## The Kervaire-Milnor Formula

The K-M formula and its ingredients are presented in

1) Bernoulli numbers and the unity of mathematics by Barry Mazur, p.14, Secs. 4, 5, and 6

2) Differential topology forty-six years later by Milnor

3) Homotopy group of spheres Wikipedia

4) Exotic sphere Wikipedia

5) J-homomorphism Wikipedia

7) Bernoulli numbers, homotopy groups, and a theorem of Rohlin by Milnor and Kervaire

The K-M formula, as presented by Mazur, is

$card [\Theta_{4k-1}] = R(k) \; card[ H_{4k-1}] \; B_{2k}/2k$

where (if I interpret Mazur, and Milnor, correctly) $\Theta_{j}$ is the group of homotopy spheres up to h-cobordism, or essentially the set of all oriented diffeomorphism classes of closed smooth homotopy $n$-spheres; $R(k)=2^{2k-2}(2^{2k-1}-1)$ for odd $k$ and twice that for even $k$; $H_{j}$ is the group of stable homotopy classes of continuous maps from the $(m+j)$-sphere to the $j$-sphere, and $B_n$ are the Bernoulli numbers.

In Refs. 2 and 4 and OEIS A001676 (also cf. A187595,  A189995, A048648, A047680,   A053381, A083420) are lists of the order of the classes:

$1, \;1, \;1,\;1,\; 1, \;1, \;28,\; 2,\; 8,\; 6,\; 992,\; 1,\; 3,\; 2,\; 16256, \;2,\; 16,\; 16,\; 523264,\; 24, \;8, \;4,\;...$

with the numbers for the dimensions $4n+3$ greater than three sticking out like sore thumbs. Ref. 4 also juxtaposes the factors of the associated Kervaire-Milnor formula.

In addition, a search on the OEIS (no commas, just spaces between numbers) for

$28 \; \; 992 \;\; 16256 \;$

and separately

$56 \; \; 992 \;\; 16256 \; \;$

reveal several related combinatorial structures and mathematical expressions, some clearly related to the K-M formula and others not.

Why do these particular dimensions stand out? What is the ultimate source of this peculiarity?

The first table in Ref. 3 shows that these salient dimensions correlate with the yellow-highlighted unstable groups, and the article demonstrates the relation to J-homomorphism (Ref. 5), linking it to the denominator of the Bernoulli numbers, as Mazur noted.

The factors

$\displaystyle \frac{2^{n+1}-4^{n+1}}{n+1}B_{n+1}=(-1)^n(2^{n+1}-4^{n+1})\zeta(-n)=-(-2)^{1+n}\eta(-n) \; ,$

reflected in the Mellin transform for the Dirichlat eta function

$\displaystyle 2^{1-s}\eta(s)= \int_{0}^{\infty} \frac{t^{s-1}}{(s-1)!}\frac{2}{e^{2t}+1}dt \; ,$

are well-known among those familiar with special integer sequences. They are the Taylor series coefficients of the e.g.f. $\frac{2}{e^{2t}+1}$ which is the sequence A155585, which, omitting the signs and zeros, are the tangent, or zag, numbers of A000182 (embedded in the Euler, or up/down numbers A000111) with associations to several combinatorial and physical models. The e.g.f. and the Dirichlet eta function are related to the Fermi-Dirac distribution , governing the statistics of fermions, and the volumes of moduli spaces in quantum guage theory in two dimensions (see the second example in the MathOverflow question “Geometric/physical/probabilistic interpretations of the Riemann zeta(n>1)?”

The odd terms of the sequence $4^n-2^n \rightarrow 2,\; 56,\; 992,\; 16256,\; 65280,\; 1047552, ...$ (A020522) enumerate walks on the cycle-graph $C_8$ (shades of Bott periodicity?) and appear (mod small factors of 2) in tables in the “Exotic sphere” where they are described (again mod small factors) in terms of the order of the non-trivial groups of $\Theta_n$ and its cyclic subgroup $bP_{n+1}$. So, the question could be rephrased

Why do these non-trivial groups leap out periodically with large orders while intervening dimensions are trivial?

Related stuff:

Complex cobordism theory for number theorists” by Ravenel

Elliptic genera of quantum field theory” by Witten

“Anomalies and modular invariance in string theory” by Schellekens and Warner

A note on the Landweber-Strong elliptic genus” by Zagier

What are some interesting problems at the intersection of algebraic number theory and algebraic topology?” MO-Q, see answer by Lawson

Iwasawa theory for chromatic localizations” by Peterson

Chromatic homotopy theory” by Lurie, course notes

What is an elliptic genus?” by Ochanine

Genus of a multiplicative sequence” Wikipedia

Algebraic topology and modular forms” by Hopkins

These last three refs above were tracked down through Drew Heard’s answer to the following MO question:

Why should I care about topological modular forms?” MathOverflow

A survey of elliptic cohomology” by Lurie

Periodic cohomology theories defined by elliptic curves” by Landweber, Ravenel, and Stong

Elliptic cohomology: A historical review” by Redden

Manifolds and Modular Forms by Hirzebruch, Berger, and Jung

A normal form for elliptic curves” by Edwards

Characteristic Classes by Milnor and Stasheff (appendix on Bernoulli numbers)

More on the Bernoulli and Euler polynomials and their umbral compositional inverses

The reciprocal integers and the Bernoulli numbers are intimately paired through the formalism of Appell Sheffer sequences, two pairs with Lie and quantum algebras written all over them, and to the algebraic topology/geometry of exotic spheres through the Kervaire-Milnor formula and multiplicative (elliptic) genera.

The Bernoulli polynomials, as any Appell sequence $P_n(x)$ with $P_n(0)=1$, can be contsructed algebraically from a base sequence of numbers, naturally the Bernoulli numbers $B_n(0)$, as

$B_n(x)=(B.(0)+x)^n = \sum_{k=0}^n B_k(0)x^{n-k} \; .$

(This extends to $B_n(x+y)=(B.(x)+y)^n$. And, the lowering op is $D=d/dx$; i.e., $D\; P_n(x) = n \; P(n-1) \; .$)

The umbral compositional inverse Appell sequence $\hat{B}_n(x)$ has the reciprocal integers $\hat{B}_n(0)=1/(n+1)$ as its base sequence.

As an umbral inverse pair,

$B_n(\hat{B}.(x)) = x^n = \hat{B}_n(B.(x)) ,$

that is, when substitiuting $\hat{B}.(x)$ for $x$ in the n-th Bernoulli polynomial and evaluating $(\hat{B}.(x))^n = B_n(x)$, the sum reduces to $x^n$, and vice versa.

Related properties (connected to surjections and the geometry of permutahedra, A133314) are that the two lower triangular matrices of coefficients of the polynomials are a matrix inverse pair, the e.g.f.s of the base sequences are multipicative inverses, and the triangles are formed by multiplying each n-th diagonal of the Pascal matrix by $B_n(0)$ or $\hat{B}_n(0)$. In fact, the e.g.f.s for the polynomials are

$e^{B.(x)t} = \frac{t}{e^t-1}e^{xt}$

and

$e^{\hat{B}.(x)t}=\frac{e^t-1}{t}e^{xt} \; ,$

so

$\hat{B}_n(x) = \frac{(x+1)^{n+1}-x^{n+1}}{n+1} \; .$

Then

$D \; x^{n+1}/(n+1) = x^n = \hat{B}_n(B.(x)) = \frac{(B.(x)+1)^{n+1}-(B.(x))^{n+1}}{n+1}$

$= \frac{(B.(x+1))^{n+1}-B.(x)^{n+1}}{n+1}= \frac{B_{n+1}(x+1)-B_{n+1}(x)}{n+1} \; ,$

implying that

$f(B.(x+1))-f(B.(x)) = D f(x)$

applied term by term to the Taylor series of any function. This can be taken as the defining property of the Bernoulli polynomials.

So, we see that the properties of the Appell sequence pair are ineluctably intertwined–the Bernoulli numbers and the reciprocal integers form an intimate couple. This is most literally reflected in the raising ops (differential generator, $R\;P_n(x)=P_{n+1}(x)$) of the two, which differ only by a sign:

for the Bernoulli polynomials,

$R_x = x - \sum_{n \ge 0} (-1)^n B_{n+1}(0)\; \hat{B}_n(0) \frac{D^n}{n!}$

and for the reciprocal polynomials,

$\hat{R}_x= x + \sum_{n \ge 0} (-1)^n B_{n+1}(0)\; \hat{B}_n(0) \frac{D^n}{n!} \; .$

There is much more to this story. This operator with the coefficients treated as general indeterminates generates the cycle index partition polynomials for the symmetric group (A036039 and A231846, MO-Q Cycling through the zeta garden: zeta functions for graphs, cycle index polynomials, and determinants) and so also generalized symmetric functions with the coefficients of $(-1)^n B_{n+1}(0)/(n+1)= \zeta(-n)$ as the power functions and the Appell e.g.f.s. as those for the analogous elementary and complete symmetric functions, making links to exterior algebra/calculus and simplices.

So, any elucidation of overarching topology has strong links to other important areas of analysis.

Note that the normalized Bernoulli numbers above, $BN_n(0)$, are related to another pair of Appell sequences:

$e^{BN.(x)t} = \frac{2}{e^{2t}+1}e^{tx}= [1-t+ 2 t^3/3! - 16 t^5/5! + 272 t^7/7! - \cdots]e^{xt} \; ,$

the e.g.f. of the polynomials of A081733, and

$e^{\hat{BN}.(x)t}=\frac{e^{2t}+1}{2}e^{tx} \; ,$

the e.g.f. of the polynomials of A119468, with

$\hat{BN}_n(x) = \frac{(x+2)^n + x^n}{2}$ and for $n>0$

$\hat{BN}_n(0)=2^{n-1} \; .$

The triangle associated to the last polynomials, aside from the diagonal, are half the coefficients of the f-vectors for the hypercubes A038207 and have numerous interpretations. The surjection/reciprocal e.g.f. mapping governed by the signed, refined partition face polynomials of permutahedra (A133314) can be applied to these multiples of two to generate the normalized Bernoulli numbers.

Following in Euler, Dirichlet, Riemann, and Witten’s footsteps, note the connection to the Dirichlet eta and Riemann zeta functions

$(-1)^n\frac{2^{1+n}-4^{1+n}}{n+1}B_{n+1} = (2^{1+n}-4^{1+n}) \zeta(-n) = 2^{1+n} \eta(-n)$

and interpolatinng with the Mellin transform,

$2^{1-s} \eta(s) = \int_0^\infty\frac{t^{s-1}}{(s-1)!}\frac{2}{e^{2t}+1} \; dt = (2^{1-s}-4^{1-s}) \zeta(s) = 2^{1-s} (1-2^{1-s})\zeta(s) ,$

which can be related directly to volumes of manifolds in 2-D quantum guage theories by the formula in example 2 of MO-Q: Geometric … interpretations of Riemann $\zeta(n >1)$ from Witten’s paper and the Kervaire-Milnor formula above (mentioned by Baez in his response to MO-Q: Why do the Bernoulli numbers arise everywhere?). Interpolating the polynomials gives a specialized Lerch zeta function.

The Lagrange compositional inversion (LIF) of A133437, (see also MO-Q: Why is there a connection between enumerative geometry and nonlinear waves?), based on the combinatorics of the Stasheff polytopes, or associahedra, allows the cleanest geometric mappings I know of between the reciprocals of the odd natural numbers and the normalized Bernoulli $BN_n(0)$, or Riemann zeta/Dirichlet eta values.

Removing the constant term from their e.g.f. above and changing signs give the alternate g.f. for the multiplicative elliptic genus

$\frac{e^{2t}-1}{e^{2t}+1}=\tanh(t)=t - 2 t^3/3! + 16 t^5/5! -272 t^7/7! + \cdots$

with compositional inverse (FGL logarithm)

$\frac{\log(\frac{(1+t)}{(1-t)})}{2}=\tanh^{-1}(t)= t + t^3/3 + t^5/5 + \cdots \; .$

Then applying the LIF using these reciprocals, we can ignore the even dimensional associahedra and any faces of the odd n-dimensional associahedra that map to partitions of $2(n+1)$ containing any even number to obtain the $BN_{n+2}(0)$. There are well-known mappings among the refined faces of the associahedra, trees (but not strictly binary), dissections of polygons, and numerous other combinatorial models. The e.g.f. $\tanh(t)$ enumerates certain classes of binary trees (A000182).

There are rich connections to elliptic curves, the Lorentz formal group law, and generalized Hirzebruch genera, as noted in A008292 and some drafts of A155585 currently in the queue (see the Buchstaber and Bunkova paper, pp. 35-37, in particular), and the KdV, inviscid Burgers, and Ricatti equations (see the later Elliptic Lie Triad entry at this site). The simple iterated Lie derivative/infinitesimal generator (infinigen) $((1-x^2)d/dx)^n$ underlies these associations and acting on $x$ and evaluated at the origin generates the $BN_n(0)$. It can be mapped, a la Cayley, to rooted binary trees (A145271) and weaves a web of connections between the Legendre polynomials and the family of Jacobi quartic elliptic curves (see refs above on elliptic genera and MO-Q: Geometric picture of invariant differential of an elliptic curve). The differential equation for the Legendre polynomials is

$\displaystyle [(1-x^2)D_x]^2 \; L_n(x) + (n+1)n \; (1-x^2) \; L_n(x)=0 \; .$

For more on the connections of Euler-Bernoulli numbers to algebraic geometry/topology, see On the number of connected components in the space of M-polynomials of hyperbolic functions by Shapiro and Vainshtein and the refs in the notes on the Kervaire-Milnor formula and the entry Bernoulli Appells at this site.

P. Feinsilver in “Lie algebras, representations, and analytic semigroups through dual vectors” draws connections among the Bernoulli-Appell-Sheffer formalism, Heisenberg-Weyl algebra, Meixner orthogonal polynomials, and the universal enveloping algebra.

Tempesta has also published much on Appell polynomials and their connections to number theory and the Lazard formal group law.

Stochastic interpretations enter though characterictc fcts and the Laplace transform related to e.g.f.s and classical cumulants and moments as presented by Taylor and Rota for the finite op calc (Feinsilver also) and through the Stieltjes-Cauchy transform related to o.g.f.s and the free cumulants and moments as presented by Voiculescu and Speicher. Related diagrammatics enter through the exp/log transform -Bell/cumulant generating partition polynomials for the classical cumulants (A127671, A036040) and for comp. inversion /noncrossing partitions/Dyck paths/polygon dissections for the free cumulants (A134264).