at.algebraic topology – Why the symbol map in Atiyah-Singer paper is continuous?

I am reading Index of elliptic operators:I paper, by Atiyah and Singer these days and I am trying to understand all the paper. I find it difficult to verify the following statement on page 512 :

Let X be compact and E,F are vector bundles over X, then the symbol
$$ sigma:mathcal{P}_s^m(X;E,F) rightarrow text{Symb}^m(X;E,F)$$
is continuous for the sup norm topology on the unit sphere bundle of $T^*X$; it extends by continuity to a map $$sigma_s:overline{mathcal{P}_s^m}(X;E,F) rightarrow overline {text{Symb}^m}(X;E,F)$$ with kernel equal to compact operators $H_s rightarrow H_{s-m}$.

Notation. $mathcal{P}_s^m(X;E,F)$ is space of pseudo-differntial operators of order $m$ considered as a map $$H_{s}(X;E) rightarrow H_{s-m}(X;F) $$
and $overline{mathcal{P}_s^m}(X;E,F)$ is its completion under the operator norm topology.

I know how to prove that when $m=s=0$ and I know how to show that the mentioned kernel is composed of compact operators, I can show also that it contains all differential elliptic operators of order m-1 ( I wonder if the closure of the latter is exactly the compact operators).

I will be thankful for any help, It would be great if you give me a hint how to prove this when X is euclidean domain and $E,F$ are trivial line bundles.

at.algebraic topology – Rational homotopy groups of $S^2vee S^2$

From what I understand $pi_n(S^2vee S^2)otimesmathbb{Q}neq 0$ for $ngeq 2$. My question is:

Is there a “hands-on” proof of this fact using differential forms?

I am sure I will receive answers like: that is Hilton’s theorem or use Sullivan’s minimal model or check the section in Bott and Tu about the rational homotopy theory.

However, all these answers are useless for me because I am an analyst and not topologist and in order to use this fact in my research I need a straightforward construction that I could use to get integral estimates of forms.

By explicit I mean as explicit as the degree defined as the integral of the pullback of the volume form or the Hopf invariant as the Whitehead integral formula.

at.algebraic topology – Available frameworks for homotopy type theory

I am thinking about trying to formalise some parts of classical unstable homotopy theory in homotopy type theory, especially the EHP and Toda fibrations, and some related work of Gray, Anick and Cohen-Moore-Neisendorfer. I am encouraged by the successful formalisation of the Blakers-Massey and Freudenthal theorems; I would expect to make extensive use of similar techniques. I would also expect to use the James construction, which I believe has also been formalised. Some version of localisation with respect to a prime will also be needed.

My question here is as follows: what is the current status of the various different libraries for working with HoTT? If possible, I would prefer Lean over Coq, and Coq over Agda. I am aware of https://github.com/HoTT/HoTT, which seems moderately active. I am not clear whether that should be regarded as superseding all other attempts to do HoTT in Coq such as https://github.com/UniMath. I am also unclear about how the state of the art in Lean or Agda compares with Coq.

at.algebraic topology – Commutator length of the fundamental group of some grope

A popular way to describe a grope as the direct limit $L$ of a nested sequence of compact 2-dimensional polyhedra
$L_0 to L_1 to L_2 to cdots$
obtained as follows. Take $L_0$ as some $S_g$, an oriented compact surface of positive
genus $g$ from which an open disk has been deleted. To form $L_{n+1}$ from $L_n$, for each
loop $a$ in $L_n$ that generates the first homology group $H_1(L_n)$, attach to $L_n$ some $S_{g_a}$ by identifying
the boundary of $S_{g_a}$ with the loop $a$. Since the fundamental group of $S_g$ punctured
is a free group on $2g$ generators, this procedure embeds each $pi_1(L_n)$, and thus
each finitely generated subgroup of $pi_1(L)$, as a subgroup of a free group, with each
generator $a$ of $pi_1(L_n)$ becoming a product of $g_a$ commutators in $pi_1(L_{n+1})$. Hence
$pi_1(L)$ is a countable, perfect, locally free group. See What is a grope? for some motivation.

Here we build a grope $A$ in stages. At each stage put in disks with 3 handles on all the various handle curves at that stage. In other words, grope $A$ is built using disks with 3 handles uniformly throughout.

Question. Is it possible that there is a generating set $G$ of $pi_1(A)$ such
that each element of $G$ has commutator length bounded by 2?

at.algebraic topology – What is the top cohomology group of a non-compact, non-orientable manifold?

Let $M$ be a connected, non-compact, non-orientable topological manifold of dimension $n$.
Question: Is the top singular cohomology group $H^n(M,mathbb Z)$ zero?
This naïve question does not seem to be answered in the standard algebraic topology treatises, like those by Bredon, Dold, Hatcher, Massey, Spanier, tom Dieck, Switzer,…
Some remarks.
a) Since $H_n(M,mathbb Z)=0$ (Bredon, 7.12 corollary) we deduce by the universal coefficient theorem: $$ H^n(M,mathbb Z) =operatorname {Ext}(H_{n-1}(M,mathbb Z), mathbb Z)oplus operatorname {Hom} (H_n(M,mathbb Z),mathbb Z)=operatorname {Ext}(H_{n-1}(M,mathbb Z),mathbb Z )$$
But since $H_{n-1}(M,mathbb Z)$ need not be finitely generated I see no reason why $operatorname {Ext}(H_{n-1}(M,mathbb Z),mathbb Z)$ should be zero.
b) Of course the weaker statement $H^n(M,mathbb R) =0$ is true by the universal coefficient theorem, or by De Rham theory if $M$ admits of a differentiable structure.
c) This question was asked on this site more than 8 years ago but the accepted answer is unsubstanciated since it misquotes Bredon.
Indeed, Bredon states in (7.14, page 347) that $H^n(M,mathbb Z)neq0$ for $M$ compact, orientable or not, but says nothing in the non-compact case, contrary to what the answerer claims.

at.algebraic topology – Is there a homotopy coherent analogue of Dieudonne modules?

Let $K$ be a perfect field with characteristic $>0$ and $mathcal{H}$ the category of graded connected abelian hopf algebras over $K$.
By a theorem of Schoeller there is a canonical equivalence between $mathcal{H}$ and the category of
Dieudonne modules over $K.$

Is there a homotopy-coherent version of Dieudonne modules and a homotopy-coherent version of Schoeller’s theorem that identifies a derived version of Dieudonne module with a connected $E_infty$-hopf algebra over $K$?

By definition an $E_infty$-hopf algebra over $K$ is an abelian group object (in the derived sense) in the $infty$-category of $E_infty$-coalgebras in the derived $infty$-category of $K,$ which is connected if its zeroth homology is canonically $K.$

at.algebraic topology – Homology of a fiber as a cotorsion product

Let $K$ be a field. For any differentially graded coalgebra $A$ over $K$, any differentially graded right $A$-comodule $M$ over $K$ and any differentially graded left $A$-comodule $N$ over $K$ let
$mathrm{Cotor}_A(M,N)$ denote the cotorsion product of $M$ and $N$ relative to $A$.

The graded $K$-vector space $mathrm{Cotor}_A(M,N)$ is by definition the homology of the totalization of the cosimplicial cochain complex over $K$ with $n$-th term $M otimes A^{otimes n} otimes N$,
where the tensor product is in cochain complexes over $K.$

Let $X to Y$ be a Serre fibration between connected spaces and $F$ its fiber over a given point $y$ of $Y.$

If $Y$ is simply connected, by a theorem of Eilenberg and Moore there is a canonical isomorphism
begin{equation}
H_*(F;K)cong mathrm{Cotor}_{C_*(Y; K)}(C_*(X; K),C_*(*; K)), (**)
end{equation}

where $C_*(-;K)$ are singular chains with coefficients in the field $K.$

Can we replace the condition that $Y$ is simply connected by a weaker condition?

For example, is there still a canonical isomorphism $(**)$ if $Y = BG = K(G,1) $ for $G$ a derived p-complete abelian group?

at.algebraic topology – When do two topoi have the same cohomology of constant sheaves

Recently, I have some questions for some generalizations from algebraic topology.

I learn some homotopy theory in algebraic topology. We know that, if two spaces are homotopy, then they have same cohomology groups for constant sheaves.

I want to know if there are similar theory for topoi, such as homotopy theory for topoi?

Thanks for your answers

at.algebraic topology – Endofunctors of the surface category

Let $mathrm{Cob}_2$ be the symmetric monoidal $(infty,1)$-category whose objects are closed oriented $1$-manifolds and whose morphisms are compact oriented surface bordisms. (Higher morphisms are diffeomorphisms, isotopies, etc.)

Another question asked whether symmetric monoidal $(infty,1)$-functors $mathrm{Cob}_2 to mathcal{C}$ have been classified for general $mathcal{C}$.
The answer seems to be that we have some idea of what’s going on, but we don’t have a full classification.
I’m now wondering whether we know enough to say anything at all, when the target category is $mathrm{Cob}_2$ itself:

What is the $(infty,1)$-category of (strongly) symmetric monoidal endo-functors of this $(infty,1)$-category:
$$
mathrm{Fun}_{(infty,1)}^otimes(mathrm{Cob}_2, mathrm{Cob}_2) = ?
$$

I’m asking this because $mathrm{Cob}_2$ is the simplest bordism category that is not covered by the cobordism hypothesis.
(For the extended surface category $mathrm{Cob}_{langle 0,1,2rangle}$ (an $(infty,2)$-category) this question could be answered by invoking the cobordism hypothesis.)
So this seems like a good way of testing how well we understand $infty$-categories of bordisms beyond the omnipresent cobordism hypothesis.


There are some things one can show about this category of functors by elementary means.
For example, because $mathrm{Cob}_2$ has duals, every natural transformations between symmetric monoidal functors is invertible. Hence the functor category is actually an $infty$-groupoid and one could equivalently ask: what is its homotopy type?

One can also construct a few examples. Let $A$ be compact oriented $0$-manifold (= a finite set $A$ with a map $A to {pm 1}$). Then we can construct a functor:
$$
mathcal{Z}_A: mathrm{Cob}_2 longrightarrow mathrm{Cob}_2, quad
M mapsto M times A, quad (W:M to N) mapsto (W times A:M times A to N times A).
$$

This defines a functor from the groupoid of compact oriented $0$-manifolds to the functor category I’m interested in.
One can also almost define a splitting for this, by sending an arbitrary $mathcal{Z}:mathrm{Cob}_2 to mathrm{Cob}_2$ to $pi_0(mathcal{Z}(S^1))$, though this does not recover the orientation.
(Fun fact: when looking at the homotopy category the functor $hmathrm{Cob}_2 to hmathrm{Cob}_2$
that reverses the orientation is isomorphic to the identity functor via a unique natural isomorphism.)

I would suspect that every endofunctor is of the form $mathcal{Z}_A$ and that we have an equivalence:
$$
mathrm{Fin}_{/{pm1}}^{cong} xrightarrow{ simeq }
mathrm{Fun}_{(infty,1)}^otimes(mathrm{Cob}_2, mathrm{Cob}_2),
qquad A mapsto mathcal{Z}_A.
$$

However, there seem to be no tools available to actually prove this.

at.algebraic topology – $pi_{2n-1}(operatorname{SO}(2n))$ element represents the tangent bundle $TS^{2n}$, not torsion and indivisible?

Question: Is the element $alpha$ in $pi_{2n-1}(operatorname{SO}(2n))$ representing the tangent bundle $TS^{2n}$ of the sphere $S^{2n}$ indivisible and not torsion?

My understanding so far —

An $operatorname{SO}(2n)$ bundle over $S^{2n}$ corresponds to an element in $pi_{2n}operatorname{BSO}(2n) =pi_{2n-1}operatorname{SO}(2n)$.

Not torsion: There does not exist any integer $m > 0$ such that $malpha$ is a trivial element.

Indivisible: There does not exist any integer $k > 1$ and any element $beta$ in $pi_{2n-1}operatorname{SO}(2n)$ such that $alpha=kbeta$.

Ref: Mimura, Toda: Topology of Lie groups. Chapter IV Corollary 6.14.