started fiber integration

]]>I have tried to give *algebraic topology* a better Idea-section.

A student asked “What is a cobordism?” and I checked and realized that the $n$Lab entry *cobordism* was effectively empty.

So I have now added some basic text in the Idea-section and added a bare minimum of references. Much more should be done of course, but at least now there are pointers.

]]>I’ve been thinking about generalizing the Cech-Delign double complex to the case where $U(1)$ is replaced by some Lie group, $G$, and $\mathbb{R}$ is replaced with $\mathfrak{g}$. I came across this post on Nonabelian Weak Deligne Hypercohomology by Urs a while back and was wondering if his musing was ever fully considered/resolved?

For full context of why I’m considering this: I was working on a project during my PhD that I’d like to eventually publish, but I constructed an element in a (Hochschild-like) curved dga with a Chen map to holonomy (path or surface) which is a chain map and map of algebras. It was suggested that this is not enough of a “result” unless I could find the right notion of equivalence to fully flesh out this map. I was hoping that some resolution of the referenced article above could allow me to put my work in that context.

P.S. This is my first post here so my apologies if I made a cultural error.

]]>Started to write up a homotopy-theoretic version of James construction following ideas of Brunerie’s IAS talk at filtered topological space

]]>brief entry *model structure on semi-simplicial sets*, just in order to record a recent note by Benno van den Berg.

We know locales are “better” than spaces, and that groupoids are “better” than groups (Edited to remove incorrect statement – see Mike Shulman’s comment below). I wanted to find a way of defining the fundamental groupoid that was natural from this point of view.

I’ve worked this out, and proved the Seifert-van Kampen theorem. Surprisingly, it all seems to work for lattices (with the exception of the preservation of products through the fundamental groupoid functor, which seems to only work with locales)!

I’m not in academia and I’m new to algebraic topology and category theory, so I would really appreciate feedback, and have any errors pointed out. Also, if this is interesting to anyone I would love to hear it! I have lots more ideas for extending this stuff that I haven’t worked out yet.

I’m going to use the language of locales, since that makes things more intuitive – but I will not use the distributivity, or existence of infinite joins.

Let L be a locale. A **cover** of L is a finite set of opens such that the join is $\top$, and every meet in the cover is the join of opens in the cover.

Covers form a category, where the morphisms are functions taking each open to an open containing it. I call these **clumpings**.

Opens from a cover C can overlap. In particular, an n-**overlap** is an open from C in the meet of a (multi-)set of opens from C, where the set has cardinality n.
Note: every open in C which is contained in the meet is a *distinct* n-overlap.

A **skeleton** of a cover is a groupoid constructed in the following way:

- Every 1-overlap of opens in the cover generates an element in the groupoid.
- Every 2-overlap generates an edge (i.e. isomorphism) in the groupoid. This edge goes between the elements generated from the corresponding 1-overlaps we get if we take out 1 of the opens.
- Every 3-overlap generates a composition relation between the three edges we get by looking at the edges generated from the corresponding 2-overlaps we get if we take out 1 of the opens.

For a cover C, we call this groupoid $Sk_C$.

Sk is a functor.

Finally, we make a diagram in the category of groupoids from all of the covers of L, along with their morphisms, using Sk. Then, we take the limit of this diagram, which is the **fundamental groupoid** of L. We call this $\pi L$.

$\pi$ is also a functor.

And of course, this functor preserves pushouts! I’m not going to go into the full proof, but the key idea is to use the first isomorphism theorem for frames/lattices (which we get because these are varietal).

A good thing to try all this stuff out on is the pseudocircle – of course we get a groupoid equivalent to $\mathbb{Z}$!

It’s kinda weird that we just stopped with 3-overlaps – the really cool part is that it seems to all still work if we just go to n! The main thing I’ve had trouble with in actually working this out is just understanding composition in n-groupoids enough to make sure everything is all good. My guess is that a Kan complex is the easiest kind of n-groupoid to do this for – but I am open to suggestions!

Anyway I hope this all at least approximately right! I have a paper with the proofs (and pictures!) that I can email to you if you’d like.

]]>I have put a minimum remark on the framed bordism ring, its relation to the stable stems, literature and basic examples into *Homotopy groups of spheres – Relation to framed bordism* and (the same) into *cobordism ring – Examples – Framed cobordism*.

I have started editing at *Thom’s theorem*. So far it has just the definition of the bordism ring, the statement of the theorem and some literature.

added statement and proof of the (or one version of the) Serre long exact sequence of a Serre fibration with highly connected base and fibers.

]]>I have split off an entry *classical model structure on simplicial sets* from “model structure on simplicial set”. This entry should eventually contain detailed, self-contained and polished discussion of the definition, verification and key properties of the standard Kan-Quillen model structure.

So far I have inserted fair bit of background material regarding (minimal) fibrations and geometric realizations, essentially the material in chapter 1 of Goerss-Jardine. A bunch of little proofs are spelled out, but not yet the more laborious ones. Discussion of the verification of the axioms is not yet in the entry, but the key parts of the Quillen equivalence to $Top_{Quillen}$ are (modulo relying on previous lemmas that don’t have proofs spelled out yet).

The somewhat random list of properties of $sSet_{Quillen}$ that used to be sitting at “model structure on simplicial sets” I have copied over to a section “Basic properties”, just for completenes, but this now needs re-organization to give decent logical flow.

For the moment I have to leave it at that, need to take care of something else now for a little bit.

]]>created an entry *Leray-Hirsch theorem*, so far just with the bare statement

A homotopy quantum field theory is a symmetric monoidal functor $\begin{multline} \begin{matrix} \mathrm{Cobord}(n) &\rightleftarrows\space \\tau:\mathrm{HCobord}(n,X)\to\mathrm{Vect}_{\mathbb{K}}$, with $X$ a path connected space with basepoint $\ast$. There is the following diagram (see page 15 of this ArXiv article):

A homotopy QFT restricted to domain $\pi_1\mathrm{Map}(S^n,X)$ is a contravariant functor $\pi_1\mathrm{Map}(S^n,X)\to\mathrm{Vect}_{\mathbb{K}}\hookrightarrow\mathrm{Set}$, which is a locally constant sheaf on $\pi_1\mathrm{Map}(S^n,X)$, i.e., a sheaf of sections of a covering space of $\pi_1\mathrm{Map}(S^n,X)$. Let us denote by $\mathrm{LSheaf}\pi_1\mathrm{Map}(S^n,X)$ the category of locally constant sheaves on $\pi_1\mathrm{Map}(S^n,X)$. Then, $\mathrm{LSheaf}\pi_1\mathrm{Map}(S^n,X)\hookrightarrow \mathrm{Sh}\pi_1\mathrm{Map}(S^n,X)\hookrightarrow \mathrm{PSh}\pi_1\mathrm{Map}(S^n,X)$. We know that $\mathrm{LSheaf}\pi_1\mathrm{Map}(S^n,X)\hookrightarrow \mathrm{HQFT}(n,X)$, and hence, my first question is:

Is the following statement true: $\mathrm{HQFT}(n,X)\hookrightarrow \mathrm{PSh}\pi_1\mathrm{HCobord}(n,X)$? Or is it $\mathrm{HQFT}(n,X)\hookleftarrow \mathrm{PSh}\pi_1\mathrm{HCobord}(n,X)$ (which I think is unlikely, and most probably wrong)?

We know that loops on $\mathrm{Map}(S^n,X)$ are $X$-cobordisms, since $\pi_1\mathrm{Map}(S^n,X)\hookrightarrow\mathrm{HCobord}(n,X)$.

In the following article, it is written that a string connection assigns to a loop a vector space and assigns to cobordisms between loops, linear transformations. Hence, my second and third questions are:

]]>Does it follow that string connections defined on the associated vector bundle over $\pi_1\mathrm{Map}(S^n,X)$ (what

isthe associated vector bundle over $\pi_1\mathrm{Map}(S^n,X)$? I haven’t been able to find any literature on it - so that would be helpful.) as defined in the article above are all homotopy quantum field theories restricted to domain $\pi_1\mathrm{Map}(S^n,X)$?What exactly would it mean for a string connection to be a HQFT?

Here is a question, probably for Charles (Rezk), if he sees it.

The string orientation of tmf $M String \to tmf$ refines the un-twisted Witten genus on manifolds with String-structure. More generally, the Witten genus on a Spin-manifold is twisted by a complex vector bundle (“heterotic string”) and is a modular form for “String^c-structure”, characterized by $\frac{1}{2}p_1 - c_2 = 0$.

In the article

- Qingtao Chen, Fei Han, Weiping Zhang, “Generalized Witten Genus and Vanishing Theorems” (arXiv:1003.2325)

this twisted Witten genus on $String^c$-structures is re-considered, and on p.2 an obvious question is mentioned: does the twisted Witten genus also have a “topological” lift to a map of spectra?

Now, there is indeed an obious map of spectra $M String^c \longrightarrow tmf$, namely the hocolim over the left half of the diagram

$\array{ && B String^c \\ & \swarrow && \searrow^{\mathrlap{p}} \\ \ast & \mathrlap{\swArrow_{\sigma^c}} & \downarrow^{\mathrlap{p^\ast(\frac{1}{2}p_1 - c2)}} & \mathrlap{B Spin \times B SU} \\ & \searrow && \swarrow_{\frac{1}{2}p_1 - c_2} \\ && B^3 U(1) \\ && \downarrow^{\mathrlap{\rho}} \\ && B GL_1(tmf) \\ && \downarrow \\ && tmf Mod } \,,$where $\sigma^c$ denotes the homotopy that exhibits $B String^c$ as the homotopy fiber of $\frac{1}{2}p_1 - c_2$, and where $\rho$ is the twisting map exhibiting the plain string orientation of $tmf$ as in Ando-Blumberg-Gepner 10.

My exercise is to check if that map

$\underset{\to}{\lim} \sigma^c : M String^c \to tmf$induces on homotopy groups the twisted Witten genus, correctly.

While I am slowly chewing on this, I thought I’d ask if anyone has considered this before. Quite likely this is clear to experts such as Charles.

Or rather, possibly the push should be rather along the left half of

$\array{ && B String^c \simeq B String // SU \\ & \swarrow && \searrow^{\mathrlap{p_{Spin}}} \\ (B SU \simeq \ast//SU )& \mathrlap{\swArrow_{\sigma^c}} & \downarrow^{\mathrlap{p_{Spin}^\ast(\frac{1}{2}p_1)}} & \mathrlap{B Spin } \\ & {}_{\mathllap{c_2}}\searrow && \swarrow_{\frac{1}{2}p_1} \\ && B^3 U(1) \\ && \downarrow^{\mathrlap{\rho}} \\ && B GL_1(tmf) \\ && \downarrow \\ && tmf Mod }$and hence land in $SU$-equivariant elliptic cohomology.

( Here I am using the pasting of homotopy pullbacks

$\array{ B String &\to& B String^c &\to& B Spin \\ \downarrow && \downarrow && \downarrow \\ \ast &\to& B SU &\to& B^3 U(1) }$in order to identify $B String^c \simeq B String // SU$. )

I suppose that the twisted Witten genus should land in equivariant tmf this way is something that Matthew Ando has been suggesting, though I am not sure if I have seen the place where this is stated explicitly.

]]>gave *Adams conjecture* an entry of its own

added rough description and original citation to *Adams e-invariant*

gave *Lagrangian cobordism* an Idea-section added references related to the Fukaya category and cross-linked with relevant entries.

created *red-shift conjecture*