BLOG Carl McTague mathematician, composer, photographer, fiddler

11 Jul 2017 | categories: Photographs

# Chimney Bluff

A drumlin eroding into Lake Ontario.

15 Mar 2017 | categories: Photographs

# Der Himmel über Rochester

27 Mar 2016 | categories: Photographs

# Letchworth Lower Falls

17 Dec 2014 | categories: Mathematics

# How to Recognize Generators for String Bordism

A key result of my paper:

unmentioned in the abstract or introduction—and likely of independent interest—is the following characteristic-number-theoretic criterion for a set to generate the String bordism ring (with 6 inverted).

Theorem 4. A set $S$ generates the ring $\pi_*\mathrm{MO}\langle8\rangle[1/6]$ if:

1. For each integer $n>1$, there is an element $M^{4n}$ of $S$ such that for any prime $p>3$: \begin{align*} \mathrm{ord}_p \big( \mathrm{s}_n[M^{4n}] \big) = \begin{cases} 1 & \text{if 2n=p^i-1 or 2n=p^i+p^j for some integers 0 \le i \le j} \\ 0 & \text{otherwise} \end{cases} \end{align*} where $\mathrm{s}_n[M]$ is the characteristic number corresponding to the $n$th power sum symmetric polynomial $\sum x_i^n$ in the Pontrjagin roots of the tangent bundle of $M$, and $\mathrm{ord}_p(-)$ its $p$-adic order, i.e. the heighest power of $p$ which divides it.
2. For each prime $p>3$ and each pair of integers $0<i<j$, there is an element $N^{2(p^i+p^j)}$ of $S$ such that: \begin{align*} \mathrm{s}_{(p^i+p^j)/2}[N^{2(p^i+p^j)}]&=0 \\ \mathrm{s}_{(p^i+1)/2,(p^j-1)/2}[N^{2(p^i+p^j)}] &\not\equiv 0 \mod p^2 \end{align*} where $\mathrm{s}_{m,n}[N]$ is the characteristic number corresponding to the symmetric polynomial $\sum_{i\ne j} x_i^mx_j^n$ in the Pontrjagin roots of the tangent bundle of $N$.

This is a consequence of Hovey’s calculation of:

The homotopy of $\mathrm{MString}$ and $\mathrm{MU}\langle6\rangle$ at large primes, Algebr. Geom. Topol. 8 (2008), 2401–2414.

which itself builds on Hopf-ring-theoretic work of Ravenel and Wilson. It should be compared with the analogous result for the oriented bordism ring (with 2 inverted):

Theorem (Novikov, cf. Stong p. 180). A sequence $\{M^{4n}\}_{n\ge1}$ generates the ring $\pi_*\mathrm{MSO}[1/2]$ if and only if:

• For any integer $n>0$ and any odd prime $p$: \begin{align*} \mathrm{ord}_p \big( \mathrm{s}_n[M^{4n}] \big) = \begin{cases} 1 & \text{if 2n=p^i-1 for some integer i>0} \\ 0 & \text{otherwise} \end{cases} \end{align*}

## References

13 Dec 2014 | categories: Photographs, Serpents, Archaelogy, Chronology

# After Laocoön [or Before?]

Little is known with certainty about the serpent—not even how many thousand years old it is.

# References

[It struck me later how sim­i­lar my com­po­si­tion is to Edvard Munch’s in The Scream (1893).]

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