David Roberts scratch 2014

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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)R(G)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 rr 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 Π 1\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 It is unpublished, but referred to in Treumann, Exit paths and constructible stacks.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 SS of a space XX, there should be a poset 𝒮\mathcal{S} of strata (for example II in entry dated Feb 24, arising from a GG-action), and a functor EP(X,S)𝒮EP(X,S) \to \mathcal{S} which is the posetal reflection.

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

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

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

where SS might be a finer stratification arising from the GG-action.

What I want to do is go from a global section of this constructible sheaf to a GG-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 GG-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 GG-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][0,1] by {{0},(0,1]}\{\{0\},(0,1]\} and consider a stratified map [0,1]X[0,1] \to X. Then pull back a GG-vector bundle to the interval. In fact, we should take the trivial GG-space structure on [0,1][0,1], and see if we can get a GG-map, or something like it.

In fact, by taking a slice in XX at the point to which 00 is sent, we can probably restrict to the case where XX 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 GO(V)G\to O(V) (GG compact) and a path p:[0,1]Vp\colon [0,1] \to V, ttxt\mapsto t x for some fixed xVx \in V, and want to give (I think!!) for a GG-vector bundle EE on VV a map of representations E 0E xE_0 \to E_x.

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

Then I have E×VE\times V for EE a complex vector space, write E xE_x for E×{x}E\times\{x\}, and so E 0E_0 is a representation of GG, and there are isomorphisms E xE gxE_x \stackrel{\simeq}{\to} E_{gx} for each gGg\in G which are compatible with group multiplication. Note we have the canonical isomorphisms E 0E xE_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)GStab(x) \hookrightarrow G, so a map E 0E xE_0 \to E_x such that …. what? Perhaps I’m looking at the (co)unit of the ResIndRes \dashv Ind adjunction? See also induced representation



24

Given a proper GG-space XX there is a stratification of XX by orbit type, and this is such that a stratum X (H)X_{(H)} is in the closure of a stratum X (K)X_{(K)} iff (K)<(H)(K) \lt (H), where the relation <\lt on conjugacy classes (H)(H) of GG 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,)(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,)(C G,<)(I,\lhd) \to (C_G,\lt) where C GC_G is the poset of conjugacy classes of closed subgroups of GG.


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 FF on XX, and any xXx\in X there is an open nhd UU of xx such that F| UF|_U has a finite filtration

0=F 0F 1F m=F| U 0=F_0 \subset F_1 \subset \ldots \subset F_m = F|_U

by constructible sheaves F kF_k such that F k/F k1=i !LF_k/F_{k-1} = i_! L for i:SUUi\colon S\cap U \hookrightarrow U the inclusion of the restriction of the stratum SXS \subset X to UU, and LL a locally constant sheaf on SUS\cap U.


Also note that for the inclusion of a stratum i:SXi\colon S\hookrightarrow X and a locally constant sheaf LL on SS, i !Li_! L is constructible on XX. (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 BG\mathbf{B}G for compact Lie groups GG. But this, if I insist on finite coproducts in each degree bumps up against Christoph’s conjecture about multiplicative finite good open covers of BG\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 AffAff sending Spec(R)Spec(R) to Alg(R)Alg(R) as described in Week 209?


H-spaces are simple/abelian hence nilpotent, so U 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 BU \mathbf{B}U_\otimes.



20

In HTT Lurie defines an effective epimorphism in an \infty-topos as being (1)(-1)-connected morphisms. These are the essentially those maps which induce epimorphisms on π 0\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 nn-truncated objects. From effective+epi+in+an+(oo,1)-category we have

For CC an (∞,1)-topos, a morphism f:XYf:X\to Y f:XYinCf : 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)=MaxSpec(R) is a Noetherian space of dim <d\lt d, PP(R)P\in \mathbf{P}(R) (projective modules over RR) be faithful (i.e. [P,][P,-] is exact and faithful), let n=lcm(localranks)n = lcm(local ranks). QP(R)\exists Q\in \mathbf{P}(R) such that P RQR n d+1P\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)Spec(R) \to \mathbb{Z} is everywhere positive.

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

0ker(rk)K VB(X)H 0(X,)0 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 n̲\otimes\underline{\mathbb{C}^n} n thn^{th} roots of vector bundles.


Last revised on August 17, 2016 at 01:50:20. See the history of this page for a list of all contributions to it.