My PhD thesis: The Multiplicative Structure of the Unoriented Bordism Spectrum

During my PhD, I thought about highly structured ring spectra of characteristic 2. The main focus was the unoriented bordism spectrum, which is well-understood as a plain homotopy commutative ring spectrum (it splits as a wedge sum of suspensions of Eilenberg--MacLane spectra $H\mathbb{F}_2$), but not as an $E_n$ ring spectrum. One can prove that this splitting refines to a map of $E_1$ ring spectra, and I used Dyer--Lashof operations to prove that there is an $E_2$ equivalence $$MO \simeq H\mathbb{F}_2 \wedge \bigwedge_{i = 1}^{\infty} \mathbb{P}_2(S^{2i}).$$ I reduced the existence of a similar $E_3$ equivalence $$MO \simeq H\mathbb{F}_2 \wedge \mathbb{P}_3(S^2) \wedge \bigwedge_{i = 1}^{\infty} \mathbb{P}_3(S^{4i})$$ to the existence of an $E_3$ map $H\mathbb{F}_2 \to MO$. I studied a hypothetical $E_3$ decomposition of $H\mathbb{F}_2$, reducing the existence of such a map to computations in $E_3$ centres. Though I was unable to perform these calculations, I studied centres in enriched higher category theory more generally. I proved that for an $E_1$ monoid $R$ in a monoidal $\infty$-category $\mathcal{V}$, there is an equivalence $$Z_1(R) \simeq \Omega_{\text{Id}_{BR}} [BR, BR]$$ where $BR$ denotes the one object $\mathcal{V}$-enriched $\infty$-category with $\text{Hom}_{BR}(\bullet, \bullet) = R$. This fact has a well-known 1-categorical variant, and there are similar results $$Z_n(R) \simeq \Omega_{\text{Id}}Z_{n - 1}(BR)$$ which only exist in the homotopical context. Both come from proving that $$Z_{\mathcal{C}}(X) \simeq RZ_{\mathcal{D}}(LX)$$ for a nicely behaved adjoint pair $(L, R)$. Having this in the higher-categorical context makes proving some facts about centres of $E_n$ monoids a bit easier.

I also made some headway into understanding the low-dimensional skeleta of an $E_\infty$ cell decomposition of $MO$.

The primary goal was to understand $MO$ well enough to determine whether the ring homomorphism $$\begin{gather} MO_* \to \mathbb{F}_2[t],\\ [M] \mapsto \chi(M)t^{\frac{\dim M}{2}}, \end{gather}$$ refines to a map of $E_n$ ring spectra. Using the $E_1$ and $E_2$ equivalences, I proved that this can be done for $n = 1, 2$. Furthermore, I proved that if an $E_\infty$ refinement exists, then $Eh$ (the spectrum refining the target) must be an $H\mathbb{F}_2$-algebra, and there must be an isomorphism $H_*Eh \cong H_*\text{THH} \mathbb{F}_2$ of algebras over the Dyer--Lashof operations, and that the hypothetical map $$H_*MO \to H_*\text{THH}\mathbb{F}_2 \to \pi_*\text{THH}\mathbb{F}_2$$ commutes with the Dyer--Lashof operations.

To speed up the low-dimensional computations, I wrote a Python program that can calculate

the Dyer--Lashof operations in $H_*BO$, and therefore in $H_*MO$,
the normal and tangential Stiefel--Whitney classes of projective spaces and Dold manifolds,
the Hurewicz images of their bordism classes in the standard basis of $H_*BO$,
conversions between the standard basis of $H_*BO$ and the basis of duals of monomials $w_{i_1} \cdots w_{i_n}$,

and generally perform arithmetic operations in $H_*BO$. I will make the source code available soon. I will also add a link to my thesis, and the $\LaTeX$ source, after my viva, in case Sheffield's plagarism detection apparatus thinks I've copied it verbatim from my own website.