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:

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 $\pi(\mathscr{U},x_0)$ .
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 $\varphi_{\beta}(\pi(\mathscr{U},x_0))=\pi(\mathscr{U},x_1)$ .
Refinement: If an open cover $\mathscr{V}$ refines $\mathscr{U}$ (every $V\in \mathscr{V}$ is contained in some $U\in\mathscr{U}$ ), then $\pi(\mathscr{V},x_0)\leq \pi(\mathscr{U},x_0)$ .
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.
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.

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.
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 not equivalent in general. Here’s a counterexample:

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 property 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.