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Stream: theory: algebraic topology

Topic: Deducing (stable) Serre finiteness from the Quillen-Pridd...

Brendan Murphy (Mar 29 2024 at 16:14):

The Serre finiteness theorem says that $\pi_k(\mathbb{S}^n)$ is finitely generated for all $n, k$, and furthermore it's actually finite unless $n = k$ or $n$ is even and $k = 2n-1$ (in which cases the free part has rank $1$). In particular the stable homotopy groups $\pi_k^{\mathrm{st}}(\mathbb{S})$ of the sphere spectrum are finite if $k > 0$ (and ofc we get $\mathbb{Z}$ when $k = 0$ and $0$ when $k < 0$).

Connective spectra are equivalent to grouplike $\mathbb{E}_{\infty}$ homotopy types or to symmetric monoidal $\infty$-groupoids in which every object is invertible. From this latter POV the Quillen-Priddy theorem says that if we take the symmetric monoidal groupoid of permutations, $\mathbb{P} = (\operatorname{Core}(\mathsf{FinSet}), \sqcup, \varnothing)$ and freely add an inverse to its generator to obtain a new category $\overline{\mathbb{P}}$, freely within the context of symmetric monoidal $\infty$-groupoids, then $\overline{\mathbb{P}}$ is isomorphic to the sphere spectrum.

I'm wondering if we can explicitly analyze the process of group completion/freely adding an inverse to deduce that there are finitely many $(k+1)$-automorphisms of any $k$-cell of $\overline{\mathbb{P}}$ up to $(k+2)$-isomorphism. In fact, I think it might simplify things if we truncate. It suffices to show $\operatorname{Ho}_n(\overline{\mathbb{P}})$ is locally finite for any $n < \infty$, and since both the group completion and homotopy category are characterized by mapping out properties we should have that $\operatorname{Ho}_n(\overline{\mathbb{P}})$ is the $n$-groupoid obtained by freely adding an inverse to the generator of $\mathbb{P}$ within the context of symmetric monoidal $n$-groupoids. This then seems possibly amenable to an argument that inverting an object within a locally finite symmetric monoidal $n$-groupoid produces a locally finite symmetric monoidal $n$-groupoid (if that general claim is true) by some relatively explicit construction of the localization

Brendan Murphy (Mar 29 2024 at 16:48):

Naively, thinking of how the grothendieck group of a commutative monoid is constructed, I would expect that if we have a symmetric monoidal (higher) groupoid $\mathcal{G}$ then the group completion $\overline{\mathcal{G}}$ can be constructed as a quotient of $\mathcal{G} \times \mathcal{G}$. More specifically, if $i : \overline{\mathcal{G}} \to \overline{\mathcal{G}}$ is the inversion operation and $F : \mathcal{G} \to \overline{\mathcal{G}}$ the structure map of the group completion (both strong symmetric monoidal functors, at least I think) then the composite $\mathcal{G} \times \mathcal{G} \overset{F \times F}{\to} \overline{\mathcal{G}} \times \overline{\mathcal{G}} \overset{\mathrm{id} \times i}{\to} \overline{\mathcal{G}} \times \overline{\mathcal{G}} \overset{\otimes}{\to} \overline{\mathcal{G}}$ is an effective epimorphism (of symmetric monoidal groupoids? or possibly just groupoids). But I don't have a proof of this and ofc it would just be a first step. Even for finite dimensional groupoids we can get something non locally finite from a finite colimit of locally finite groupoids, eg the circle $B\mathbb{Z}$ is the pushout of $1 \leftarrow 1 \coprod 1 \rightarrow 1$ as $1$-groupoids (or as $\infty$-groupoids)