Editing Exotic sphere (section)

=== Parallelizable manifolds===
The group <math>\Theta_n</math>  has a cyclic subgroup

:<math>bP_{n+1}</math>

represented by ''n''-spheres that bound [[parallelizable manifold]]s. The structures of <math>bP_{n+1}</math> and the quotient

:<math>\Theta_n/bP_{n+1}</math>

are described separately in the paper {{harvs|authorlink1=Michel Kervaire|last1=Kervaire|last2=Milnor|authorlink2=John Milnor|year=1963}}, which was influential in the development of [[surgery theory]]. In fact, these calculations can be formulated in a modern language in terms of the [[surgery exact sequence]] as indicated [[surgery exact sequence#Examples|here]].

The group <math>bP_{n+1}</math> is a cyclic group, and is trivial or order 2 except in case <math>n = 4k+3</math>, in which case it can be large, with its order related to the [[Bernoulli number]]s. It is trivial if ''n'' is even. If ''n'' is 1 mod 4 it has order 1 or 2; in particular it has order 1 if ''n'' is 1, 5,  13,  29, or 61, and {{harvs|txt|first=William|last=Browder|authorlink=William Browder (mathematician)|year=1969}} proved that it has order 2 if <math>n = 1</math> mod 4 is not of the form <math>2^k - 3</math>. It follows from the now almost completely resolved [[Kervaire invariant]] problem that it has order 2 for all ''n'' bigger than 126; the case <math>n = 126</math> is still open. The order of <math>bP_{4k}</math> for <math>k\ge 2</math> is

:<math>2^{2k-2}(2^{2k-1}-1)B,</math>

where ''B'' is the numerator of <math>4B_{2k}/k</math>, and <math>B_{2k}</math> is a [[Bernoulli number]]. (The formula in the topological literature differs slightly because topologists use a different convention for naming Bernoulli numbers; this article uses the number theorists' convention.)