## 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 – \$mathbb Z\$-formality of spheres

A topological space $$X$$ is $$mathbb Z$$-formal, if the singular cochain complex $$C^*(X,mathbb Z)$$ is
quasi-isomorphic to $$H^*(X, mathbb Z)$$ as an augmented differential graded ring.

It’s quite simple to write down specific quasi-isomorphisms to show that the Spheres $$S^n$$ are $$mathbb Q$$-formal spaces by fixing a volume form $$v in Omega^n(S^n)$$ and considering the maps $$H^*(S^n)=operatorname{span}(1,(v)) to Omega^*(S^n)$$ sending $$1$$ to the $$1$$-form and $$(v)$$ to $$v$$
and the canonical map $$C^*(S^n) to Omega^*$$.

Is it also possible to show the $$mathbb Z$$-formality of the Spheres $$S^n$$ by writing down specific quasi-isomorphisms, or is it easier to use another method for showing $$mathbb Z$$-formality?

## at.algebraic topology – Why does \$iota_4^2 in H^8(K(mathbb Z/2,4);mathbb Z/2)\$ not come from \$H^8(K(mathbb Z/2,4);mathbb Z)\$?

In Hatcher’s Chapter 5 (https://pi.math.cornell.edu/~hatcher/AT/ATch5.pdf) on page 574 (page 57 in the pdf), he states that $$iota_4^2 in H^8(K(mathbb Z/2,4);mathbb Z/2)$$ is not in the image of $$H^8(K(mathbb Z/2,4);mathbb Z)to H^8(K(mathbb Z/2,4);mathbb Z/2)$$. The argument for the lower classes ($$iota_4, Sq^2iota_4, Sq^2Sq^1iota_4$$) which don’t come from $$H^*(K(mathbb Z/2,4);mathbb Z)$$ is that the Bockstein homomorphism $$Sq^1colon H^*(K(mathbb Z/2,4);mathbb Z/2) to H^{*+1}(K(mathbb Z/2,4);mathbb Z/2)$$ applied to those classes is nonzero, so it can’t come from $$H^*(K(mathbb Z/2,4);mathbb Z)$$. This argument doesn’t work for $$iota_4^2 = Sq^4 iota_4$$ because $$Sq^1Sq^4 iota_4 = Sq^5 iota_4 = 0$$.

How can one argue that $$iota_4^2 in H^8(K(mathbb Z/2,4);mathbb Z/2)$$ doesn’t come from $$H^8(K(mathbb Z/2,4);mathbb Z)$$? Is it even true?

## at.algebraic topology – The homotopy class of the loop \$(S^1,ast)to (Omega^2(S^2),id_{S^2})\$ that rotates the sphere

Let $$Omega^2(S^2,x)$$ be the set of basepoint-preserving maps $$S^2to S^2$$ (with basepoint $$x=(1,0,0)$$). Instead of taking the constant map $$S^2to S^2$$ as the basepoint of $$Omega^2(S^2)$$, let’s take the identity map $$id_{S^2}$$ to be the basepoint.

Certainly, it is the case that $$pi_3(S^2,x)cong pi_1(Omega^2(S^2),id_{S^2})$$ since loop spaces have homotopy equivalent path components. Also, if $$X$$ is the path component of $$id_{S^2}$$, in $$Omega^2(S^2)$$, then $$pi_1(Omega^2(S^2),id_{S^2})cong H_1(X)$$ by the Hurewicz map. Hence, $$pi_1(Omega^2(S^2),id_{S^2})=H_1(X)$$ is infinite cyclic.

Let $$f:S^1to Omega^2(S^2)$$ be the loop based at $$id_{S^2}$$, which rotates $$S^2$$ once around the $$x$$-axis. This map is a low-dimensional case of the $$J$$-homomorphism and its homotopy class apparently generates $$pi_1(Omega^2(S^2),id_{S^2})$$.

The map $$f$$ itself is very simple and highly geometric but the connection to the Hopf map gets a bit muddled. What is a direct, “elementary” proof of why the homotopy class of $$f$$ generates $$pi_1(Omega^2(S^2),id_{S^2})$$? By “elementary,” I mean some argument using elementary tools from homology theory, geometry, or homotopy theory that perhaps connects back to the Hopf map (or not) and could be understood by someone who does not know general results characterizing the image of the $$J$$-homomorphism.

## at.algebraic topology – Universal cover of finetely connected surface with boundary

Let $$M$$ be a finetely connected orientable surface with compact boundary. This means $$M$$ is homeomorphic to a compact orientable surface $$Sigma$$ of genus $$g geq 0$$ minus $$r geq 1$$ points and minus $$k geq 1$$ open discs.

Is there a detailed description of the universal cover of $$M$$? What if $$k = 1$$?

## at.algebraic topology – Quasifibrations and transfinite filtrations

This question takes place in the category $$mathrm{CGWH}$$
of compactly generated weak Hausdorff spaces.

Let $$lambda$$ be a limit ordinal, and suppose we have
a diagram $$Phi: lambda to mathrm{CGWH}$$, as indicated
$$X_0 hookrightarrow X_1 hookrightarrow cdots X_xi hookrightarrow X_{xi+1} hookrightarrow cdots .$$
We’ll assume the inclusion maps are as nice as could be reasonably hoped for: they are all obtained by pushouts from closed cofibrations. Let’s also assume that if $$xi< lambda$$ is a limit ordinal, then $$X_xi = mathrm{colim}, Phi|_xi$$. Write $$Y = mathrm{colim}, Phi$$.

Now suppose we have a map $$p: Eto Y$$, and we hope to prove that it is a quasifibration. If $$lambda = omega$$, then the diagram $$Phi$$ can be taken to be $$mathbb{N}$$-indexed, and there is a “classical” theorem with various technical conditions, whose heuristic import is that if all of the pullback maps $$p_n : E_n to X_n$$ are quasifibrations, the so is $$p$$ (see, for example Theorem 2.7 in Peter May’s paper “Weak equivalences and quasifibrations”, available at https://www.math.uchicago.edu/~may/PAPERS/67.pdf).

Can this be extended to the more general ordinal-indexed case, possibly at the expense of imposing some additional conditions?

## at.algebraic topology – Persistent homotopy groups

Everybody in algebraic topology loves homology and cohomology, but sometimes we like homotopy groups also, since they detect different things (think about spheres) .

An interesting and recent application of topology is topological data analysis, when one is given a filtration of topological spaces $${X_r}_{r in (0,1)}$$ with a finite number of “critical values”. A number $$t in (0,1)$$ is called critical if for every $$epsilon >0$$ the map $$X_{r-epsilon} to X_{r+epsilon}$$ is not a (weak) homotopy equivalence.

There is a cool theorem that classifies the possible filtrations of abelian groups $${H_k(X_r) }$$ that can arise if $$X_r$$ have finite dimensional homology in each degree. Morally it is a superposition of “bars”: there is a basis in which each generator born at some time and dies at some other time. Correct me if I am wrong in the generality of this theorem.

I was thinking if it is possible to define persistent homotopy groups and to generalize this classification theorem. Persistent homotopy groups are defined exactly the same way. In the case $$pi_k, kge 2$$, I guess the theorem still holds, but this is a question: cam $${pi_k(X_r)}$$ be decomposed in a finite number of bars?

In the $$k=1$$ case, maybe there is a presentation in terms of resolutions with finite free groups where each generator/relation/relation among relations born at some time and dies at another. Here I always assume finite number of critical values.

Bonus: is there a “persistent homotopy diagram” in which one can nicely represent a $${pi_1(x_r) }$$ persistent group?

Bonus 2: if the theory goes through without great changes, why homotopy is not that used? Is it harder to compute?

Edit: to start with something simpler, you can substitute abelian groups with vector spaces over Q, that is to ignore torsion at the moment.

## 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 – Exotic analtytic triangulations of \$S^5\$?

I would like to understand a bit better the nature of bad triangulations of $$S^5$$, discussed in two Lectures of Lurie

https://www.math.ias.edu/~lurie/937notes/937Lecture2.pdf

http://www-math.mit.edu/~lurie/937notes/937Lecture3.pdf

The example in the lectures is based on the fact that the double suspension of the Poincare 3-sphere is homeomorphic to $$S^5$$.

I would like to fiddle a bit with Definition 1 from lecture 3. Here is the definition.

Definition 1. Let $$K$$ be a polyhedron and M a smooth manifold. We say that a map $$f : K to M$$ piecewise differentiable (PD) if there exists a triangulation of $$K$$ such that the restriction of $$f$$ to each simplex is smooth. We will say that $$f$$ is a PD homeomorphism if $$f$$ is piecewise differentiable, a homeomorphism, and the restriction of $$f$$ to each simplex has injective differential at each point.

In the situation that I want to consider, I would like to impose the condition instead that $$f$$ restricts analytically to each simplex (stronger condition). But only ask that the restriction has injective differential in the interior of each simplex (weaker condition). So here is a question:

Question. Suppose $$X$$ is a simplicial complex that is homemorphic to $$S^5$$. Suppose that the homeomoprhism $$varphi: Xto S^5$$ can be realised so that its restriction to the interior of each simplex in $$X$$ is an analytic diffeomorphism onto its image. Is it true then that $$X$$ is $$PL$$ homeomorphic to the standard sphere? (This would mean that there is a map $$varphi’: Xto S^5$$ that is $$PL$$ on each simplex of $$X$$ for a standard $$PL$$ structure on $$S^5$$.)

## at.algebraic topology – Cohomology of BG, G non-connected Lie group, and spectral sequence relating to classifying space of connected component of the identity

Suppose $$G$$ is a Lie group, with $$pi_0(G)$$ not necessarily finite, but might as well assume $$G_0$$, the connected component of the identity, is compact.

In the case that $$pi_0(G)$$ is finite, then we know that there is an injection $$H^*(BG,mathbb{Q})to H^*(BG_0,mathbb{Q})$$, and this can apparently be seen via a spectral sequence argument, using the fact that the rational cohomology of $$Bpi_0(G)$$ is concentrated in degree zero. So this is some kind of Leray–Serre spectral sequence argument on either $$pi_0(G)to BG_0to BG$$ or $$BG_0to BGto Bpi_0(G)$$ (and I suspect the latter), probably using the degeneration and some kind of “edge homomorphism is injective” argument.

I suspect that in the case that we know something strong about the rational cohomology of $$Bpi_0(G)$$, then we might be able to say something in the case where $$pi_0(G)$$ is not finite.

Unfortunately my spectral sequence knowledge is limited, and I can’t find a treatment of spectral sequences that seems general enough to deal with this setup in general (namely non-simply-connected base, and possibly non-connected fibre, plus non-finiteness issues, depending on which fibration is used).

Is my intuition correct, that $$H^*(Bpi_0(G),mathbb{Q}) = H^0(Bpi_0(G),mathbb{Q})$$ can let us conclude something about how the cohomology of $$BG$$ relates to that of $$BG_0$$?

Also, what would be a good reference that covers a general-enough version of the relevant spectral sequence?