$$ \usepackage{amssymb} \newcommand{\N}{\mathbb{N}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ZZ}{\ooalign{Z\cr\hidewidth\kern0.1em\raisebox{-0.5ex}{Z}\hidewidth\cr}} \newcommand{\colim}{\text{colim}} \newcommand{\weaktopo}{\tau_\text{weak}} \newcommand{\strongtopo}{\tau_\text{strong}} \newcommand{\normtopo}{\tau_\text{norm}} \newcommand{\green}[1]{\textcolor{ForestGreen}{#1}} \newcommand{\red}[1]{\textcolor{red}{#1}} \newcommand{\blue}[1]{\textcolor{blue}{#1}} \newcommand{\orange}[1]{\textcolor{orange}{#1}} \newcommand{\tr}{\text{tr}} \newcommand{\id}{\text{id}} \newcommand{\im}{\text{im}\>} \newcommand{\res}{\text{res}} \newcommand{\TopTwo}{\underline{\text{Top}^{(2)}}} \newcommand{\CW}[1]{\underline{#1\text{-CW}}} \newcommand{\ZZ}{% \ooalign{Z\cr\hidewidth\raisebox{-0.5ex}{Z}\hidewidth\cr}% } % specific for this document \newcommand{\cellOne}{\textcolor{green}{1}} \newcommand{\cellTwo}{\textcolor{red}{2}} \newcommand{\cellThree}{\textcolor{brown}{3}} \newcommand{\cellFour}{\textcolor{YellowOrange}{4}} $$

\(\ell^2\)-Invariants

This article is about the \(\ell^2\)-Invariants of \(G\)-CW-complexes. We will start by constructing the \(\ell^2\)-completion of cellular chain complexes and then take a look at some applications of the \(\ell^2\)-completion of cellular chain complexes. We will calculate the \(\ell^2\)-homology groups and the \(\ell^2\)-betti numbers of \(S^\infty := \colim_n S^n\) with the group \(G:=\prod_{n=0}^\infty \Z/2\Z\) acting on it. We will also take a look at the \(\ell^2\)-betti numbers of the torus and the \(\ell^2\)-betti numbers of the universal covering space of the torus. Finally, we will take a look at Lück’s theorem and its implications for the \(\ell^2\)-betti numbers of the universal covering space of a connected, compact CW-complex whose fundamental group is residually finite.

mathematics
topology
algebra
group theory
ring theory
measure theory
functional analysis
differential geometry
differential topology
algebraic topology
Author

Luca Leon Happel

Published

February 4, 2024

Modified

February 5, 2024

WARNING: This is a work in progress. I am currently writing this article and it is not finished yet. I will update it regularly and remove this warning once it is finished.

Introduction

This semester (WiSe 2023/24) I took a course about \(\ell^2\)-Invariants(see Jun.-Prof. Dr. H. Kammeyer 2024) at the University of Düsseldorf. As a reminder, \(\ell^2\) is the Hibert space of square-summable sequences of complex numbers. Meaning, a sequence \((a_n\in\C)_{n=0}^\infty\) is in \(\ell^2\) if and only if \(\sum_{n=0}^\infty |a_n|^2 < \infty\). By defining an inner product on \(\ell^2\) as \(\langle a, b \rangle = \sum_{n=0}^\infty a_n \overline{b_n}\), we make \(\ell^2\) to a complete complex inner product space, i.e. a Hilbert space.

For context, this course was part of my master’s degree in mathematics and it was the culmination of a series of courses about Algebraic Topology, which I attended for the past two and a half years. Previous courses concentrated primarily on the basics of algebraic topology, such as homology and cohomology, and their applications to the classification of manifolds and the computation of homotopy groups and cohomology rings.

The tools in the form of algebraic structures and theorems that we learned in these coureses, like homology, work well with finite CW-complexes and their finite coverings, after all we are just dealing with finitely generated modules in these cases and concepts like determinants and betti numbers are well defined. However, when we want to study infinite CW-complexes, we need to use more sophisticated tools, such as \(\ell^2\)-Invariants, to understand the topology of these spaces. Something interesting happens, when we look at covering spaces of finite CW-complexes. By the Galois correspondence, we can associate a covering space \(\bar X\) with a subgroup \(H\) of the fundamental group \(G\) of a base space \(X\).(see Hatcher 2010a, Proposition 1.38) Now, we have a group action of \(G\) on \(\bar X\) and this added structure allows us to define the \(\ell^2\) betti numbers \(b_i^{(2)}(H \curvearrowright \bar X)\), \(\ell^2\) homology groups \(H_i^{(2)}(H \curvearrowright \bar X)\) and much more.

Let us build some intuition before we proceed though. Consider the base space \(X=S^1\), and a \(d\) sheeted covering \(\bar X_d\). The fundamental group of \(X\) is \(\Z\) and the fundamental group of \(\bar X_d\) is \(\Z/d\Z\). An early result in the study of \(\ell^2\)-Invariants is that for \(d\)-sheeted coverings \(\bar X_d \to X\), the \(\ell^2\)-betti numbers of \(\bar X_d\) are \(b_i^{(2)}(\Z/d\Z\curvearrowright\bar X_d) = b_i(\bar X_d) / |\Z/d\Z| = b_i(\bar X_d) / d\). This is quite astounding, as betti numbers are commonly seen as “counting the number of \(i\)-dimensional holes” in a space, but here we see that the \(\ell^2\)-betti numbers are not integers, but rational numbers. So, geometrically, what does it mean for a space to have “one third of a hole”? Let us investigate for the case \(d=3\).

Show the code
# 3d plotly plot of a a circle and a 3-sheeted covering right next to it
import plotly.graph_objects as go
import numpy as np
# S^1
def s1()->(np.ndarray, np.ndarray, np.ndarray):
    theta = np.linspace(0, 2*np.pi, 100)
    x = np.cos(theta)
    y = np.sin(theta)
    z = np.zeros(100)
    return x, y, z
# 3-sheeted covering
def covering(sheets: int, handlestart: float = 1/8, subsections=100):
    theta = np.linspace(0, (sheets-handlestart)*2*np.pi, subsections)
    x = np.cos(theta)
    y = np.sin(theta)
    z = theta/(2*np.pi)
    # Add a small handle from (1,0,0) to (1,0,`sheets`) in a small bow.
    # This handle is just here to show that this covering is still homeomorphic to S^1.
    # The preimages of (1,0,0) therefor would still be a discrete set of `sheets` points, not a line.
    x = np.append(x, ( lambda v : (1-v**6)/10 )(np.linspace(-1,1, subsections)) + np.cos(np.linspace((sheets-handlestart)*2*np.pi, sheets*2*np.pi, subsections)))
    y = np.append(y, ( lambda v : (1-v**6)/10 )(np.linspace(-1,1, subsections)) + np.sin(np.linspace((sheets-handlestart)*2*np.pi, sheets*2*np.pi, subsections)))
    z = np.append(z, ( lambda v : (1-np.arctan(10*v)/np.arctan(10))*(sheets-handlestart)/2 )( np.linspace(-1, 1, subsections) ))
    return x,y,z
# draw the plot
fig = go.Figure()
fig.add_trace((lambda v: go.Scatter3d(x=v[0], y=v[1], z=v[2], mode='lines', name='S^1'))(s1()))
fig.add_trace((lambda v: go.Scatter3d(x=v[0], y=v[1], z=v[2], mode='lines', name=f'{3}-sheeted covering'))(covering(3)))
fig.show()

It is clear, that the betti numbers of the circle \(S^1\) are \(b_0(S^1) = 1\) and \(b_1(S^1) = 1\). The same is the case for any \(n\)-sheeted covering. In particular, we have \(b_0(\bar X_3) = 1\) and \(b_1(\bar X_3) = 1\). When we take a look at the \(\ell^2\)-betti numbers, we must take into account the action of the deck transformation group(see Hatcher 2010b) \(\text{Deck}(p) = \Z/3\Z\) on \(\bar X_3\), where \(p: \bar X_d \to X\) is the canonical projection.

Show the code
import plotly.express as px
import numpy as np
import pandas as pd

# Function to generate circle coordinates for a given t of rotation around the x-axis
def get_frame(t):
    x,y,z = covering(3)
    x = (1-t)*x+t*x/np.hypot(x,y)
    y = (1-t)*y+t*y/np.hypot(x,y)
    z = z*(1-t)
    return x, y, z

# Generate data for each frame
ts = np.linspace(0, 1, 20)
df = pd.DataFrame()

for t in ts:
    x, y, z = get_frame(t)
    temp_df = pd.DataFrame({'x': x, 'y': y, 'z': z, 't': t})
    df = pd.concat([df, temp_df])

# Initial plot using Plotly Express
fig = px.line_3d(df, x='x', y='y', z='z', animation_frame='t')

# Fixing the axis ranges
fig.update_layout(
    scene=dict(
        xaxis=dict(range=[-1.3,1.3], autorange=False),
        yaxis=dict(range=[-1.3,1.3], autorange=False),
        zaxis=dict(range=[0,3], autorange=False),
        aspectmode='cube',  # This ensures equal aspect ratio for all axes
    ),
    title="Homotopy between id and p",
    updatemenus=[{
        "buttons": [
            {
                "args": [None, {"frame": {"duration": 50, "redraw": True}, "fromcurrent": True}],
                "label": "Play",
                "method": "animate"
            },
            {
                "args": [[None], {"frame": {"duration": 0, "redraw": True}, "mode": "immediate", "transition": {"duration": 0}}],
                "label": "Pause",
                "method": "animate"
            }
        ],
        "direction": "left",
        "pad": {"r": 10, "t": 87},
        "showactive": False,
        "type": "buttons",
        "x": 0.1,
        "xanchor": "right",
        "y": 0,
        "yanchor": "top"
    }],
    sliders=[{
        "steps": [{"args": [[f"{t}"], {"frame": {"duration": 50, "redraw": True}, "mode": "immediate"}], 
                   "label": f"{t}", "method": "animate"} for t in ts],
    }]
)
# Show figure
fig.show()

Maybe it is already somewhat obvious, that there is a canonical group action from \(\Z/3\Z\) on \(\bar X_3\). This action, the deck transformation, permutes the three preimages of each point in \(S^1\) using Cayley’s theorem. The result of applying the deck transformation to \(\bar X_3\) therefore is a 3-sheeted covering of \(S^1\) again and is an isomorphism of covering spaces from \(\bar X_3\) to \(\bar X_3\). As an example, let us consider some point \(x\) on \(\bar X_3\). \(\text{Deck}(p) \cong \Z/3\Z \cong \langle g | g^3 = 0\rangle\) acts on \(x\) by \(g.x = x + 1\), if \(x\) is not in the topmost sheet, otherwise it pushes \(x\) down to the bottom sheet, assuming we embed \(\bar X_3\) into \(\R^3\) in the obvious way. In the following plot, we can see what happens to both a single point and the whole covering space under the action of the deck transformation. From left to right, we apply \(g\) not at all, once and then twice.

Show the code
from plotly.subplots import make_subplots
# the individual point we want to track under the action of the deck transformation
angle = np.pi*7/8
point = ( 
    np.cos(angle),
    np.sin(angle),
    angle/(2*np.pi)
 )
# group action. Because Z/3Z is cyclic, we only need to define the generator
g = lambda v: (v[0],v[1],v[2]+1 if v[2] < 2 else 0)

# covering [0]
fig = make_subplots(rows=1, cols=3, specs=[[{'type': 'scene'}, {'type': 'scene'}, {'type': 'scene'}]])
fig.add_trace((lambda v: go.Scatter3d(x=v[0], y=v[1], z=v[2], mode='lines', name='[0]'))(covering(3)), row=1, col=1)
# add a little red dot to show what happens to one individual point
fig.add_trace(go.Scatter3d(x=[point[0]], y=[point[1]], z=[point[2]], mode='markers', name="x", marker=dict(size=5, color='red')), row=1, col=1)

# covering [1]
fig.add_trace((lambda v: go.Scatter3d(x=v[0], y=v[1], z=v[2], mode='lines', name='[1]'))(covering(3, 0, 100)), row=1, col=2)
fig.add_trace(go.Scatter3d(x=[g(point)[0]], y=[g(point)[1]], z=[g(point)[2]], mode='markers', name="g.x", marker=dict(size=5, color='red')), row=1, col=2)

# covering [2]
fig.add_trace((lambda v: go.Scatter3d(x=v[0], y=v[1], z=v[2], mode='lines', name='[2]'))(covering(3, 1/8, 100)), row=1, col=3)
fig.add_trace(go.Scatter3d(x=[g(g(point))[0]], y=[g(g(point))[1]], z=[g(g(point))[2]], mode='markers', name="g.g.x", marker=dict(size=5, color='red')), row=1, col=3)


# legend
fig.update_layout(
    title="Group action of the cyclic group with three elements on our covering space",
)
fig.show()

It seems that \(\ell^2\)-betti numbers somehow measure “how much each symmetry contributes to the betti-number”, but we still need to make this precise. In this example however, there are three symmetries, \(b_1(X_3) = 1\) and \(b_1(\Z/3\Z\curvearrowright X_3) = 1/3\), so our intuition kinda makes sense.

Futher Motivation

One of the main goals in this lecture was for us to reach Lück’s theorem:

Lück’s theorem: Let \(X\) be a connected, compact CW-complex whose fundamental group \(G=\pi_1(X)\) is residually finite. Then for every residual chain \((G_i)\) in \(G\) and every \(n\ge 0\) we have \[\lim_{i\to\infty} \frac{b_n^{(2)}(\bar X_i)}{[G:G_i]} = b_n^{(2)}(G\curvearrowright \tilde X)\] where \(\bar X_i \to X\) is the covering space associated with \(G_i\) and \(\tilde X\) is the universal covering space of \(X\).

For clearance, the definition of a residual chain is as follows:

Definition 1 A sequence \(G=G_0 \ge G_1 \ge G_2 \ge \dots\) of subgroups of a group \(G\) is called a residual chain if for each \(i\ge 0\) the index \([G:G_i]\) is finite and the intersection \(\bigcap_{i\ge 0} G_i = \{1\}\).

So a simple and fitting example of a residual chain would be the sequence \(G_i = \Z/2^i\Z\) for \(i\ge 1\) and \(G_0 = \Z\) in the case of \(G=\Z\), as we had before with \(S^1\) and its universal covering space \(\tilde X = \R\). By plugging in Lück’s theorem, we get:

\[ \begin{align*} & \lim_{i\to\infty} \frac{b_n^{(2)}(\bar X_i)}{[G:G_i]} &= b_n^{(2)}(G\curvearrowright \tilde X) \\ \Leftrightarrow& \lim_{i\to\infty} \frac{b_n^{(2)}(\bar X_i)}{2^i} &= b_n^{(2)}(\Z\curvearrowright \R) \\ \Leftrightarrow& \lim_{i\to\infty} \frac{\frac{1}{2^i}}{2^i} &= b_n^{(2)}(\Z\curvearrowright \R) \\ \Leftrightarrow& 0 \end{align*} \]

In fact, we can generalize this to \(b_n^{(2)}(\tilde { \mathbb{T}^k }) = 0\), where \(\mathbb{T}^k\) is the \(k\)-dimensional torus and therefor \(\tilde { \mathbb{T}^k } \cong \R^k\), which is in stark contrast to the usual betti numbers \(b_n(\tilde { \mathbb{T}^k }) = \binom{k}{n}\).

Constructing The \(\ell^2\)-Completion Of Cellular Chain Complexes

Instead of talking only about coverings of CW-complexes, we will now talk about general CW-complexes, which come equipped with a special kind of group action. We will call these \(G\)-CW-complexes. The definition of a \(G\)-CW-complex is as follows:

Definition 2 A \(G\)-CW-complex is a CW-complex \(X\) together with an action by a discrete group \(G\) such that each open cell \(E\) of \(X\) is mapped to another open cell \(gE\) by the action of \(g\in G\). If \(gE\cap E \neq \emptyset\), then \(g\) must fix \(E\) pointwise.

This is a generalisation of the observations we made earlier, as the deck transformation group \(\text{Deck}(p)\) is a discrete group and the action of \(\text{Deck}(p)\) on \(\bar X_d\) is a \(G\)-action which satisfies the conditions of the definition.

By this definition, we know that open \(i\)-cells of any \(G\)-CW-complex \(X\) are mapped onto open \(i\)-cells. Let us recall that a CW-complex by definition allows for a filtration into \(i\)-skeletons \(\emptyset = X_{-1} \subseteq X_0 \subseteq X_1 \subseteq \dots \subseteq X_n = X\), where each \(X_i\) is obtained by attaching \(i\)-cells to \(X_{i-1}\). Let us also recall the definition of \(\TopTwo\):

Definition of \TopTwo Definition of \(\TopTwo\) from (H. Kammeyer 2022)

Because each CW complex also is a topological space and by the inclusion of the \(( i-1 )\)-skeletons into the \(i\)-skeletons, we can define the cellular chain complex \(C_*(X)\) of a \(G\)-CW-complex \(X\) as \(C_i(X) = H_i(X_i, X_{i-1})\). This construction is similar to how one defines cellular homology for CW-complexes(see H. Kammeyer 2022, Definition 6.21) and the idea here is that we use the fact that \(X_{i-1} \to X_i\) is a cofibration to get an isomorphism between \(H_i(X_i, X_{i-1})\) and \(H_i(X_i/X_{i-1}, X_{i-1}/X_{i-1}) \cong \tilde H_i(X_i/X_{i-1})\) (see H. Kammeyer 2022, Proposition 5.6). In pictures, we can think of \(C_i(X)\) as counting the number of \(i\)-dimensional holes in \(X_i\), after we collapse the \((i-1)\)-skeleton to a point as is illustrated in Figure 1 with a torus.

Figure 1: Construction of \(C_2(\mathbb{T})\) given \(\mathbb{T}\) as a CW-complex with four 2-cells, eight 1-cells and four 0-cells.

Notice that for each \(g\in G\) we have a homeomorphism in \(\TopTwo\): \(g: (X_i, X_{i-1}) \xrightarrow{\sim} (X_i, X_{i-1})\). This self-homomorphism induces an automorphism on the homology group \(H_i(X_i, X_{i-1})\), because by Definition 2 the action of \(g\) on \(X_i\) is cellular, meaning \(g\) sends each \(i\)-cell to exactly one other \(i\)-cell or itself. This means, that we can apply each \(g\) to some cell \(i\)-cells \(c_i\) and the result will again be in \(C_i(X)\). Furthermore, by construction \(C_i(X)\) is an abelian group and as such it is a \(\Z\)-module. Pairing this with the group action, we have that \(C_i(X)\) is a left \(\Z G\)-module, where \(\Z G\) is the group ring of \(G\) over \(\Z\). This means, for any \(c\) in \(C_i(X)\), we can define \(\left(\sum_g \lambda_g g\right).c \in C_i(X)\) for \(\lambda_g\in\Z\) and \(g\in G\).

Independence of the Choice of Filtration

Implicitly, we made a choice in our construction. When we chose a specific filtration of \(X\) into \(i\)-skeletons, we also chose a specific cellular chain complex \(C_*(X)\). Here comes our group action into play. In Figure 1 as an example we may chose \(G = C_2\) to be the cyclic group with two elements and the action of \(C_2\) on \(\mathbb{T}\) to be the permutation of the green cell with the brown cell and the red cell with the yellow cell. Esentially \(C_2\) acts by mirroring the \(x\)-axis as seen in Figure 2.

Figure 2: The action of \(C_2\) on \(\mathbb{T}\)

This gives us the following pushout diagram as in Theorem 3.2 (see H. Kammeyer 2019):

\[ \begin{CD} \coprod_{i\in I_n} G/H_i \times S^{n-1} @>q_n>> X_{n-1} \\ @Vi_nVV @Vj_nVV\\ \coprod_{i\in I_n} G/H_i \times D^n @>Q_n>> X_n \end{CD} \]

This pushout diagram just tells us that the torus is constructed by attaching \(G\)-equivant cells. Here \(q_n\) is the map, that attaches the boundary of \(n\)-disks to our \((n-1)\)-Skeleton. \(i_n\) and \(j_n\) are the respective inclusion maps. \(Q_n\) is the attachment map determining how the \(n\)-cells are glued into the \(n\)-skeleton. Let us understand what \(I_n\) and \(H_i\) are next. For that, we first label all our \(n\) cells. In this example, we only really care for the \(n=2\) case, so let us do that.

Figure 3: Labeling of the 2-cells of the torus

Labeling each \(n\)-cell gives us an index set \(J_n := \{\cellOne, \cellTwo, \cellThree, \cellFour\}\). By cayleys theorem, we know that \(G\) is isomorphic to a subgroup of the symmetric group \(S_4\) and as such permutes the index set \(J_n\), which means we can take a look at the orbit set \(I_n := G\backslash J_n = \{\cellOne,\cellThree\}, \{\cellTwo,\cellFour\}\). Well, why would we even want to look at this orbit set? The answer is, that this reduces the number of unique \(n\)-cells by taking into account, that we can access all \(n\)-cells in the same orbit just by applying our group action!

The action of \(C_2 = \langle g \rangle\) on the 2-cells of the torus

Representative of the orbit \(\{\cellOne,\cellThree\}\).

Representative of the orbit \(\{\cellTwo,\cellFour\}\).

How can we reconstruct our space from this information now? Well, each of the representatives is just a copy of \(D^2\), a \(n\)-disk. If we were to multiply this by \(G\), then we might end up with too many \(n\)-disks. If, for example, we would have chosen \(G=V_4=\langle r, s\rangle\) to be the Klein four group, and we would let \(\langle r \rangle \cong C_2\) act just as before and \(\langle s\rangle\) trivially, then by \(V_4 \cellTwo = \{\cellTwo, \cellFour\}\), but \(V_4 \times D^2 \cong \coprod^4 D^2\) and we would end up with four \(2\)-disks instead of the two we would have hoped for. To avoid this, we need to take the quotient of \(G\) acting on \(D^2\) by the stabilizer \(H_k\) of a representative of the orbit \(k\). In our example with \(V_4\), we have \(H_{\{\cellOne, \cellThree\}} = H_{\{\cellTwo, \cellFour\}} = \langle s\rangle\) and therefor \(V_4/H_{\{\cellOne, \cellThree\}} \times D^2 \cong \coprod^2 D^2\). This is the reason, why we have the orbit set \(I_n\) and the stabilizer groups \(H_i\) in our pushout diagram.

So, we really only need one one \(n\)-cell for each \(G\)-orbit. If we were to do that, we will end up with a cellular basis of our \(G\)-CW complex. In our example with \(G=C_2\), this would be the case if we were to only have two \(2\)-cells: \(\cellOne\cup\cellTwo\) and \(\cellThree\cup\cellFour\). This yields a \(\Z G\)-isomorphism \(\bigoplus_{i\in I_n} \Z(G/H_i) \cong C_*(X)\) (see H. Kammeyer 2019, Proposition 3.6).

The \(\ell^2\)-Completion

Finally we can take a look at the \(\ell^2\)-chain completion of \(C_*(X)\), which is defined as \(\ell^2 G \otimes_{\Z G} C_*(X)\). The \(\ell^2\)-completion of a \(\Z G\)-module \(M\) is defined as \(\ell^2G \otimes_{\Z G} M\), which is functorial and extends itself to the \(\ell^2\)-chain completion of a \(G\)-CW complex \(X\). Here, it is important that \(C_*\) is functorial also, and as such \(\ell^2 G \otimes_{\Z G} C_*(X)\) is functorial as well. This in turn is important, because it means, that the differentials \(d_*^{(2)}\) are given as \(\id \otimes d_*\).

Applications

Let us now take a look at some applications of the \(\ell^2\)-completion of cellular chain complexes. We will start with the \(\ell^2\)-homology groups and the \(\ell^2\)-betti numbers of \(S^\infty := \colim_n S^n\) with the group \(G:=\prod_{n=0}^\infty \Z/2\Z\) acting on it. It can easily be seen that \(S^\infty\) is a \(G\)-CW complex, where \(g = (g_1, g_2, \dots ) \in G\) acts on \(S^\infty\) by letting \(g_i\in \Z/2\Z\) permute the two \(i\)-cells.

Figure 4: \(\colim_n S^n\). Here it is easy to see how \(G\) acts on \(S^\infty\)

By following the familiar pattern of defining homology using our differentials, we can define \(\ell^2\)-homology the same way:

\[ H_n^{(2)}(G\curvearrowright S^\infty) := \ker d_n^{(2)} / \im d_{n+1}^{(2)} \]

We wish to use proposition 3.6 (see H. Kammeyer 2019) to calculate the \(\ell^2\)-homology groups of \(S^\infty\). For that, we need to find a cellular basis of \(S^\infty\). We can do that by taking a look at the orbit set \(I_n = G\backslash J_n\) and the stabilizer groups \(H_i\). In this case, we have \(J_n = \{\cellOne_n, \cellTwo_n\}\) and \(I_n = \{\{\cellOne_n, \cellTwo_n\}\}\), where \(\cellOne_n\) and \(\cellTwo_n\) are the two \(n\)-cells of \(S^\infty\). For each \(n\), we therefor only have one orbit. This also tells us what our stabilizer groups are: \(H_{\{\cellOne_n, \cellTwo_n\}} = \prod_{k=1}^{n-1} \Z / 2\Z \times \{e\} \times \prod_{k=n+1}^\infty \Z / 2\Z\). One can see this visually quite well: We mirror the \(n\)-th basis vector of \(\R^\infty\) using the \(n\)-th component of \(G\) and leave the other components untouched. Now, we have two \(n\)-cells for each basis vector and these are permuted exactly by the \(n\)-th component of \(G\). All other components of \(G\) leave the \(n\)-th basis vector untouched and therefor also the \(n\)-th \(n\)-cell. Thus, the stabilizer group of \(G\) that leaves the \(n\)-cells untouched is exactly the subgroup of \(G\) that leaves the \(n\)-th basis vector untouched.

By proposition 3.6, we now have \(C_n(S^\infty) \cong \bigoplus_{i\in I_n} \Z(G/H_i) = \Z ( \prod_{k=1}^\infty ( \Z / 2\Z ) / ( \prod_{k=1}^{n-1} \Z / 2\Z \times \{e\} \times \prod_{k=n+1}^\infty \Z / 2\Z ) ) \cong \Z(\Z / 2\Z )\). Now, we only need to calculate the \(\ell^2\)-completion by taking the tensor product with \(\ell^2 G\). We thus get by proposition 3.7 (see H. Kammeyer 2019) that \(\ell^2 G \otimes_{\Z G} C_n(S^\infty) \cong \ell^2 G \otimes_{\Z G} \Z(\Z / 2\Z ) \cong \ell^2(\Z / 2\Z ) \cong \C[\iota]/(\iota^2-1)\). The differentials \(d_n^{(2)}\) are given as \(\id \otimes d_n\). For \(n>0\) we know, that \(d_n\) cannot be surjective, because \(d_n(\cellOne_n)\) must be one of \(\pm\cellOne_{n-1}+\cellTwo_{n-1}\) or \(\pm\cellOne_{n-1}-\cellTwo_{n-1}\) and the same holds for \(d_n(\cellTwo_n)\). No matter which of these options holds for \(\cellOne_n\) and \(\cellTwo_n\), we have that \(\cellOne_{n-1}, \cellTwo_{n-1} \notin \im d_n\). For the sake of convenience, we may choose the CW structure of \(S^\infty\) such that \(d_n(\cellOne_n) = d_n(\cellTwo_n) = -\cellOne_{n-1}+\cellTwo_{n-1}\) for \(n\ge 1\).

Show the code
import plotly.graph_objects as go

def get_frame_line1(t):
    t = t if t <= 1 else 1
    t = t if t >= 0 else 0
    x = -np.sin(np.linspace(-np.pi,t*np.pi-np.pi,100))
    y = 0*np.linspace(-np.pi,t*np.pi-np.pi,100)
    z = np.cos(np.linspace(-np.pi,t*np.pi-np.pi,100))
    return x, y, z

def get_frame_line2(t):
    t = t if t <= 1 else 1
    t = t if t >= 0 else 0
    x = np.sin(np.linspace(-np.pi,t*np.pi-np.pi,100))
    y = 0*np.linspace(-np.pi,t*np.pi-np.pi,100)
    z = np.cos(np.linspace(-np.pi,t*np.pi-np.pi,100))
    return x, y, z

def get_frame_surface1(t):
    t = t if t <= 1 else 1
    t = t if t >= 0 else 0
    if t == 0:
        x = 0*np.linspace(-1, 1, 100)
        z = 0*np.linspace(-1, 1, 100)
        x, z = np.meshgrid(x, z)
        y = 0 * x
        return x, y, z
    # slice of a disk that slowly increases its angle
    # kinda like a pizza slice
    # top is (0,0,1), bottom is (0,0,0)
    # grows along the surface spanned by (1,0,0) and (0,0,1)

    # Define the radial and angular grids
    theta = np.linspace(-t*np.pi*2 -np.pi/2, -np.pi/2, 100)
    phi = np.linspace(0, np.pi / 2, 100)
    theta, phi = np.meshgrid(theta, phi)

    # Spherical to Cartesian coordinates conversion
    x = np.sin(phi) * np.cos(theta)
    y = np.cos(phi)
    z = np.sin(phi) * np.sin(theta)
    return x, y, z


def get_frame_surface2(t):
    x,y,z = get_frame_surface1(t)
    y = -y
    return x,y,z

# Generate data for each frame
ts = np.linspace(0, 1, 100)
line_df = pd.DataFrame()
surface_df = pd.DataFrame()

for t in ts:
    x1, y1, z1 = get_frame_line1(2*t)
    x2, y2, z2 = get_frame_line2(2*t)
    x3, y3, z3 = get_frame_surface1(4*(t-0.5))
    x4, y4, z4 = get_frame_surface2(4*(t-0.75))

    temp_line_df = pd.DataFrame({
        'x1': x1, 'y1': y1, 'z1': z1,
        'x2': x2, 'y2': y2, 'z2': z2,
        't': t})
    line_df = pd.concat([line_df, temp_line_df])

    temp_surface_df = pd.DataFrame({
        'x3': x3.flatten(), 'y3': y3.flatten(), 'z3': z3.flatten(),
        'x4': x4.flatten(), 'y4': y4.flatten(), 'z4': z4.flatten(),
        't': [t] * 10000})  # 10000 = 100x100
    surface_df = pd.concat([surface_df, temp_surface_df])

frames = []
for t in ts:
    line_row = line_df[line_df['t'] == t]
    surface_row = surface_df[surface_df['t'] == t]
    frames.append(go.Frame(
        data=[
            go.Scatter3d(x=line_row['x1'], y=line_row['y1'], z=line_row['z1'], mode='lines', line=dict(color='green', width=2)),
            go.Scatter3d(x=line_row['x2'], y=line_row['y2'], z=line_row['z2'], mode='lines', line=dict(color='red', width=2)),
            go.Surface(x=surface_row['x3'].values.reshape(100, 100), y=surface_row['y3'].values.reshape(100, 100), z=surface_row['z3'].values.reshape(100, 100), colorscale='reds', showscale=False),
            go.Surface(x=surface_row['x4'].values.reshape(100, 100), y=surface_row['y4'].values.reshape(100, 100), z=surface_row['z4'].values.reshape(100, 100), colorscale='greens', showscale=False)
        ],
        name=str(t)
    ))

# Create a figure using plotly.graph_objects
fig = go.Figure(
    layout=go.Layout(
        updatemenus=[dict(
            type="buttons",
            buttons=[dict(label="Play",
                          method="animate",
                          args=[None, {"frame": {"duration": 75, "redraw": True},
                                       "fromcurrent": True}]),
                    dict(label="Pause",
                         method="animate",
                         args=[[None], {"frame": {"duration": 0, "redraw": False},
                                        "mode": "immediate",
                                        "transition": {"duration": 0}}])],
            
        )],
        sliders=[
            dict(
                active=0,
                steps=[dict(method="animate",
                            args=[[f"{t}"], {"frame": {"duration": 75, "redraw": True},
                                             "mode": "immediate",
                                             "transition": {"duration": 0}}]) for t in ts]
            )
        ]
    ),
    frames=frames
)

# Add initial data
fig.add_trace(go.Scatter3d(
    x=line_df['x1'].iloc[0:100], y=line_df['y1'].iloc[0:100], z=line_df['z1'].iloc[0:100],
    mode='lines', line=dict(color='green', width=2)))
fig.add_trace(go.Scatter3d(
    x=line_df['x2'].iloc[0:100], y=line_df['y2'].iloc[0:100], z=line_df['z2'].iloc[0:100],
    mode='lines', line=dict(color='red', width=2)))
# surface plots
fig.add_trace(go.Surface(
    x=surface_df['x3'].iloc[0:10000].values.reshape(100, 100),
    y=surface_df['y3'].iloc[0:10000].values.reshape(100, 100),
    z=surface_df['z3'].iloc[0:10000].values.reshape(100, 100),
    colorscale='reds', showscale=False))
fig.add_trace(go.Surface(
    x=surface_df['x4'].iloc[0:10000].values.reshape(100, 100),
    y=surface_df['y4'].iloc[0:10000].values.reshape(100, 100),
    z=surface_df['z4'].iloc[0:10000].values.reshape(100, 100),
    colorscale='greens', showscale=False))

# Layout configuration
fig.update_layout(
    scene=dict(
        xaxis=dict(range=[-1, 1], autorange=False),
        yaxis=dict(range=[-1, 1], autorange=False),
        zaxis=dict(range=[-1, 1], autorange=False),
    ),
    title_text="Attachment maps for the 1 and 2 cells",
)

# Show the figure
fig.show()