Stable homotopy theory and generalised cohomology

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 $\pi_6^s$?

Why is complex K-theory $KU^*$ not isomorphic to ordinary cohomology $H^*(-; A)$ for any abelian group $A$?

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?

Which of these closed manifolds are boundaries of compact manifolds? (Choose all that apply.)

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?

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?

Which Lean statement formalises 'the multiplicative formal group law has height 1 at the prime $p$' as it is proved in this lesson?

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 $\dim \mathfrak{N}_6$?

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 $k = 1, 2, 3, 4$.