Homotopy groups of spheres
Type the sphere dimension and the stem . You get everything this page knows about : the group when it is known, otherwise its order or bounds on it, its -primary parts, known summands and exponents. Every statement cites the theorem, table or paper it comes from.
The table
Row , column holds , written as in Toda’s table: means and means . The dashed line is the Freudenthal boundary , a line of slope one. To its right a row no longer changes, so only its first stable entry is written. Click a cell for all the details and references.
Stable stems for
From Isaksen–Wang–Xu (2023, Table 1), with the 2024 corrections of Burklund–Isaksen–Xu in stems 70 and 71. Rows marked “open” list the possible 2-primary alternatives. Click a row to open it in the calculator.
References
All sources used on this page. Theorem, table and section numbers in the cards refer to these editions. The arXiv items from 2024–2026 are preprints.
What the data covers
- Integral groups, , all : Toda’s table.
- 2-primary parts, stems , all , plus the known groups in stem 33: the 2026 summary of Hirato–Lee–Miyauchi–Mukai–Ohara–Yang (Appendix B), which collects Toda, Mimura–Toda, Mimura, Mimura–Mori–Oda, Oda, Inoue–Miyauchi–Mukai, Miyauchi–Mukai and Yang–Wu. Orders through stem 50 come from Baer’s 2026 machine computation. The two sources agree everywhere they overlap.
- 3-primary parts, stems , all : Toda’s 2003 tables (odd spheres), extended to even spheres by Serre’s splitting.
- Stable stems : Isaksen–Wang–Xu and Burklund–Isaksen–Xu. Odd primes: Ravenel’s tables (3-primary through stem 108) and the image of .
- Everything else comes from general theorems (Freudenthal, Serre, Adams, James, Cohen–Moore–Neisendorfer, Hopf fibrations, Thomeier…) applied to the data above. The cards say which theorem was used.