math. 26 (2020) 81 MR4182837
35	M Krannich, On characteristic classes of exotic manifold bundles, Math. Ann. 379 (2021) 1 MR4211081
36	M Kreck, Isotopy classes of diffeomorphisms of (k − 1)–connected almost-parallelizable 2k–manifolds, from: "Algebraic topology" (editors J L Dupont, I H Madsen), Lecture Notes in Math. 763, Springer (1979) 643 MR0561244
37	N J Kuhn, Localization of André–Quillen–Goodwillie towers, and the periodic homology of infinite loopspaces, Adv. Math. 201 (2006) 318 MR2211532
38	R Lampe, Diffeomorphismen auf Sphären und die Milnor–Paarung, Diplomarbeit, Universität Mainz (1981)
39	T C Lawson, Remarks on the pairings of Bredon, Milnor, and Milnor–Munkres–Novikov, Indiana Univ. Math. J. 22 (1973) 833 MR0312506
40	J P Levine, Lectures on groups of homotopy spheres, from: "Algebraic and geometric topology" (editors A Ranicki, N Levitt, F Quinn), Lecture Notes in Math. 1126, Springer (1985) 62 MR0802786
41	J Lurie, Elliptic cohomology, II: Orientations, preprint (2018)
42	M Mahowald, M Hopkins, The structure of 24 dimensional manifolds having normal bundles which lift to BO[8], from: "Recent progress in homotopy theory" (editors D M Davis, J Morava, G Nishida, W S Wilson, N Yagita), Contemp. Math. 293, Amer. Math. Soc. (2002) 89 MR1887530
43	M Mahowald, M Tangora, Some differentials in the Adams spectral sequence, Topology 6 (1967) 349 MR0214072
44	J Milnor, Classification of (n−1)–connected 2n–dimensional manifolds and the discovery of exotic spheres, from: "Surveys on surgery theory, I" (editors S Cappell, A Ranicki, J Rosenberg), Ann. of Math. Stud. 145, Princeton Univ. Press (2000) 25 MR1747528
45	R M F Moss, Secondary compositions and the Adams spectral sequence, Math. Z. 115 (1970) 283 MR0266216
46	O Nakamura, Some differentials in the mod 3 Adams spectral sequence, Bull. Sci. Engrg. Div. Univ. Ryukyus Math. Natur. Sci. (1975) 1 MR0385852
47	C Nassau, Cohomology charts, electronic reference (2000)
48	S Oka, The stable homotopy groups of spheres, II, Hiroshima Math. J. 2 (1972) 99 MR0322865
49	P Pstrągowski, Synthetic spectra and the cellular motivic category, Invent. Math. 232 (2023) 553 MR4574661
50	D Quillen, The Adams conjecture, Topology 10 (1971) 67 MR0279804
51	D C Ravenel, Complex cobordism and stable homotopy groups of spheres, 121, Academic (1986) MR0860042
52	C Rezk, Supplementary notes for Math 512, lecture notes (2002)
53	J Roitberg, On a construction of Bredon, Proc. Amer. Math. Soc. 33 (1972) 623 MR0295370
54	R Schultz, Composition constructions on diffeomorphisms of S^p × S^q, Pacific J. Math. 42 (1972) 739 MR0315731
55	S Stolz, Hochzusammenhängende Mannigfaltigkeiten und ihre Ränder, 1116, Springer (1985) MR0871476
56	S Stolz, A note on the bP–component of (4n−1)–dimensional homotopy spheres, Proc. Amer. Math. Soc. 99 (1987) 581 MR0875404
57	S Stolz, P Teichner, What is an elliptic object?, from: "Topology, geometry and quantum field theory" (editor U Tillmann), Lond. Math. Soc. Lect. Note Ser. 308, Cambridge Univ. Press (2004) 247 MR2079378
58	M C Tangora, On the cohomology of the Steenrod algebra, Math. Z. 116 (1970) 18 MR0266205
59	P Teichner, Elliptic cohomology via conformal field theory, course notes (2007)
60	C T C Wall, Classification of (n−1)–connected 2n–manifolds, Ann. of Math. 75 (1962) 163 MR0145540
61	C T C Wall, Classification problems in differential topology, VI : Classification of (s−1)–connected (2s+1)–manifolds, Topology 6 (1967) 273 MR0216510