# Problems in homotopy theory Unstable homotopy theory

• (J. Neisendorfer) Bott periodicity says that the homotopy groups of the infinite dimensional unitary group $U$ are periodic. The filtration of $U$ by the finite dimensional unitary groups $U\left(n\right)$ and the fibration sequences $U\left(n-1\right)\to U\left(n\right)\to {S}^{2n-1}$ yield a spectral sequence converging to the very well understood homotopy of $U$ with ${E}^{1}$ term given by the homotopy groups of the odd dimensional spheres. Find some explanation for this or, at least, draw some corollaries of this. There must be a lot of cancellation in this spectral sequence. Is any of it systematic in some as yet unknown way?

• (J. Neisendorfer) The previous problem asked to explain the differentials in the spectral sequence which begins with the homotopy groups of the odd dimensional spheres and abuts to the homotopy of the infinite unitary group $U$. This problem is related to complex Bott periodicity. Of course, there is an analogous question related to the real Bott periodicity of the infinite orthogonal group $O$.

• (J. Neisendorfer) Given a space $X$ localized at a prime $p$ with mod $p$ homology groups $H$, when is there a homotopy commutative H-space $Y$ and a map $X\to Y$ such that the mod $p$ homology of $Y$ is the free commutative algebra generated by the vector space $H$? The best work in this direction, dealing with the case $X$ = a Moore space, was begun by Anick and is continued by Theriault and Gray.

• (J. Neisendorfer) In the case of $p$ primary Moore spaces, which we will denote by ${P}^{n}\left({p}^{r}\right)$ for the space with one nontrivial integral cohomology group isomorphic to $ℤ/{p}^{r}ℤ$ in dimension $n$, the existence and properties of multiplications, such as Samelson products, in mod ${p}^{r}$ homotopy theory depend on the existence of maps ${P}^{n+m}\left({p}^{r}\right)\to {P}^{n}\left({p}^{r}\right)\wedge {P}^{m}\left({p}^{r}\right)$ which lead to homotopy decompositions of the smash ${P}^{n}\left({p}^{r}\right)\wedge {P}^{m}\left({p}^{r}\right)$ into a bouquet ${P}^{n+m}\left({p}^{r}\right)\vee {P}^{n+m-1}\left({p}^{r}\right)$.

At odd primes, these maps always exist and are homotopy commutative up to sign. At primes greater than 3 or if $p=3$ and $r$ is greater than 1, then these maps are also homotopy associative. If $p=3$ and $r=1$, then they are not homotopy associative. (In low dimensions the homotopy commutativity and homotopy associativity are only mod Whitehead products. But in all applications, the homotopy theory is of maps into loop spaces. Hence the Whitehead products all vanish.)

If $p=2$ and $r=1$, the maps cannot even exist but, if $p=2$ and $r$ is greater than 1, they do exist. The question is: for what values of $r$, if any, do these maps become homotopy commutative up to sign and homotopy associative. Progress on these questions would be of interest to those who study algebraic K-theory mod powers of 2. There is a good chance that these questions are answerable with diligent effort.

• (B. Gray) The question of whether ${W}_{n}$, the fiber of the double suspension, is a double loop space localized away from 2 is open. It has a lot of implications.

• (B. Gray) ${S}^{3}$ has a ${v}_{1}$ periodic self map. What is the order of the ${v}_{1}$ torsion? Conjecture: for $p>3$ every ${v}_{1}$ torsion class has order ${v}_{1}$.

• (M. Mandell) Structures on cochain complexes of spaces. Since the operation ${\mathrm{Sq}}^{0}$ never vanishes on the cohomology of spaces, the cochain complexes ${C}^{*}\left(X;{𝔽}_{p}\right)$ can’t ever have “formality” properties like the corresponding ones in rational homotopy theory. However, one could forget information and just remember the cochain-level structure as an ${E}_{n}$-algebra, and then one does obtain a number of examples (such as $n$-fold suspensions). Are there interesting things we can say about homotopy theory with this weaker structure? What constructions are preserved, what can one calculate, etc? (There are slides up that say more.)

• (W. S. Wilson) The Johnson-Yosimura question on unstable $\mathrm{BP}$-homology. Given a space $X$ and an element $x\in {\mathrm{BP}}_{n}\left(X\right)$, show that $x$ is ${v}_{n}$-torsion free.

• (M. Bendersky) What is the general version of Cohen-Moore-Neisendorfer? Specifically, they give the bound on the $p$-torsion in the homotopy of a finite sphere. What is the largest possible order of v_1 torsion on the mod $p$ Moore space, with obvious generalizations to the ${v}_{n}$ torsion on the sphere mod $\left({p}^{k},{v}_{1}^{s},\dots \right)$? (By homotopy I mean in the unstable novikov spectral sequence.)

These numbers show up in the unstable chromatic spectral sequence.

• (M. Bendersky) A better understanding of the chromatic Hopf invariant.

Here are examples of what I mean by this.

At the prime 2 $\eta$ desuspends to ${S}^{3}$ as a ${v}_{1}$ periodic class. But $\eta$ further desuspends to ${S}^{2}$, but as a ${v}_{0}$ class.

At $p=3$ ${\beta }_{1}$ desuspends to ${S}^{9}$ as a ${v}_{2}$ periodic class, but goes back to ${S}^{5}$ as a v_1 periodic class (I may be off on the spheres of origin). The obstruction to desuspending with given periodicity is what I call the chromatic Hopf invariant. It is a natural construction in the unstable chromatic spectral sequence. It is understood for ${v}_{0}$, ${v}_{1}$ and ${v}_{2}$ (again on the level of the novikov ${E}_{2}$). Very little is known (to me at least) for higher periodicity - (something is known of the greek letter elements).

• (B. Gray) The Barratt Conjecture: Suppose that $X$ is a double suspension and the identity map of $X$ has order ${p}^{r}$. Then every element of the homotopy of X has order dividing ${p}^{\left(}r+1\right)$.

Fred Cohen has worked on this and at times claimed he was close to proving that it is true; to the best of my knowledge he has stopped working on it.

• (B. Gray) Determine the maximum order of the 2-torsion in the homotopy of ${S}^{2n+1}$.

• (B. Gray) Let ${W}_{n}$ be the fiber of the double suspension:

${W}_{n}\to {S}^{2n-1}\to \Omega {S}^{2n+1};$W_n \to S^{2n-1} \to \Omega S^{2n+1};

Question 1: ${W}_{n}$ is known to be homotopy equivalent to the loops on a homotopy commutative homotopy associative H-space localized at $p>2$. Is ${W}_{n}$ a double loop space?

Any H-space has the property that there is a self map which induces multiplication by $p={v}_{0}$ in homotopy. There is a self map:

$v\left(1\right):{W}_{n}\to {\Omega }^{2p-2}{W}_{n}$v(1): W_n \to \Omega^{2p-2} W_n

which induces multiplication by ${v}_{1}$ in ${v}_{1}$ periodic homotopy.

Question 2: For which space $X$ is there a map

$X\to {\Omega }^{d}X$X \to \Omega^d X

which induces an isomorphism in ${v}_{1}$ periodic homotopy? If such maps exist, to what extent are they unique?

Question 3: It is known that there are homotopy classes in ${W}_{n}$ on which $v\left(1{\right)}^{n}$ is zero. Do all $v\left(1\right)$ torsion classes in ${W}_{n}$ have order $v\left(1{\right)}^{n}$ ? (False if $p=3$)

• (J. Neisendorfer) Let $p$ be a prime and consider the Bousfield-Dror Farjoun localization in which the local objects are those spaces $X$ for which the space of pointed maps ${\mathrm{map}}_{*}\left(Bℤ/pℤ,X\right)$ is weakly contractible. Haynes Miller’s version of the Sullivan conjecture says that all finite complexes are local in this sense. More generally, any $X$ which has mod $p$ cohomology with a locally finite action of the mod $p$ Steenrod algebra is also local in this sense. Conjecture, this is the complete class of local spaces. By definition the map from $Bℤ/pℤ$ to a point is a local equivalence in this sense. What exactly is the class of all local equivalences?

• (J. Wu) Let $𝒞$ be a category of certain spaces and let $𝒟$ be another category of certain spaces (or any other category). A functor $F:𝒞\to 𝒟$ is called rigid if $F\left(X\right)\simeq F\left(Y\right)$ implies $X\simeq Y$ for any $X,Y\in 𝒞$.

In geometry, there are some special manifolds having the property that their cohomology rings determine their homeomorphism or homotopy type. We can consider other functors concerning their rigidity.

There are some very basic functors in homotopy theory: The loop functor $\Omega$, the suspension functor $\Sigma$

Consider the functor $X\to \Sigma \Omega X$, so $F=\Sigma \Omega$. This functor is not rigid in general. For instance, if $X={S}^{2}$ and $Y={\mathrm{ℂℙ}}^{\infty }×{S}^{3}$, then $\Omega X\simeq \Omega Y$ but $X$ is not homotopy equivalent to $Y$.

However the functor $F=\Sigma \Omega$ is rigid from the homotopy category of simply connected finite co-H-spaces to homotopy category of spaces. Namely for simply connected finite co-H-spaces $X$ and $Y$, $\Sigma \Omega X\simeq \Sigma \Omega Y$ implies $X\simeq Y$.

I guess that this statement is true for any simply connected co-H-spaces of finite type. (But our method can not be applied for infinite co-H-spaces.)

The dual question: Is the functor $\Omega \Sigma$ rigid for simply connected H-spaces?

The (big) space $\Omega \Sigma X$ might be regarded as a “free deformation” of an H-space $X$ in the sense that the multiplication on X may be too tight but the multiplication on $\Omega \Sigma X$ can be “freed up” using the James construction. From this view, one may have more questions concerning the multiplication properties of H-spaces.

• (D. Kahn) A pointed space is called $textit\mathrm{rigid}$ if the group of self eqivalences is trivial.It is easy to see that the only rigid Eilenberg-Maclane space is $K\left(Z/2,n\right)$. But there are spaces with 2 non-trivial homotopy groups AND non trivial rational homology, which are rigid. In some (possible science fiction) version of homotopy theory, these would be the elementary particles.

Question: find the rigid spaces.

• (J. Wu) In homotopy theory, one can decompose $H$-spaces as a product of “homotopy indecomposable $H$-spaces”, and co-$H$-spaces as a wedge of “homotopy indecomposable co-$H$-spaces”. (For stable homotopy, the suspensions of all spaces are co-H-spaces.) Theoretically there is a collection of atomic $H$-spaces or atomic co-$H$-spaces. The first basic question may be to determine the homology of atomic spaces. At least a collection of certain atomic spaces.

Here is a concrete question: Determine the homology of the stable atomic pieces of the self-smash products of the projective spaces (localized at $p$).

More precisely, let $X={\mathrm{ℝℙ}}^{n}$, ${\mathrm{ℂℙ}}^{n}$ or ${\mathrm{ℍℙ}}^{n}$. Consider a $k$-fold self smash product ${X}^{\wedge k}$ of $X$. Assume that one can take suspensions (and so one can work in stable category). Decompose ${X}^{\wedge k}$ into atomic pieces and determine the homology of its atomic pieces.

For the case when $X$ is a projective plane, the answer has been given with a nice connection to the modular representation theory of symmetric groups. The next case may be ${\mathrm{ℝℙ}}^{4}$, ${\mathrm{ℂℙ}}^{4}$ and ${\mathrm{ℍℙ}}^{4}$ localized at $2$.

Here is a point: We do not know how the general atomic (co-$H$ or $H$-) spaces look like in general. However we do know that the self-smash products of co-H spaces admit decompositions. So starting from an explcit space $X$ (say a projective space), we would obtain more atomic spaces from the self-smash products of $X$. This gives a way to understand more and more atomic spaces.

Remark: There is fundamental connection between decompositions of self-smash products and the modular representation theory of symmetric groups (through the tensor representations).

• (J. Wu) Let $F\to E\to B$ be a fibration (localized at $p$) such that both $F$ and $B$ are finite H-spaces. Find a criterion when the total space $E$ is an H-space.

The total space $E$ may not be an H-space in general. It seems not easy to see when $E$ is an H-space. A classical question is whether the total space of an spherical bundle over a sphere, with both fibre and the base are odd dimensional, is an H-space. The answer is affirmative for $p>3$, and it becomes a tricky question for the case $p=3$.

For the case $p>3$, the first tricky case is that: $F$ is an odd dimensional sphere and $B$ is an H-space with its homology given as an exterior algebra on $p-2$ generators. The homology of $E$ is the exterior algebra on $p-1$ generators in odd dimensions. There are special properties of these spaces similar to 2-local spheres. The obstructions for the existence of a multiplication on $E$ are similar to that for the Hopf invariant one question.

It seems difficult to me if the homology of $E$ is an exterior algebra on $p$ generators or more.

• (J. Stasheff) Let $F\to E\to B$ be a principal fibration such that $F$ is a double loop space or loops on an $H$-space. Find a criterion when the classifying map is an $H$-map.
• (J. Wu) Structure of homotopy groups.

There are a lot of connections between homotopy groups and geometric groups. There would be many questions concerning the structures of the homotopy groups with connections to geometric groups.

### Problems on exponents

• (J. Neisendorfer) What are the precise 2 primary exponents for the homotopy groups of the spheres ${S}^{n}$?

There is an upper bound, due to James, namely ${4}^{n}$ for the spheres ${S}^{2n+1}$, but that is not the best possible. Some improvements have been made by Selick but no one believes his results are best possible. Barratt and Mahowald have made a conjecture on the best possible 2 primary result.

• (B. Gray) Moore’s conjecture:

Given a finite simply connected complex in which the rational homology of the loop space has at most polynomial growth, then for each prime $p$, the $p$ torsion in the homotopy has a bound on its exponent.

• (A. Salch) Try to mimic Cohen-Moore-Neisendorfer to get exponent bounds on motivic, parameterized, and/or equivariant unstable homotopy groups.

• (J. Wu) Let $X$ be a simply connected space of finite type. Then each homotopy group ${\pi }_{n}\left(X\right)$ is a finitely generated abelian group and so its $p$-torsion component has an exponent, say ${p}^{{e}_{n}}$. Now let ${p}^{{f}_{X}\left(n\right)}$ be the exponent of the homotopy groups ${\pi }_{1}\left(X\right)$, ${\pi }_{2}\left(X\right)$, …, up to ${\pi }_{n}\left(X\right)$.

Given any $X$, $\left({f}_{X}\left(n\right)\right)$ is a monotone increasing sequence.

Question: Study the growth of the sequence $\left({f}_{X}\left(n\right)\right)$.

If $X$ is a double suspension with degree ${p}^{r}$ map null homotopic, the Barratt conjecture says that the sequence $\left({f}_{X}\left(n\right)\right)$ is bounded above by $r+1$.

If $X$ satisfies the hypothesis of the Moore conjecture, the Moore conjecture states that the sequence $\left({f}_{X}\left(n\right)\right)$ is bounded above.

There are some spaces $X$ that do not satisfy the hopothesis of the Barratt conjecture or the Moore conjecture. For instance if $X$ is a wedge product of ${S}^{2}$ with itself. In this case, the sequence $\left({f}_{X}\left(n\right)\right)$ is not bounded above. But one can look at its growth such as whether it has linear growth, polynomial growth,…

Assume that in the future people may discover algorithms for computing homotopy groups using computer programs. If the growth of the exponents is exponential, it may “too bad”. If the growth of the exponents is polynomial or even linear, we may be happier for the situation.

• (F. Muro) There are algorithms to compute homotopy groups of simply connected CW-complexes with finitely many cells (e.g. spheres!), using Serre and Curtis spectral sequences, etc.

MR0184231 Curtis, Edward B. “Some relations between homotopy and homology.” Ann. of Math. (2) 82 1965 386–413.

For non-simply connected spaces the word problem shows up, so there cannot be a general algorithm. The problem is that homotopy theory has very hard computational complexity, even rational homotopy theory, which some people regard as ‘the easy part’:

MR1710993 Lechuga, Luis; Murillo, Aniceto “Complexity in rational homotopy.” Topology 39 (2000), no. 1, 89–94.