Spanier Groups: a modern take on vintage covering space theory

Edwin H. Spanier’s 1966 Algebraic Topology book is a true classic. Well-written and precise, I still find myself referring to it regularly even though it is really “old.” Spanier takes a unique approach to covering space theory that I haven’t seen anywhere else. I’ve found his approach to covering space theory so much more intuitive and general than modern books, that I use it when I teach covering space theory to my students. In particular, Spanier defined subgroups \pi(\mathscr{U},x_0) of a fundamental group \pi_1(X,x_0) to “detect” when a covering map p:Y\to X exists that corresponds to a given subgroup H\leq \pi_1(X,x_0). In particular,

Modern Interpretation of Spanier’s Classification Theoreom: A path-connected, locally path-connected space X has a covering space classifying (unique up to equivalence) a subgroup H\leq \pi_1(X,x_0) if and only if H is open in the Spanier topology on \pi_1(X,x_0).

Notice that X doesn’t have to be semilocally simply connected (SLSC) for this to work. The term “Spanier group(s)” was coined in [1]; it immediately stuck and has become fairly standard.

Spanier groups with respect to open covers

Notation: \alpha\cdot\beta denotes path-concatenation and \alpha^{-} denotes the reverse path of \alpha. We’ll work in the fundamental groupoid and write [\alpha][\beta]=[\alpha\cdot\beta] for the operation on path-homotopy classes.

Definition: Let \mathscr{U} be an open cover of a space X with basepoint x_0\in X. The Spanier group of (X,x_0) with respect to \mathscr{U} is the subgroup of \pi_1(X,x_0) generated by all homotopy classes of loops based at x_0 of the “lasso” form \alpha\cdot\gamma\cdot\alpha^{-} where Im(\gamma)\subseteq U for some U\in\mathscr{U}. In short, it’s

\pi(\mathscr{U},x_0)=\langle [\alpha\cdot\gamma\cdot\alpha^{-}]\in \pi_1(X,x_0)\mid \exists U\in\mathscr{U}\,\,Im(\gamma)\subseteq U\rangle

spanier

This means that a generic element of \pi(\mathscr{U},x_0) has the form \prod_{i=1}^{n}[\alpha_i][\gamma_i][\alpha_{i}^{-}] where \alpha_i:([0,1],0)\to (X,x_0) are paths and each \gamma_i is a loop with image in some member of \mathscr{U}.

Let’s first make some basic observations:

  1. Normality: The Spanier group \pi(\mathscr{U},x_0) is always a normal subgroup of \pi_1(X,x_0) since the conjugate [\beta][\alpha][\gamma][\alpha^{-}][\beta]^{-}=[\beta\cdot\alpha][\gamma][(\beta\cdot\alpha)^{-}] of a generator [\alpha][\gamma][\alpha^{-}] still has the form of a generator of \pi(\mathscr{U},x_0).
  2. Invariant under basepoint-change: if x_1\in X is another point and \beta:[0,1]\to X is a path from x_1 to x_0 and \varphi_{\beta}:\pi_1(X,x_0)\to\pi_1(X,x_1), \varphi_{\beta}([\eta])=[\beta\cdot\eta\cdot\beta^{-}] is the basepoint-change isomorphism, then \varphi_{\beta}(\pi(\mathscr{U},x_0))=\pi(\mathscr{U},x_1).
  3. Refinement: If an open cover \mathscr{V} refines \mathscr{U} (every V\in \mathscr{V} is contained in some U\in\mathscr{U}), then \pi(\mathscr{V},x_0)\leq \pi(\mathscr{U},x_0).
  4. They generate a topology on \pi_1: The cosets of Spanier groups form a basis for a topology on \pi_1(X,x_0) sometimes called the Spanier topology. In particular, a set W\subseteq \pi_1(X,x_0) is open if for every g\in W, there exists an open cover \mathscr{U} of X such that g\pi(\mathscr{U},x_0)\subseteq W. There are many topologies one can put on \pi_1. This one is closely related to covering space theory as we’ll see below.
  5. Continuity: If f:(Y,y_0)\to (X,x_0) is a continuous function, then f^{-1}\mathscr{U}=\{f^{-1}(U)\mid U\in\mathscr{U}\} is an open cover of Y such that the induced homomorphism f_{\#}:\pi_1(Y,y_0)\to\pi_1(X,x_0) satisfies f_{\#}(\pi(f^{-1}\mathscr{U},y_0))\leq\pi(\mathscr{U},x_0). I use the term continuity here because this is equivalent to saying that the induced homomorphisms f_{\#} are continuous with respect to the Spanier topology. Hence, the Spanier topology gives us one example of a fundamental group functor \pi_1:\mathbf{Top}_{\ast}\to \mathbf{TopGrp} to the category of topological groups.

Some intuition may be lost in the definition of the Spanier group \pi(\mathscr{U},x_0) in terms of it’s generators. I’ll give here an example of a more complicated element of \pi(\mathscr{U},x_0):

bcs

Let \Delta_2 be a 2-simplex with basepoint vertex d_0 and (bd_{k}\Delta_2)_1 be the 1-skeleton of some barycentric subdivision of \Delta_2 (the union of the black edges in the image on the right). Let f:((bd_{k}\Delta_2)_1,d_0)\to (X,x_0) be a map such that for every 2-simplex \tau of bd_{k}\Delta_2, we have f(\partial \tau)\subseteq U for some U\in\mathscr{U}. In other words, each small black triangle gets mapped into an element of the cover. With some careful choices of conjugating paths, the inclusion loop \partial \Delta_2\to (bd_{k}\Delta_2)_1 of the outermost triangle factors as a huge product of “lasso” loops where the hoops of the lasso are the boundaries of the 2-simplices in the subdivision bd_{k}\Delta_2. Since lassos map to lassos, the “large” loop f|_{\partial \Delta_2}:\partial \Delta_2\to X factors as a huge product of Spanier group generators and therefore represents an element \pi(\mathscr{U},x_0).

In my view, this construction really helps to clarify which homotopy classes end up in the Spanier groups and which ones do not.

Two definitions of “semilocally simply connected”

Spanier basically used these groups to provide an alternative definition of the “semilocally simply connected” property. But one oversight in Spanier’s book is that there are two competing notions that are not always the same if your spaces are not necessarily locally path connected.

Red Alert: There are two non-equivalent definitions of the “semilocally simply connected” property that differ by the addition of a single quantifier.

  1. A space X is based semilocally simply connected at x\in X if there exists an open neighborhood U of X such that the induced homomorphism \pi_1(U,x)\to \pi_1(X,x) is trivial.
  2. A space X is unbased semilocally simply connected at x\in X if there exists an open neighborhood U of X such that for all y\in U, the induced homomorphism \pi_1(U,y)\to \pi_1(X,y) is trivial.

Certainly the unbased property implies the based property and these are equivalent for locally path-connected spaces. However, these two definitions are equivalent in general. Here’s a counterexample:

lassospace

A one-dimensional compact space that is based SLSC but not unbased SLSC.

In particular, any small neighborhood of the right endpoint of the horizontal line contains circles but contains no path from that point to those circles.

The following lemma is why the unbased SLSC is typically used – it is a necessary condition for having a simply connected covering space.

Lemma 1: If X admits a simply connected covering space, then X is unbased SLSC.

Proof. This lemma is a special case of Theorem 4 below (where p_{\#}(\pi_1(Y,y_0))=1 since the covering space Y is simply connected and p_{\#} is inejctive) but this is a nice exercise to work out on it’s own. \square

So…even though the space pictured above is based SLSC, it can’t possibly have a simply connected covering space.

Lemma 2: Suppose X is locally path-connected. Then X is unbased SLSC if and only if there exists an open cover \mathscr{U} of X such that \pi(\mathscr{U},x_0)=1.

Proof. Suppose X is (unbased) SLSC. For each x\in X, let U_x be an open neighborhood of x such that for every y\in U_x, the inclusion induces the trivial homomorphism \pi_1(U_x,x)\to \pi_1(X,x), i.e. every loop in U_x based at x is null-homotopic in X. Since X is locally path-connected, we may replace each U_x with a smaller path-connected neighborhood that necessarily has the same property. Now \mathscr{U}=\{U_x\mid x\in X\} is an open cover of X. Consider a generator [\alpha\cdot\gamma\cdot\alpha^{-}] of \pi(\mathscr{U},x_0) where \gamma has image in some U_x. Since U_x is path-connected, there is a path \beta:[0,1]\to U_x from x to \gamma(0). Now [\alpha\cdot\gamma\cdot\alpha^{-}]=[\alpha\cdot\beta][\beta\cdot\gamma\cdot\beta^{-}][\alpha\cdot\beta]^{-1} where \beta\cdot\gamma\cdot\beta^{-} is a loop based a x and thus is null-homotopic in X. Thus [\alpha\cdot\gamma\cdot\alpha^{-}]=[\alpha\cdot\beta][\beta\cdot\gamma\cdot\beta^{-}][\alpha\cdot\beta]^{-1}=[\alpha\cdot\beta][\alpha\cdot\beta]^{-1}=1. Since all the generators of \pi(\mathscr{U},x_0) are trivial, this subgroup is the trivial subgroup.

Conversely, suppose there exists an open cover \mathscr{U} of X such that \pi(\mathscr{U},x_0)=1. For each x\in X, let V_x be a path-connected neighborhood of x such that V_x\subseteq U for some U\in\mathscr{U}. Since \mathscr{V}=\{V_x\mid x\in X\} refines \mathscr{U}, we have \pi(\mathscr{V},x_0)\leq \pi(\mathscr{U},x_0) and thus \pi(\mathscr{V},x_0)=1. We check that the inclusion (V_x,x)\to (X,x) induces the trivial homomorphism on fundamental groups. If \gamma:([0,1],\{0,1\})\to (V_x,x) is a loop, consider any path \alpha:[0,1]\to X from x_0 to x. Then [\alpha\cdot\gamma\cdot\alpha] is generator of \pi(\mathscr{V},x_0) and must therefore, represent the identity element in \pi_1(X,x_0). Since [\alpha\cdot\gamma\cdot\alpha]=1 in \pi_1(X,x_0), path conjugation in the fundamental groupoid gives [\gamma]=1 in \pi_1(X,x). Thus \gamma is null-homotopic in X, completing the proof. \square

Notice that local path-connectivity is necessary for both directions of the above proof. If you want to do away with this, you’ll need to use “based Spanier groups” which are also defined in [1]. Or, if you want to put on your categorical fancy pants you can use the locally path-connected coreflection lpc(X).

Theorem 3: lpc(X) is unbased SLSC if and only if there exists an open cover \mathscr{U} of X such that \pi(\mathscr{U},x_0)=1.

Lemma 2 also says something about the Spanier topology.

Corollary 4: If X is path-connected and locally path-connected, then X is SLSC if and only if \pi_1(X,x_0) is discrete with the Spanier topology.

Spanier Groups and Covering Spaces

The following theorem has no hypotheses in the spaces involved except for path connectivity.

Theorem 5: If p:Y\to X is a covering map of path-connected spaces, then there exists an open cover \mathscr{U} of X such that for any choice of basepoints p(y_0)=x_0, we have \pi(\mathscr{U},x_0)\leq p_{\#}(\pi_1(Y,y_0)).

Proof. Let \mathscr{U} be the open cover of X by neighborhoods that are evenly covered by p. Let y_0\in Y and p(y_0)=x_0. Consider a generator [\alpha\cdot\gamma\cdot\alpha^{-}] of the Spanier group \pi(\mathscr{U},x_0) where \gamma is a loop with image in U\in\mathscr{U}. Let y=\widetilde{\alpha}(1) be the end point of the lift \widetilde{\alpha}:([0,1],0\to (Y,y_0) of \alpha and let V be an open subset of p^{-1}(U) containing y that is mapped homeomorphically onto U by p. Since \gamma has image in U, the lift \widetilde{\gamma} of \gamma starting at y is a loop in V based at y. Thus \widetilde{\alpha}\cdot\widetilde{\gamma}\cdot \widetilde{\alpha}^{-} is a loop in Y based at y_0. We have [\alpha\cdot\gamma\cdot\alpha^{-}]=p_{\#}([\widetilde{\alpha}\cdot\widetilde{\gamma}\cdot \widetilde{\alpha}^{-}])\in p_{\#}(\pi_1(Y,y_0)). Since p_{\#}(\pi_1(Y,y_0)) contains all generators of \pi(\mathscr{U},x_0), we have \pi(\mathscr{U},x_0)\leq p_{\#}(\pi_1(Y,y_0)). \square

The following classification of coverings blows many “standard” approaches out of the water because it includes the entire lattice of subgroups classified by covering maps even if the space in question is not SLSC. Even though spaces like the Hawaiian earring or Menger cube don’t have simply connected coverings, they do have lots and lots of intermediate covering spaces and I often find myself in need of these. Weaker classifications, e.g. in Munkres and Hatcher assume SLSC and say nothing about these intermediate coverings that Spanier’s approach includes.

Spanier’s Covering Space Classification Theorem: Suppose X is path-connected and locally path-connected and H\leq \pi_1(X,x_0) is a subgroup. Then there exists a covering map p:(Y,y_0)\to (X,x_0) such that p_{\#}(\pi_1(Y,y_0))=H if and only if there exists an open covering \mathscr{U} such that \pi(\mathscr{U},x_0)\leq H.

Proof. Theorem 5 above is the only if direction that holds generally. For the converse, suppose the hypotheses on X and that there exists an open covering \mathscr{U} such that \pi(\mathscr{U},x_0)\leq H. I’m not going to show all of the nitty gritty details here, but I’ll give all the ingredients for how to build a corresponding covering map.

Let \widetilde{X}_H be the set of “groupoid cosets” H[\alpha]=\{h[\alpha]\mid h\in H\} for paths \alpha:([0,1],0)\to (X,x_0) starting at a fixed basepoint x_0\in X. Notice that H[\alpha]=H[\beta] if and only if \alpha(1)=\beta(1) and [\alpha\cdot\beta^{-}]\in H. Give \widetilde{X}_H the topology generated by the sets B(H[\alpha],U)=\{H[\alpha\cdot\epsilon]\mid Im(\epsilon)\subseteq U\} where U is an open neighborhood of \alpha(1) in X. Let p_H:\widetilde{X}_H\to X, p_H(H[\alpha])=\alpha(1) be the endpoint projection map. Since X is path-connected, p_H is onto and if V is a path-connected open set in X, then p_H(B(H[\alpha],V))=V. Hence, p_H is an open map since X is locally path-connected.

By assumption, we may find an open cover \mathscr{U} of X such that \pi(\mathscr{U},x_0)\leq H. By refining \mathscr{U}, we may assume each element of U is path-connected. Given x\in X pick U\in\mathscr{U} containing x. It’s easy to see that p_{H}^{-1}(H)=\bigcup_{\alpha(1)=x}B(H[\alpha],U). Also, some routine arguments (using the path-connectivity of U) show that for any two paths \alpha,\beta:([0,1],0,1)\to (X,x_0,x), the open sets B(H[\alpha],U) and B(H[\beta],U) are either disjoint are equal. Since we already know that p_H is an open map, it suffices to show that p_H is injective on B(H[\alpha],U). Suppose p_{H}(H[\alpha\cdot\epsilon])=\alpha\cdot\epsilon(1)=\alpha\cdot\delta(1)=p_{H}(H[\alpha\cdot\delta]) for paths \epsilon,\delta in U. Now \alpha\cdot(\epsilon\cdot\delta^{-})\cdot\alpha^{-} is a well-defined “lasso” loop where \epsilon\cdot\delta^{-} has image in U\in\mathscr{U}. Therefore, [\alpha\cdot(\epsilon\cdot\delta^{-})\cdot\alpha^{-}]\in \pi(\mathscr{U},x_0)\leq H, which implies H[\alpha\cdot\epsilon]=H[\alpha\cdot\delta], proving injectivity. \square

Notice that the subgroup condition \pi(\mathscr{U},x_0)\leq H in the statement of the classification is precisely what is needed to verify the locally injective part of the definition of a covering map. The typical arguments for uniqueness don’t require SLSC so equivalent based coverings still correspond to conjugate subgroups of \pi_1(X,x_0) and so on and so forth.

The classification theorem at the start of the post is a slick restatement of this theorem.

Corollary 6: Suppose X is path connected and locally path connected. The lattice of subgroups corresponding to covering maps over X is upward closed and closed under finite intersection.

Now, if X is also SLSC, then, as we proved above, \pi_1(X,x_0) is discrete with the Spanier topology, which means all subgroups are open, which means all subgroups are classified by covering spaces! This case gives you back the specific classification theorem you might be used to for SLSC spaces.

References

[1] H. Fischer, D. Repovs, Z. Virk, A. Zastrow, On semilocally simply-connected spaces, Topology Appl. 158 (2011) 397-408.

[2] E.H. Spanier, Algebraic Topology, McGraw-Hill, 1966.

Posted in Uncategorized | Leave a comment

Higher Dimensional Hawaiian Earrings

2d Hawaiian Earring 03

The (1-dimensional) Hawaiian earring \mathbb{H} is a 1-dimensional Peano continuum (connected, locally path-connected, compact metric space) constructed by adjoining a shrinking sequence of circles at a single point.

Hawaiian earring space

The Hawaiian earring space

The importance of \mathbb{H} stems from the fact that this is the prototypical space that has infinite products in it’s fundamental group. This makes it a useful test-space for detecting the existence of non-trivial infinite products (or algebraic wildness) in other fundamental groups. The Hawaiian earring group is not a free group but it is basically an infinitary (i.e. having infinite products) analogue of a free group. Moreover, \mathbb{H} is aspherical, i.e. \pi_k(\mathbb{H},b_0)=0 for k\geq 2. In this post, I’m going to give a summary about their higher dimensional analogues \mathbb{H}_n\subseteq \mathbb{R}^{n+1}, the n-dimensional Hawaiian earring.

Constructing \mathbb{H}_n

Definition: let n\geq 0 and C_{k}^{n}=\{\mathbf{x}\in\mathbb{R}^{n+1}\mid \|\mathbf{x}-(1/k,0,0,\dots ,0)\|=1/k\} be the n-sphere of radius 1/k centered at (1/k,0,0,\dots,0). The n-dimensional Hawaiian earring is the set

\mathbb{H}_n=\bigcup_{k\in\mathbb{N}}C_{k}^{n}

with the subspace topology inherited from \mathbb{R}^{n+1}. Let b_0=(0,0,\dots,0) denote the origin, which we typically take as the basepoint.

2dim

Two views of \mathbb{H}_2, the 2-dimensional Hawaiian earring

The 0-dimensional Hawaiian earring \mathbb{H}_0=\{0,\dots,1/3,2/5,1/2,2/3,1,2\} is simply a convergent sequence and \mathbb{H}=\mathbb{H}_1 is the usual Hawaiian earring. We can build \mathbb{H}_n using a variety of constructions:

  1. One-point compactification: \mathbb{H}_n is homeomorphic to the one-point compactification of a countable disjoint union of open n-cells \coprod_{k\in\mathbb{N}}(0,1)^n, i.e. \mathbb{H}_n\cong \coprod_{k\in\mathbb{N}}[0,1]^n/\coprod_{k\in\mathbb{N}}\partial [0,1]^n.
  2. Subspace of a product: \mathbb{H}_n is homeomorphic to the wedge sum \bigvee_{k\in\mathbb{N}}S^n viewed naturally as a subspace of \prod_{k\in\mathbb{N}}S^n with the product topology. To make this identification, the m-th wedge summand is identified with the set \prod_{k\in\mathbb{N}}A_k where A_k=\{b_0\} if k\neq m and A_m=S^n.
  3. Inverse Limit: Let X_k=\bigcup_{i=1}^{k}C^{n}_{i} be the union of the first k n-spheres and r_{k+1,k}:X_{k+1}\to X_k be the retraction that collapses C_{k+1}^{n} to b_0 and is the identity elsewhere. Then the natural retractions r_k:\mathbb{H}_n\to X_k collapsing \bigcup_{i>k}C_{i}^{n} to b_0 induce a homeomorphism \mathbb{H}\cong\varprojlim_{k}(X_k,r_{k+1,k}).
  4. Reduced Suspension: For n\geq 1, the n-dimensional Hawaiian earring \mathbb{H}_{n} is homeomorphic to the reduced suspension \Sigma \mathbb{H}_{n-1} of the (n-1)-dimensional Hawaiian earring. By iteration, we obtain a formula similar to that for ordinary spheres: \mathbb{H}_{n+m}\cong\Sigma^m\mathbb{H}_n for all m,n\geq 0. It is important here that we use the reduced suspension and not the unreduced suspension. The undreduced suspension S\mathbb{H}_{n-1} is not homotopy equivalent to \Sigma \mathbb{H}_{n-1}. This means that your typical tricks, including Mayer Vietoris Sequences, won’t be helpful for computing the singular homology groups of \mathbb{H}_{n}.

The 2-dimensional Hawaiian earring \mathbb{H}_2, was shown to have some interesting properties in the famous paper [1] by Barratt and Milnor. To put their observation into context, recall that if we consider an ordinary wedge of 2-spheres, elementary computations show that

\displaystyle\widetilde{H}_k\left(\bigvee_{\mathbb{N}}S^2\right)=\begin{cases} \bigoplus_{\mathbb{N}}\mathbb{Z} & k=2 \\ 0 & k\neq 2 \end{cases}.

It might seem that \mathbb{H}_2 should be similar. In particular, \mathbb{H}_2 is simply connected and locally simply connected. However, Barratt and Milnor showed that there are infinitely many k>2 such that H_k(\mathbb{H}_2) is uncountable! This seems really strange and “anomalous” until one reads the paper to see how it’s done.

I’ve said it once and I’ll say it again: the “wildness” in wild algebraic topology is really just about infinite products. The reason why higher homology groups of \mathbb{H}_2 are often non-trivial is precisely because one can consider infinite sums \sum_{m=1}^{\infty}[f_m] of elements of [f_m]\in \pi_k(\mathbb{H}_2), e.g. given by shrinking sequences of non-trival Whitehead products. Even though [f_m] is trivial in homology of any finite wedge of the spheres C_{k}^{2}, the k-cycle represented by the infinite \sum_{m=1}^{\infty}[f_m] (apply the Hurewicz homomorphism) is not the boundary of a finite sum of n+1-chains and therefore represents a non-trivial homology class.

The homotopy groups \pi_k(\mathbb{H}_n), k\leq n

Recall that a space (X,x_0) is n-connected if \pi_m(X,x_0) is trivial for 0\leq m\leq n.

Theorem (Eda-Kawamura): \mathbb{H}_n is (n-1)-connected and locally (n-1)-connected. Moreover, the embedding \varphi:\mathbb{H}_n\to \prod_{k=1}^{\infty}C_{k}^{n} induces an isomorphism on \pi_n.

This means that

\pi_n(\mathbb{H}_n,b_0)\cong \prod_{k=1}^{\infty}\mathbb{Z}

is the Baer-Specker group.

Easy part of the proof: \varphi_{\#}:\pi_n(\mathbb{H}_n,b_0)\to\prod_{k=1}^{\infty}\mathbb{Z} is onto.

Identify \pi_n(C_{k}^{n},b_0) with \mathbb{Z}. Given a sequence of integers (a_1,a_2,a_3,\dots)\in\prod_{k=1}^{\infty}\mathbb{Z}, let f_k:(I^n,\partial I^n)\to (C_{k}^{n},b_0) be a map such that [f_k]=a_k\in \pi_n(C_{k}^{n},b_0). Define f=\prod_{k=1}^{\infty}f_k as the infinite concatenation of n-loops. Explicitly, f:(I^n,\partial I^n)\to (\mathbb{H}_n,b_0) is defined so that the restriction f:\left[\frac{k-1}{k},\frac{k}{k+1}\right]\times I^{n-1}\to\mathbb{H}_n is f(t_1,t_2,\dots, t_n)=f_k((1 + k) (1 + k (t_1-1)),t_2,t_3,\dots,t_n), for all k\geq 1 and f(1,t_2,t_3,\dots,t_n)=b_0.

higherinfproduct

The map f as an infinite concatenation of 2-loops

It’s now easy to see that \varphi_{\#}([f])=(a_1,a_2,a_3,\dots ), proving that \varphi_{\#} is surjective.

Hard part of the proof: \varphi_{\#} is injective.

The Eda-Kawamura proof in [2] is inspired but is written in a very technical fashion and has scared off many-a-reader. Essentially, the key idea of the proof is to simultaneously perform simplicial/cellular approximation on an infinite sequence of disjoint polyhedral domains in I^n. Then you work to perform an infinite Eckmann-Hilton type argument, to show that every map (I^n,\partial I^n)\to (\mathbb{H}_n,b_0) is homotopic to one of the form f=\prod_{k=1}^{\infty}f_k as described above, i.e. where f_k has image in C_{k}^{n}. Then if \varphi_{\#}([f])=(0,0,0,\dots), each f_k has degree 0 as a map S^n\to C_{k}^{n} and thus is null-homotopic in C_{k}^{n}. Let H_k:I^n\times I\to C_{k}^{n} be such a null-homotopy where H_{k}(\mathbf{t},0)=f_k(\mathbf{t}) and H_k(\mathbf{t},1)=b_0. Then we can define a null-homotopy H:I^n\times I\to \mathbb{H}_n of f so that H(\mathbf{t},s)=H_k(\mathbf{t},(1 + k) (1 + k (s-1))) for s\in \left[\frac{k-1}{k},\frac{k}{k+1}\right] and H(\mathbf{t},1)=b_0.

I do wonder if there is a simpler inverse limit type approach to proving that \varphi_{\#} is injective where the null-homotopy H=\varprojlim_{k}G_k is built as an inverse limit of null-homotopies G_k:I^n\times I\to \bigcup_{j\geq k}C_{j}^{n}. It would still be highly non-trivial, but would lend some insight into dealing with higher dimensional homotopy groups. If possible, one would likely have to inductively build the maps G_k by applying simplicial approximation at each step.

The homotopy groups \pi_k(\mathbb{H}_n), k> n

As noted in my problem list, these groups are still unknown. Of course, the higher homotopy groups of spheres are mysterious themselves here so the ultimate goal would be to express \pi_k(\mathbb{H}_n) in terms of homotopy groups of spheres.

From above, we may consider \mathbb{H}_n as a closed subspace of the direct product (S^n)^{\infty}=\prod_{\mathbb{N}}S^n over the naturals. The key is to notice that the argument for the “easy” part of the proof above applies to any dimension: for any k\in\mathbb{N}, the inclusion \mathbb{H}_n\to (S^n)^{\infty} induces a surjective homomorphism \varphi_{\#}:\pi_k(\mathbb{H}_n)\twoheadrightarrow \pi_k((S^n)^{\infty})=\prod_{\mathbb{N}}\pi_k(S^n). Moreover, by applying infinite commutativity, i.e. infinite Eckmann-Hilton, this surjection is a split epi.

Let’s look at the homotopy long exact sequence of the pair ((S^n)^{\infty},\mathbb{H}_n):

les3

The surjections tells us that we really have

les4

which splits up into short exact sequences:

ses3

Since the epi \varphi_{\#}:\pi_k(\mathbb{H}_n)\twoheadrightarrow\prod_{k=1}^{\infty}\pi_k(S^n) always splits canonically, we have

\pi_k(\mathbb{H}_n)\cong \pi_{k+1}((S^n)^{\infty},\mathbb{H}_n)\oplus \prod_{\mathbb{N}}\pi_k(S^n)

This tells us at least that explicitly computing \pi_k(\mathbb{H}_n) requires computing \pi_k(S^n). Notice that the Eda-Kawamura computation only tells us that \pi_{n+1}((S^n)^{\infty},\mathbb{H}_n)=0.

An interesting case is

\pi_{3}(\mathbb{H}_2)\cong \pi_{4}((S^2)^{\infty},\mathbb{H}_2)\oplus \prod_{\mathbb{N}}\pi_{3}(S^2)\cong \pi_{4}((S^2)^{\infty},\mathbb{H}_2)\oplus \prod_{\mathbb{N}}\mathbb{Z}

since \pi_{3}(S^2)\cong \mathbb{Z} is generated by the Hopf map.

So the cheesesteak-worthy question is: what is \pi_{k+1}((S^n)^{\infty},\mathbb{H}_n) when k>n?

It will typically be non-trivial because of Whitehead products and infinite products of Whitehead products…just like in Barratt-Milnor! I have my own ideas about what the answer is in the case k=n+1, but there is one technical hurdle in the way from making it precise. There is a piece of technology missing that needs to be developed.

References:

[1] M.G. Barratt, J. Milnor, An example of anomalous singular theory, Proc. Amer. Math. Soc. 13 (1962) 293-297.

[2] K. Eda, K. Kawamura, Homotopy and Homology Groups of the n-dimensional Hawaiian earring, Fund. Math. 165 (2000) 17-28.

 

Posted in Homotopy theory | Tagged , , | Leave a comment

New Problem List

Check out the new Problem List page!

Do you know of any problems I should add?

Posted in Uncategorized | Leave a comment