OFFSET
0,4
COMMENTS
a(1) = 1 because the Poincaré conjecture is true.
For all odd m+1, the order of bP_{m+1} is known to be 1. This sequence is known for all n except n = 62, where a(n) = order of bP_126 is either 1 or 2. - Riley Moriss, Mar 27 2026
REFERENCES
M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres: I. Ann. of Math. (2) 77 1963 504-537. Gives erroneous a(9).
J. P. Levine, Lectures on groups of homotopy spheres. In Algebraic and geometric topology (New Brunswick, NJ, 1983), 62-95, Lecture Notes in Math., 1126, Springer, Berlin, 1985.
LINKS
Wolfgang Lück, A Basic Introduction to Surgery Theory
Manifold Atlas, Page on exotic spheres
John W. Milnor, Differential Topology Forty-six Years Later, Notices Amer. Math. Soc. 58 (2011), 804-809.
PROG
(Python)
from sympy import bernoulli
def epsilon(k):
return (3 - (-1)**k)//2
def mersene(k):
return int(2**(2*k - 2) * (2**(2*k - 1) - 1))
def bP_4k(k):
bern = (-1)**(k+1)*bernoulli(2*k) # Topologists Bernoulli numbers
bern = bern/(4*k)
num_bern = bern.p
return epsilon(k)*mersene(k)*num_bern
def a(n):
if n in [0, 1, 2, 6, 14, 30]: return 1
elif n == 62: raise ValueError("Unknown")
elif n % 2 == 0: return 2
else: return bP_4k((n+1)//2)
# Riley Moriss, Mar 27 2026
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
N. J. A. Sloane, Feb 03 2002
EXTENSIONS
a(0)-a(1) from Andrei Zabolotskii, Feb 02 2018
a(9) corrected by and a(10)-a(30) from Riley Moriss, Mar 27 2026
STATUS
approved