The Lumenary · Research pathways

Stable homotopy theory and generalised cohomology

Name: ______________________

Date: ______________________

You will learn to

Explain stabilisation through the Freudenthal suspension theorem, read off stable stems from the table of homotopy groups of spheres, and explain why stable groups are abelian and (in positive degrees) finite
Define spectra and generalised cohomology theories, state Brown representability, and give ordinary cohomology, K-theory and cobordism as examples that are not ordinary cohomology with any coefficients
Prove that Stiefel–Whitney numbers are cobordism invariants, describe Thom's identification of cobordism with the homotopy of Thom spectra, and use it to decide which real projective spaces bound
Describe the J-homomorphism and the Adams spectral sequence, and connect Bernoulli numbers to the stable stems and to exotic spheres
Define formal group laws and their height, compute the heights of the additive and multiplicative laws, and outline chromatic homotopy theory, separating established results from open problems

1.
By the Freudenthal suspension theorem, $\pi_{n+k}(S^n) \to \pi_{n+k+1}(S^{n+1})$ is an isomorphism when $n > k + 1$ . What is the smallest $n$ for which $\pi_{n+6}(S^n)$ is guaranteed to equal the stable stem $\pi_6^s$ ?
2.
Why is complex K-theory $KU^*$ not isomorphic to ordinary cohomology $H^*(-; A)$ for any abelian group $A$ ?
- A. $KU^{-2}(\mathrm{pt}) \cong \mathbb{Z} \ne 0$ , whereas $H^{n}(\mathrm{pt}; A) = 0$ for every $n \ne 0$ .
- B.K-theory fails the exactness axiom.
- C.K-theory has a product and ordinary cohomology does not.
- D. $KU^0(X)$ is infinite for every non-empty $X$ , and $H^0(X; A)$ is finite.
3.
In the proof that Stiefel–Whitney numbers of a boundary $M = \partial W$ vanish, we wrote $w_I[M] = \langle i^* w_I(W), \partial[W, M] \rangle = \langle \delta i^* w_I(W), [W, M] \rangle = 0$ . Which fact justifies the final equality?
- A.In the long exact sequence of the pair $(W, M)$ , the composite $\delta \circ i^*$ of consecutive maps is zero.
- B.The Whitney sum formula.
- C. $M$ is orientable.
- D. $H^{n+1}(M) = 0$ because $M$ has dimension $n$ .
4.
Which of these closed manifolds are boundaries of compact manifolds? (Choose all that apply.)
- A. $\mathbb{R}P^3$
- B. $\mathbb{R}P^4$
- C. $\mathbb{R}P^2 \times \mathbb{R}P^2$
- D. $\mathbb{R}P^2 \times \mathbb{R}P^1$
5.
The image of the J-homomorphism in $\pi_{11}^s$ is cyclic of order the denominator of $B_6 / 12$ , where $B_6 = \tfrac{1}{42}$ . What is this order?
6.
Over $\mathbb{F}_3$ , what is the $3$ -series $[3]_F(x)$ of the multiplicative formal group law $F(x, y) = x + y + xy$ , and what is its height?
- A. $[3](x) = x^3$ ; height 1.
- B. $[3](x) = 3x = 0$ ; height $\infty$ .
- C. $[3](x) = x^9$ ; height 2.
- D. $[3](x) = 3x + 3x^2 + x^3$ ; height 0.
7.
Which Lean statement formalises 'the multiplicative formal group law has height 1 at the prime $p$ ' as it is proved in this lesson?
- A.nSeries mulFGL p (X : (ZMod p)[X]) = X ^ p
- B.nSeries mulFGL p (X : ℤ[X]) = X ^ p
- C.nSeries addFGL p (X : (ZMod p)[X]) = X ^ p
- D.IsFormalGroupLaw (mulFGL (A := ZMod p))
8.
By Thom's theorem, $\dim_{\mathbb{F}_2} \mathfrak{N}_n$ is the number of partitions of $n$ into parts not of the form $2^k - 1$ (that is, avoiding $1, 3, 7, 15, \dots$ ). What is $\dim \mathfrak{N}_6$ ?
9.
Write image_of_j(k) returning the denominator of $B_{2k}/(4k)$ , computing the Bernoulli numbers exactly with fractions.Fraction from the recurrence $\sum_{j=0}^{m} \binom{m+1}{j} B_j = 0$ ( $m \ge 1$ , $B_0 = 1$ ). Check that it gives $24, 240, 504, 480$ for $k = 1, 2, 3, 4$ .
Computational — Practice

Answer key — Stable homotopy theory and generalised cohomology

1.
With $k = 6$ the condition $n > k + 1$ means $n \ge 8$ , so $\pi_{14}(S^8) \cong \pi_6^s \cong \mathbb{Z}/2$ . In the table, the bold entries are exactly those with $n > k + 1$ .
2.
$KU^*(\mathrm{pt}) = \pi_{-*}(KU) = \mathbb{Z}[\beta^{\pm1}]$ is $\mathbb{Z}$ in every even degree, while $H^*(\mathrm{pt}; A)$ is concentrated in degree 0. So no choice of $A$ works. The same argument shows cobordism ( $\mathfrak{N}_2 \ne 0$ ) is not ordinary cohomology.
3.
The cohomology long exact sequence of the pair contains $H^n(W) \xrightarrow{i^*} H^n(M) \xrightarrow{\delta} H^{n+1}(W, M)$ , and exactness gives $\delta \circ i^* = 0$ . So $\delta i^* w_I(W) = 0$ and the pairing vanishes.
4.
$\mathbb{R}P^3$ and $\mathbb{R}P^2 \times \mathbb{R}P^1$ bound. $\mathbb{R}P^4$ and $\mathbb{R}P^2 \times \mathbb{R}P^2$ do not: each has a non-zero Stiefel–Whitney number (the top one, which is the Euler characteristic mod 2). By Thom, $[\mathbb{R}P^4]$ and $[\mathbb{R}P^2]^2$ form a basis of $\mathfrak{N}_4 \cong \mathbb{F}_2^2$ .
5.
$B_6 / 12 = \frac{1}{42 \cdot 12} = \frac{1}{504}$ , so the image of J has order $504$ . In fact $\pi_{11}^s \cong \mathbb{Z}/504$ : the image of J is the whole group.
6.
$1 + [3](x) = (1+x)^3 = 1 + 3x + 3x^2 + x^3 \equiv 1 + x^3 \pmod 3$ , so $[3](x) = x^3 = x^{3^1}$ : height 1. This is why complex K-theory lies at chromatic height 1 at every prime.
7.
Height 1 at $p$ means $[p]_F(x)$ has leading term $x^{p}$ over $\mathbb{F}_p$ . The Lean theorem mul_pSeries_charP proves the stronger statement $[p]_F(X) = X^p$ in $\mathbb{F}_p[X]$ , using $(1 + X)^p = 1 + X^p$ in characteristic $p$ .
8.
Partitions of 6 into parts from $\{2, 4, 5, 6\}$ : $6$ , $4 + 2$ , $2 + 2 + 2$ (note $5 + 1$ uses $1$ ). So $\dim \mathfrak{N}_6 = 3$ , with basis $[\mathbb{R}P^6]$ , $[\mathbb{R}P^4 \times \mathbb{R}P^2]$ , $[\mathbb{R}P^2]^3$ — as the experiment confirms with Stiefel–Whitney numbers.
9.
```
from fractions import Fraction
from math import comb

def image_of_j(k):
    b = [Fraction(1)]
    for m in range(1, 2 * k + 1):
        s = sum(comb(m + 1, j) * b[j] for j in range(m))
        b.append(-s / (m + 1))
    return (b[2 * k] / (4 * k)).denominator
```
The recurrence gives $B_2 = \tfrac16$ , $B_4 = -\tfrac1{30}$ , $B_6 = \tfrac1{42}$ , $B_8 = -\tfrac1{30}$ , $B_{12} = -\tfrac{691}{2730}$ . By Adams' theorem these denominators are the orders of the image of J in $\pi_3^s, \pi_7^s, \pi_{11}^s, \pi_{15}^s, \pi_{23}^s$ .