David Roberts scratch 2014

This page is intended to be a dumping ground for thoughts. No guarantee is made for correctness. On the other hand, I would be happy to discuss stuff from here if you have ideas. See David Roberts for my contact details.

See also: scratch 2015, scratch 2016

---

2014

February

28

What is the ‘integration map’ $K_G(X) \to R(G)$ of Atiyah-Singer? (section 1.3 of http://www.math.jussieu.fr/~vergne/pageperso/articles/CONGEURO.pdf only refers to the series of papers ‘The index of elliptic operators’ in toto)

---

Some references:

http://arxiv.org/abs/0809.1273v2 The Atiyah–Segal completion theorem in twisted K-theory

http://arxiv.org/abs/0712.0702v2 Pontrjagin-Thom maps and the homology of the moduli stack of stable curves (constructs a stack of virtual vector bundles)

http://arxiv.org/abs/math/0511244 Twisted sheaves and the period-index problem

---

---

26

Drinfeld: rank $r$ shtukas form an algebraic stack.

---

---

---

25

Note that the exit path category of a (Whintey?) stratification is an analogue of the fundamental groupoid. Within each stratum is looks just like $\Pi_1$, but paths can only move from a stratum to a higher one (or one of the same depth). There is a classification by \MacPherson of constructible sheaves in terms of representations of the exit path category analogous to that of covering spaces. It is unpublished, but referred to in Treumann, Exit paths and constructible stacks.

Given a stratification $S$ of a space $X$, there should be a poset $\mathcal{S}$ of strata (for example $I$ in entry dated Feb 24, arising from a $G$-action), and a functor $EP(X,S) \to \mathcal{S}$ which is the posetal reflection.

Given a $G$-space $X$ with the orbit type stratification I can compose the posetal reflection with various functors, say $I \to C_G$, as below, or $I \to R(G)-mod_f$, sending a stratum $X_{(H)}$ to the finitely generated $R(G)$-module $R(H)$. More generally one could (in fact should) have $EP(X,S) \to R(G)-mod_f$ sending $G_x=Stab(x)$ to $R(G_x)$.

Then a $G$-vector bundle on $X$ (in fact an element of equivariant K-theory) gives rise to a global section of the constructible sheaf corresponding to

$$EP(X,S) \to R(G)-mod_f$$

where $S$ might be a finer stratification arising from the $G$-action.

What I want to do is go from a global section of this constructible sheaf to a $G$-vector bundle. Will need some details about induction etc here I think.

Need to think a bit about prop 1.3 in Segal’s Equivariant K-theory, that is a $G$-equivariant version that homotopic maps (with compact domain) induce isomorphic bundles. The trick here is if I have an exit path, then I want a directed version of this, namely a map of representations, not necessarily an isomorphism, as when the path is in a $G$-space with trivial action as in Segal’s example.

So I suppose we take an exit path, leaving one stratum and entering another; in fact stratify $[0,1]$ by $\{\{0\},(0,1]\}$ and consider a stratified map $[0,1] \to X$. Then pull back a $G$-vector bundle to the interval. In fact, we should take the trivial $G$-space structure on $[0,1]$, and see if we can get a $G$-map, or something like it.

In fact, by taking a slice in $X$ at the point to which $0$ is sent, we can probably restrict to the case where $X$ is a ball about the origin in a linear action, and probably the case that the path is radial, since we only want to go up one stratum, most likely in the stratification by orbits.

So we have $G\to O(V)$ ($G$ compact) and a path $p\colon [0,1] \to V$, $t\mapsto t x$ for some fixed $x \in V$, and want to give (I think!!) for a $G$-vector bundle $E$ on $V$ a map of representations $E_0 \to E_x$.

Note that I can probably assume the action irreducible. And can I also assume $G\to O(V)$ is injective? Because otherwise I can take $G/H$ and then go trivially back up the quotient.

Then I have $E\times V$ for $E$ a complex vector space, write $E_x$ for $E\times\{x\}$, and so $E_0$ is a representation of $G$, and there are isomorphisms $E_x \stackrel{\simeq}{\to} E_{gx}$ for each $g\in G$ which are compatible with group multiplication. Note we have the canonical isomorphisms $E_0 \stackrel{\simeq}{\to} E_x$ as vector spaces, but these are in no way compatible with the group actions. Can I average or something? Note that I want compatibility with the inclusion $Stab(x) \hookrightarrow G$, so a map $E_0 \to E_x$ such that …. what? Perhaps I’m looking at the (co)unit of the $Res \dashv Ind$ adjunction? See also induced representation

---

---

24

Given a proper $G$-space $X$ there is a stratification of $X$ by orbit type, and this is such that a stratum $X_{(H)}$ is in the closure of a stratum $X_{(K)}$ iff $(K) \lt (H)$, where the relation $\lt$ on conjugacy classes $(H)$ of $G$ is such that there are representatives with one a subgroup of the other. (see the proof of theorem 1.30 in Meinrenken, Group actions on manifolds, Lecture Notes, University of Toronto, Spring 2003

Thus for $(I,\lhd)$ the partial order of the stratification (which for a compact group acting on a compact mfld, or a linear action of a compact group, is finite), there is a functor $(I,\lhd) \to (C_G,\lt)$ where $C_G$ is the poset of conjugacy classes of closed subgroups of $G$.

---

Constructible sheaves

(Reference: Section 4.1 of Alexandru Dimca, Sheaves in Topology)

The main result I’m interested in (corollary 4.1.8) is in fact only stated for stratifications by complex analytic subspaces, so I’m not sure it goes through. But here goes…

For any constructible sheaf $F$ on $X$, and any $x\in X$ there is an open nhd $U$ of $x$ such that $F|_U$ has a finite filtration

$$0=F_0 \subset F_1 \subset \ldots \subset F_m = F|_U$$

by constructible sheaves $F_k$ such that $F_k/F_{k-1} = i_! L$ for $i\colon S\cap U \hookrightarrow U$ the inclusion of the restriction of the stratum $S \subset X$ to $U$, and $L$ a locally constant sheaf on $S\cap U$.

---

Also note that for the inclusion of a stratum $i\colon S\hookrightarrow X$ and a locally constant sheaf $L$ on $S$, $i_! L$ is constructible on $X$. (Again, not sure if this works in the more general smooth setting I want)

---

---

21

Q: Given a 2-compact Lie groupoid, or in fact the canonical form of such given by actions on balls by compact Lie groups, is there a finite height hypercover of its nerve that is cofibrant in the projective local model structure on simplicial presheaves on manifolds? And how about other sorts of finiteness conditions? (e.g. finite coproduct in each degree)

For instance this includes deloopings $\mathbf{B}G$ for compact Lie groups $G$. But this, if I insist on finite coproducts in each degree bumps up against Christoph’s conjecture about multiplicative finite good open covers of $\mathbf{B}G$, which may not be true for all compact Lie groups.

---

The key step in the proof seems to be the divisibility properties of vector bundles under tensor product over appropriately finite-dimensional/compact spaces.

---

Q: Is there an analogue of Urs’ result from dcct section 3.6.4 (Compact objects) detailing compact objects as those whose corepresentable co-1-sheaf preserves monofiltered colimits, but for a (2,1)-topos of stacks of groupoids?

---

Has anyone thought about the presheaf of monoidal bicategories on $Aff$ sending $Spec(R)$ to $Alg(R)$ as described in Week 209?

---

H-spaces are simple/abelian hence nilpotent, so $U_\otimes$ is, and it is a rational homotopy type. What can we say about groupoids in simple/nilpotent rational homotopy types? Here I only care about $\mathbf{B}U_\otimes$.

---

---

20

In HTT Lurie defines an effective epimorphism in an $\infty$-topos as being $(-1)$-connected morphisms. These are the essentially those maps which induce epimorphisms on $\pi_0$-sheaves.

This is not the sort of thing I was hoping for, but in analogy with dcct section 3.6.4 I think there should probably be a tower of notions, relating to descent for presheaves representable by $n$-truncated objects. From effective+epi+in+an+(oo,1)-category we have

For $C$ an (∞,1)-topos, a morphism $f:X\to Y$ $f : X \to Y in C$ is effective epi precisely if the induced morphism on subobjects ((∞,1)-monos, they form actually a small set) by (∞,1)-pullback

Again this is reminiscent of Urs’ result about monofiltered colimits commuting with compact objects.

---

From Bass: Suppose X $=MaxSpec(R)$ is a Noetherian space of dim $\lt d$, $P\in \mathbf{P}(R)$ (projective modules over $R$) be faithful (i.e. $[P,-]$ is exact and faithful), let $n = lcm(local ranks)$. $\exists Q\in \mathbf{P}(R)$ such that $P\otimes_R Q \simeq R^{n^{d+1}}$

So there should be an analogue of this for vector bundles. The proof uses the fact ker(rank) is a nil-ideal of the K-theory and that stably isomorphic vector bundles of large enough rank are actually isomorphic (due to being able to split off trivial summands), and that virtual vector bundles of sufficiently large rank are isomorphic to actual vector bundles. Note that a faithful projective module is the same as one where the rank function $Spec(R) \to \mathbb{Z}$ is everywhere positive.

The proof in Bass is simple enough that it goes through for manifolds $X$ with finitely many components - recall that in this case we still have the split exact sequence

$$0 \to ker(rk) \to K^{VB}(X) \leftrightarrows H^0(X,\mathbb{Z}) \to 0$$

So now I just need the result that I can take, up to stabilising with $\otimes\underline{\mathbb{C}^n}$ $n^{th}$ roots of vector bundles.

---