Doomsday conjecture

In algebraic topology, the doomsday conjecture was a conjecture about Ext groups over the Steenrod algebra made by Joel Cohen, named by Michael Barratt, published by Milgram (1971, conjecture 73) and disproved by Mahowald (1977). Minami (1995) stated a modified version called the new doomsday conjecture.

The original doomsday conjecture was that for any prime p and positive integer s there are only a finite number of permanent cycles in

${\displaystyle {\text{Ext}}_{A_{*}}^{s,*}(Z/pZ,Z/pZ).\,}$

Mahowald (1977) found an infinite number of permanent cycles for p = s = 2, disproving the conjecture. Minami's new doomsday conjecture is a weaker form stating (in the case p = 2) that there are no nontrivial permanent cycles in the image of (Sq⁰)ⁿ for n sufficiently large depending on s.

References

• Mahowald, Mark (1977), "A new infinite family in ${\displaystyle {}_{2}\pi _{*}{}^{s}}$", Topology, 16 (3): 249–256, doi:10.1016/0040-9383(77)90005-2, ISSN 0040-9383, MR 0445498
• Milgram, R. James (1971), "Problems presented to the 1970 AMS symposium on algebraic topology", in Liulevicus, Arunas (ed.), Algebraic Topology, Proc. Symp. Pure Math, 22, pp. 187–201
• Minami, Norihiko (1995), "The Adams spectral sequence and the triple transfer", American Journal of Mathematics, 117 (4): 965–985, doi:10.2307/2374955, ISSN 0002-9327, JSTOR 2374955, MR 1342837
• Minami, Norihiko (1998), "On the Kervaire invariant problem", in Mahowald, Mark E.; Priddy, Stewart (eds.), Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997), Contemp. Math., 220, Providence, R.I.: Amer. Math. Soc., ISBN 978-0-8218-0805-4, MR 1642897
• Minami, Norihiko (1999), "The iterated transfer analogue of the new doomsday conjecture", Transactions of the American Mathematical Society, 351 (6): 2325–2351, doi:10.1090/S0002-9947-99-02037-1, ISSN 0002-9947, MR 1443884