%!TEX root = ../dissertation.tex



\section{Atiyah-Hirzebruch decomposition of general crystalline SPT phases}
\label{sec:Atiyah-Hirzebruch}

In this section we will consider crystalline symmetries $G$ and the classification of crystalline SPT phases in general. We will use a homological formulation that we suspect is equivalent in some sense to the cohomological formulation of Chapter \ref{chap:minimalist} by some sort of Poincar\'e duality. Furthermore, we will allow for boundaries in a physical system by considering \emph{relative} generalized homology. Throughout the section, we assume $G$ is purely spatial. However, our discussions extend immediately to direct products $G' \times G$ for any $G'$ that acts internally.

\subsection{Relative SPT phenomena}
\label{sec:genehomo}

We will speak of \emph{SPT phenomena of degree $n$} for integer $n$, which we loosely define as follows. SPT phenomena of degree $0$ are short-range entangled states. SPT phenomena of degree $-1$ are anomalies of short-range entangled states. SPT phenomena of degree $-2$ are boundaries of anomalies of short-range entangled states, i.e.\,anomalies of anomalies. Inductively, we define SPT phenomena of degree $n$ for $n < 0$ as anomalies of SPT phenomena of degree $n + 1$. As for positive degrees, we define SPT phenomena of degree $1$ to be loops of short-range entangled states. We define SPT phenomena of degree $2$ to be loops of loops of short-range entangled states, i.e.\, families of short-range entangled states parameterized by the 2-sphere $\SS^2$. In general, we define SPT phenomena of degree $n$ for $n > 0$ to be families of short-range entangled states parameterized by $\SS^n$. We will not make it precise what the terms ``anomalies" or ``anomalies of anomalies" mean, but only demand that they satisfy certain constraints described below.\footnote{There may be a clear meaning to anomalies themselves, but we will need to know the topology of the space of anomalies, and for one thing, there is the question of whether a deformation means a deformation of anomaly itself or a deformation of the parent short-range entangled state.}

Now, let $G$ be a discrete group, $X$ be a $G$-CW complex, and $Y$ be a $G$-CW subcomplex. Let us consider the classification of deformation classes of short-range entangled states on $X$ that respect $G$ that can have anomalies on $Y$. We will denote this by
\begin{equation}
\SPT_0\paren{X, Y}, \label{SPT_0(X,Y)}
\end{equation}
and refer to it as the classification of \emph{degree-$0$ SPT phenomena on $X$ relative $Y$}. For example, if $X$ is a manifold with boundary and $Y$ is its boundary, then we can put any SPT phase on the entire $X$. By contrast, if $X$ is a 2-sphere and $Y$ is its equator, then we can put any SPT phase on one hemisphere of $X$ and an unrelated SPT phase on the other hemisphere. Note that, unlike in the notation $\SPT^d\paren{G, \phi}$ where symmetry and dimensionality are explicit, in Eq.\,(\ref{SPT_0(X,Y)}) they are implicit in $X$. We posit that $\SPT_0\paren{X, Y}$ can be given by the relative equivariant generalized homology group
\begin{equation}
\SPT_0(X, Y) \isomorphic h^G_0\paren{X, Y}, \label{homology_0}
\end{equation}
where $h^G_\bullet$ is the \emph{equivariant generalized homology theory} \cite{Whi,KonoTamaki} defined by the $\Omega$-spectrum of Chapter \ref{chap:minimalist}.

We can generalize this to other degrees. We will denote by
\begin{equation}
\SPT_n\paren{X, Y} \label{SPT_n(X,Y)}
\end{equation}
the classification of deformation classes of SPT phenomena of degree $n$ on $X$ that respect $G$ that can have SPT phenomena of degree $n-1$ on $Y$, or more succinctly, the classification of \emph{degree-$n$ SPT phenomena on $X$ relative $Y$}. We posit that this, in turn, can be given by the $n$th relative equivariant generalized homology group
\begin{equation}
\SPT_n(X, Y) \isomorphic h^G_n\paren{X, Y}. \label{homology_n}
\end{equation}

As mentioned earlier, we did not make precise the meaning of anomalies, anomalies of anomalies, etc., and thus the meaning of the classification of degree-$n$ SPT phenomena on $X$ relative $Y$ is accordingly vague. However, in order for Eqs.\,(\ref{homology_0})(\ref{homology_n}) to make sense, the meaning of these terms must satisfy the following constraint. Consider the case of trivial $G$, and let $X$ be the $d$-dimensional disk $D^d$ and $Y$ be boundary $\partial D^d$. If we write $\tilde h_\bullet$ and $\tilde h^\bullet$ for the \emph{reduced} version of $h_\bullet$ and $h^\bullet$, respectively, then we have \cite{Adams2}
\begin{equation}
h_n\paren{D^d, \partial D^d} = \tilde h_n\paren{\SS^d} \isomorphic \tilde h^{d-n}\paren{\SS^0} = h^{d-n}\paren{\pt} \isomorphic \SPT^{d-n}\paren{0},
\end{equation}
the classification of $(d-n)$-dimensional SPT phases without symmetry. In particular,
\begin{equation}
h_0\paren{D^d, \partial D^d} \isomorphic h^d\paren{\pt} \isomorphic \SPT^d\paren{0},
\end{equation}
the classification of $d$-dimensional SPT phases without symmetry. Therefore, the meaning of degree-$n$ SPT phenomena and the meaning of degree-$n$ SPT phenomena on $X$ relative $Y$ must be such that, for any integer $d$ and $n$, the classification of $(d-n)$-dimensional SPT phases without symmetry is exactly the same as the classification of degree-$n$ SPT phenomena on $D^d$ relative $\partial D^d$. For a few special cases,
\begin{itemize}
\item $n = 0$: the classification of $d$-dimensional SPT phases without symmetry must be the same as the classification of SPT phases on $D^d$ that can have anomalies on $\partial D^d$. This makes sense as these seem to correspond the same phase on an infinite system in one case and a finite system with boundary in the other.

\item $n = -1$: the classification of $d$-dimensional SPT phases without symmetry must be the same as the classification of anomalies on $D^{d-1}$ that can have anomalies of anomalies on $\partial D^{d-1}$. One can envision a $d$-dimensional SPT phase living on the $d$-dimensional ball $D^d$, and think of $D^{d-1}$ as the northern hemisphere of the ball and $\partial D^{d-1}$ as the equator. The idea is that the information of the SPT phase on $D^d$ ought to be encoded in the anomaly on its surface $\SS^{d-1}$, and knowing one hemisphere of $\SS^{d-1}$ ought to be as good as knowing the entire $\SS^{d-1}$.

\item $n = -2$: the classification of $d$-dimensional SPT phases without symmetry must be the same as the classification of anomalies of anomalies on $D^{d-2}$ that can have anomalies of anomalies of anomalies on $\partial D^{d-2}$. Similarly to the previous case, the idea is that knowing one half of the equator of $\SS^{d-1}$ ought to be as good as knowing the entire equator, which ought to be as good as knowing one hemisphere of $\SS^{d-1}$, which ought to be as good as knowing the entire sphere $\SS^{d-1}$, which ought to be as good as knowing an SPT phase on $D^d$ itself.

\item $n = 1$: the classification of $d$-dimensional SPT phases without symmetry must be the same as the classification of loops of short-range entangled states on $D^{d+1}$ that can pump SPT phases to $\partial D^{d+1}$. This makes sense according to the interpretation of $\Omega$-spectrum in Sec.\,\ref{subsubsec:pumping}.

\item $n = 2$: the classification of $d$-dimensional SPT phases without symmetry must be the same as the classification of loops of loops of short-range entangled states on $D^{d+2}$ that can pump loops of short-range entangled states to $\partial D^{d+2}$. This similarly makes sense according to Sec.\,\ref{subsubsec:pumping}.
\end{itemize}

In the more general case of nontrivial $G$ or $\paren{X, Y} \neq \paren{D^d, \partial D^d}$, we suspect, by some more general form of Poincar\'e duality, that it might be possible to write
\begin{equation}
h^{\phi + d}_G \paren{\pt} \label{co_formulation}
\end{equation}
as
\begin{equation}
h^G_0\paren{X, Y} \label{ho_formulation}
\end{equation}
for some $G$-CW complex $X$ and $G$-CW subcomplex $Y$. Note that, unlike Eq.\,(\ref{co_formulation}), which is twisted if $G$ does not preserve the spatial orientation, Eq.\,(\ref{ho_formulation}) is not twisted in any case.





\subsection{Bulk-boundary correspondence}
\label{subsec:bulk-boundary-correspondence}

Every generalized equivariant homology theory admits a long exact sequence of a pair:
\begin{equation}
\cdots \fromto h^G_{n+1}\paren{X, Y} \xfromto{\partial} h^G_n\paren{Y} \xfromto{\alpha} h^G_n\paren{X} \xfromto{\beta} h^G_n\paren{X, Y} \xfromto{\partial} h^G_{n-1}\paren{Y} \fromto \cdots
\end{equation}
Here, $\alpha$ and $\beta$ are induced by the inclusions $Y \fromto X$ and $(X, \emptyset) \fromto (X, Y)$, respectively. In the spirit of the functoriality of Chapter \ref{chap:minimalist}, we shall interpret these as physically being the embedding maps. To wit, we interpret $\alpha$ as embedding an SPT phenomena on $Y$ into $X$, and $\beta$ as reinterpreting an SPT phenomena on $X$ as an SPT phenomena on $X$ relative $Y$ (with the trivial anomaly on $Y$). Equivalently, for $\beta$, we can cut open the SPT phenomena on $X$ along $Y$ and define the resulting state representing as the element of $h^G_n\paren{X, Y}$ that $\beta$ maps to.

As for $\partial: h^G_n\paren{X, Y} \fromto h^G_{n-1}\paren{Y}$, we interpret it as the anomaly extraction map. Namely, if $[a]$ is a degree-$n$ SPT phenomena on $X$ relative $Y$, then $\partial\paren{[a]}$ will be the boundary anomaly of $[a]$. This is a form of the bulk-boundary correspondence.\footnote{In order for the interpretation of $h^G_n\paren{X} \coloneq h^G_n\paren{X, \emptyset}$ [resp.\,$h^G_n\paren{Y} \coloneq h^G_n\paren{Y, \emptyset}$] as degree-$n$ SPT phenomena on $X$ (resp.\,$Y$) relative $\emptyset$ to be meaningful, $X$ (resp.\,$Y$) must not have a ``boundary" itself. For example, $X$ could be a manifold without boundary and $Y$ could be a submanifold of $X$ without boundary. If either $X$ or $Y$ has a ``boundary," then one must clarify what $h^G_n\paren{X}$ or $h^G_n\paren{Y}$ represents physically.}

Every generalized equivariant homology theory also admits a long exact sequence of a triple $\paren{X, Y, Z}$:
\begin{equation}
\cdots \fromto h^G_{n+1}\paren{X, Y} \xfromto{\partial} h^G_n\paren{Y, Z} \xfromto{\alpha} h^G_n\paren{X, Z} \xfromto{\beta} h^G_n\paren{X, Y} \xfromto{\partial} h^G_{n-1}\paren{Y, Z} \fromto \cdots
\end{equation}
This can be interpreted in an analogous way.\footnote{In order for the physical interpretation of all terms to be meaningful, one must either ensure $Z$ contains the ``boundaries" of $X$ and $Y$ or else clarify the meaning of the terms.}








\subsection{Atiyah-Hirzebruch spectral sequence}
\label{sec:ahss}

The Atiyah-Hirzebruch spectral sequence is a tool for computing generalized homology and cohomology groups. Here we are interested in the relative equivariant generalized homology group $h^G_n(X,Y)$. The general structure of the Atiyah-Hirzebruch spectral sequence is as follows. We have abelian groups $E^r_{p, q}$ indexed by a positive integer $r$ (called \emph{page}), along with integers $p$ and $q$, which provide a double grading on $E^r$. There are doubly graded homomorphisms called \emph{differentials}:
\begin{equation}
d^r_{p, q}: E^r_{p, q} \fromto E^r_{p - r, q + r - 1}.
\end{equation}
In general, either the $E^1$-page or the $E^2$-page is given, and one defines higher pages inductively by
\begin{equation}
E^{r+1}_{p, q} \coloneq \kernel \paren{d^r_{p, q}} / \image \paren{d^r_{p + r, q - r + 1}}.
\end{equation}
That is, $E^{r+1}$ is the homology of $E^r$ with respect to the differential $d^r$. If there exists some non-negative integer $d$ such that $E^r_{p, q} = 0$ for all $p < 0$ or $p > d$, then the spectral sequence stabilizes; that is, $E^r = E^{r+1}$ for large enough $r$. Writing $E^\infty \coloneq \lim_{r\fromto \infty} E^r$, we then have a filtration
\begin{equation}
0 \eqcolon F_{-1} h^G_n(X,Y) \subset F_0 h^G_n(X,Y) \subset F_1 h^G_n(X,Y) \subset \cdots \subset F_d h^G_n(X,Y) \coloneq h^G_n(X,Y)
\end{equation}
such that
\begin{equation}
F_{p} h^G_n(X,Y) / F_{p - 1} h^G_n(X,Y) \isomorphic E^\infty_{p, n - p}.
\end{equation}
Equivalently, we have abelian group extensions
\begin{equation}
0 \fromto F_{p - 1} h^G_n(X,Y) \fromto F_{p} h^G_n(X,Y) \fromto E^\infty_{p, n - p} \fromto 0.
\end{equation}
By solving these abelian group extensions, one can determine $h^G_n(X,Y)$ itself.

In what follows we will present an Atiyah-Hirzebruch spectral sequence for $h^G_n(X,Y)$ and interpret it in terms of crystalline SPT phenomena. We will see that the first differentials underlie the constructions of Refs.\,\cite{Hermele_torsor, Huang_dimensional_reduction, TB18}. However, there are also higher differentials, and these have not been previously studied. The structure of a filtration of SPT phases in the presence of crystalline symmetries and its relation to higher-order SPT phases (see Sec.~IV F 2 of our work \cite{Shiozaki2018}) have been pointed out previously in Ref.\,\cite{TB18}.





\subsubsection{\texorpdfstring{$E^1$}{E1}-page}
\label{sec:e1}

Let us assume $X$ is finite-dimensional and let $d$ be its dimension. We denote by $I_p$ the set of inequivalent $p$-cells of $X$ that are not in $Y$. If $D_j^p$ is a $p$-cell, we denote by $G_{D_j^p}$ its little group, i.e.\,the group of elements of $G$ that fix any and hence all points in the interior of $D^p_j$. We shall consider the skeletons of $X$,
\begin{align}
X_0 \subset X_1 \subset \cdots \subset X_d = X, \label{skeleton}
\end{align}
where $X_p$ is the subcomplex made of all cells of dimension $\leq p$.

The Atiyah-Hirzebruch spectral sequence for $h^G_n(X,Y)$ associated to the filtration (\ref{skeleton}) has the following $E^1$-page:
\begin{align}
E^1_{p, q} 
= h_{p + q}^G(X_p \cup Y, X_{p-1} \cup Y).
\end{align}
Using the cellular structure of $X$ and $Y$, we can rewrite this as
\begin{equation}\begin{split}
E^1_{p, q} 
= \bigoplus_{j\in I_p} h^{G_{D^p_j}}_{p + q}(D^p_j,\p D^p_j)
= \bigoplus_{j \in I_p} h^{- q}_{G_{D^p_j}}(\pt).
\end{split}\end{equation}
The summand
\begin{equation}
h^{- q}_{G_{D^p_j}}(\pt) \isomorphic \SPT^{-q} \paren{ G_{D^p_j} }
\end{equation}
can be interpreted as ``$(-q)$-dimensional SPT phases on the $p$-cell $D^p_j$ with unitary internal symmetry $G_{D^p_j}$." The meaning of this when $p + q \neq 0$ as well as when $p + q = 0$ will be explained in subsequent sections. The $E^1$-page describes the local data of crystalline SPT phenomena. The correspondence between $E^1_{p, q}$ and SPT phases is summarized in Table \ref{table:E1}.

{\small
\begin{spacing}{\dnormalspacing}
\begin{longtable}[c]{c|ccccc}
\caption[Interpretation of $E^1$-page of Atiyah-Hirzebruch spectral sequence.]{The interpretation of the $E^1$-page of the Atiyah-Hirzebruch spectral sequence.\label{table:E1}} \\
\endfirsthead
\caption[]{(Continued).} \\
\endhead
\endfoot
$\vdots$ & $\vdots$                                & $\vdots$                                & ~           & $\vdots$ \\
$q=2$    & (-2)D SPT on $0$-cells & (-2)D SPT on $1$-cells & $\cdots$ & (-2)D SPT on $d$-cells \\
$q=1$    & (-1)D SPT on $0$-cells & (-1)D SPT on $1$-cells & $\cdots$ & (-1)D SPT on $d$-cells \\
$q=0$    & 0D SPT on $0$-cells     & 0D SPT on $1$-cells    & $\cdots$ & 0D SPT on $d$-cells \\
$q=-1$   & 1D SPT on $0$-cells     & 1D SPT on $1$-cells    & $\cdots$ & 1D SPT on $d$-cells \\
$q=-2$   & 2D SPT on $0$-cells     & 2D SPT on $1$-cells    & $\cdots$ & 2D SPT on $d$-cells \\
$\vdots$ & $\vdots$                                & $\vdots$                                & ~           & $\vdots$ \\
\nobreakhline 
$E^1_{p,q}$ & $p=0$                            & $p=1$                                   & $\cdots$  & $p=d$
\end{longtable}
\end{spacing}
}






\subsubsection{First differentials}
\label{sec:d1}

The first differentials
\begin{equation}
d^1_{p, q}: E^1_{p, q} \fromto E^1_{p - 1, q}
\end{equation}
map
\begin{equation}
d^1_{p, q}: \mbox{``$(-q)$D SPT phases on $p$-cells"} \fromto \mbox{``$(-q)$D SPT phases on $(p-1)$-cells"}.
\end{equation}
They relate the $E^1$- and $E^2$-pages via
\begin{align}
E^2_{p, q}:= \kernel \paren{d^1_{p, q}} / \image \paren{d^1_{p + 1, q}}.
\end{align}
This is the first step towards converging to $E^\infty$. Let us examine the physical meaning of $d^1_{p, q}$ in low degrees.

\paragraph{$d^1_{1, 0}$: ``0D SPT phases on 1-cells" $\fromto$ ``0D SPT phases on 0-cells"}

This is the process of creating a $0$D SPT phase and its conjugate in the middle of a $1$-cell and pumping them to the two ends of the $1$-cell. This leads to the equivalence relation $E^1_{0, 0} / \image \paren{d^1_{1, 0}}$ among $0$D SPT phases on $0$-cells.

\paragraph{$d^1_{2, -1}$: ``1D SPT phases on 2-cells" $\fromto$ ``1D SPT phases on 1-cells"}

This is the process of creating a $1$D SPT phase that lives on a small circle in the interior of a $2$-cell and expanding it to the edge. This leads to the equivalence relation $E^1_{1, -1} / \image \paren{d^1_{1, 0}}$ among $1$D SPT phases on $1$-cells.

\paragraph{$d^1_{1, -1}$: ``1D SPT phases on 1-cells" $\fromto$ ``1D SPT phases on 0-cells"}

This is the map that extracts the edge anomaly of a $1$D SPT phase that lives on a $1$-cell. This expresses the gluing condition $\kernel \paren{d^1_{1, -1}} \subset E^1_{1, -1}$ of $1$D SPT phases on $1$-cells.

\paragraph{$d^1_{2, -2}$: ``2D SPT phases on 2-cells" $\fromto$ ``2D SPT phases on 1-cells"}

This is the map that extracts the boundary anomaly of a $2$D SPT phase that lives on a $2$-cell. This expresses the gluing condition $\kernel \paren{d^1_{2, -2}} \subset E^1_{2, -2}$ of $2$D SPT phases on $2$-cells.

\paragraph{$d^1_{1, -2}$: ``2D SPT phases on 1-cells" $\fromto$ ``2D SPT phases on 0-cells"}

Given an SPT anomaly that lives on a $1$-cell, this is the map that extracts its boundary. In the classification of SPT anomalies on $X$, this differential expresses the gluing condition $\kernel \paren{d^1_{1, -2}} \subset E^1_{1, -2}$ of SPT anomalies on $1$-cells.

We thus see that the first differentials are the mathematical structure behind Refs.\,\cite{Hermele_torsor, Huang_dimensional_reduction, TB18}.





\subsubsection{Higher differentials}
\label{subsubsec:higher_differentials}

The higher differentials
\begin{equation}
d^r_{p, q}: E^r_{p, q} \fromto E^1_{p - r, q + r - 1}
\end{equation}
map
\begin{equation}
d^r_{p, q}: \mbox{``$(-q)$D SPT phases on $p$-cells"} \fromto \mbox{``$(- q - r + 1)$D SPT phases on $(p - r)$-cells"}.
\end{equation}
They relate the $E^r$- and $E^{r+1}$-pages via
\begin{align}
E^{r+1}_{p, q}:= \kernel \paren{d^r_{p, q}} / \image \paren{d^r_{p + r, q - r + 1}}.
\end{align}
Let us explore the physical meaning of these higher differentials.

\paragraph{$d^2_{2, -1}$: ``1D SPT phases on 2-cells" $\fromto$ ``0D SPT phases on 0-cells"}

This is the process of creating $1$D SPT phases on small circles in $2$-cells, expanding them to the boundaries of the $2$-cells, canceling them on the bounding $1$-cells, but possibly leaving behind some $0$D SPT phases on $0$-cells that border the $2$-cells. This leads to the equivalence relation $E^2_{0, 0} / \image\paren{d^2_{2, -1}}$ among $0$D SPT phases on $0$-cells.

Note that any element in the domain $E^2_{2, -1}$ of $d^2_{2, -1}$ must be representable by an element in the kernel $\kernel\paren{d^1_{2, -1}}$ of $d^1_{2, -1}$. Since $d^1_{2, -1}$ maps $1$D SPT phases on $2$-cells to $1$D SPT on $1$-cells, the subset $\kernel\paren{d^1_{2, -1}} \subset E^1_{2, -1}$ must represent those configurations of $1$D SPT phases on $2$-cells that satisfy the gluing condition along $1$-cells. Thus it is indeed possible to cancel elements of $E^2_{2, -1}$ on the $1$-cells as we proposed to do in the previous paragraph.

See Sec.~V B 2 of our work \cite{Shiozaki2018} for a nontrivial example of $d^2_{2,-1}$. 

\paragraph{$d^2_{2, -2}$: ``2D SPT phases on 2-cells" $\fromto$ ``1D SPT phases on 0-cells"}

Given a configuration of $2$D SPT phases on $2$-cells, this map extracts the possible SPT anomaly left behind on $0$-cells while gluing the them along the bounding $1$-cells. This gives the higher gluing condition $\kernel\paren{d^2_{2, -2}} \subset E^2_{2, -2}$.

Note that any element of $E^2_{2, -2}$ must be representable by an element of $\kernel \paren{d^1_{2, -2}} \subset E^1_{2, -2}$. Since $d^1_{2, -2}$ maps $2$D SPT phases on $2$-cells to $2$D SPT phases on $1$-cells, the subset $\kernel \paren{d^1_{2, -2}} \subset E^1_{2, -2}$ must consist of configurations that satisfy the gluing condition along $1$-cells. Thus it is indeed possible to glue elements of $E^2_{2, -2}$ along the $1$-cells as we proposed to do in the previous paragraph.

See Secs.~V B 2, V G and V H 1 of our work \cite{Shiozaki2018} for nontrivial examples of $d^2_{2,-2}$. 

\paragraph{$d^3_{3, -2}$: ``2D SPT phases on 3-cells" $\fromto$ ``0D SPT phases on 0-cells"}

This is the process of creating $2$D SPT phases on small spheres in $3$-cells, expanding them to the boundaries of the $3$-cells, canceling them on the the bounding $2$-cells and $1$-cells, but possibly leaving behind some $0$D SPT phases on $0$-cells that border the $3$-cells. This leads to the equivalence relation $E^3_{0, 0} / \image\paren{d^3_{3, -2}}$ among $0$D SPT phases on $0$-cells.

Note that any element of $E^3_{3, -2}$ must be representable by an element of $\kernel \paren{d^2_{3, -2}} \subset E^2_{3, -2}$ and an element of $\kernel \paren{d^1_{3, -2}} \subset E^1_{3, -2}$. Since $d^1_{3, -2}$ maps $2$D SPT phases on $3$-cells to $2$D SPT phases on $2$-cells, an element of $\kernel d^1_{3, -2} \subset E^1_{3, -2}$ must represent a configuration that satisfies the gluing condition along $2$-cells. Similarly, since $d^2_{3, -2}$ maps $2$D SPT phases on $3$-cells to $1$D SPT on $1$-cells, an element of $\kernel\paren{d^1_{2, -1}} \subset E^1_{2, -1}$ must represent a configuration that satisfies the gluing condition along $1$-cells. Thus it is indeed possible to cancel elements of $E^e_{3, -2}$ along the $2$-cells and $1$-cells as we proposed to do in the previous paragraph.

See Sec.~VI G of our work \cite{Shiozaki2018} for an example of nontrivial $d^2_{3,-2}$. 

\paragraph{$d^3_{3, -3}$: ``3D SPT phases on 3-cells" $\fromto$ ``1D SPT phases on 0-cells"}

Given a configuration of $3$D SPT phases on $3$-cells, this map extracts the possible SPT anomaly left behind on $0$-cells while gluing the them along $2$-cells and $1$-cells. This gives the higher gluing condition $\kernel\paren{d^3_{3, -3}} \subset E^3_{3, -3}$.

Note that any element of $E^3_{3, -3}$ must be representable by an element of $\kernel \paren{d^2_{3, -3}} \subset E^2_{3, -3}$ and an element of $\kernel \paren{d^1_{3, -3}} \subset E^1_{3, -3}$. Since $d^1_{3, -3}$ maps $3$D SPT phases on $3$-cells to SPT anomalies on $2$-cells, the subset $\kernel \paren{d^1_{2, -2}} \subset E^1_{2, -2}$ must consist of configurations that satisfy the gluing condition along $2$-cells. Similarly, since $d^2_{3, -3}$ maps $3$D SPT phases on $3$-cells to SPT anomalies $1$-cells, the subset $\kernel \paren{d^1_{3, -3}} \subset E^1_{3, -3}$ must consist of configurations that satisfy the gluing condition along $1$-cells. Thus it is indeed possible to glue elements of $E^3_{3, -3}$ along $2$-cells and $1$-cells as we proposed to do in the previous paragraph.










\subsubsection{\texorpdfstring{$E^\infty$-page}{E infty page}} 
\label{subsubsec:E_inf}

Since $X$ is $d$-dimensional, we have $E^r_{p, q} = 0$ for all $p < 0$ or $p > d$. Consequently,
\begin{equation}
d^r_{p, q} = 0 ~~ \forall r > d.
\end{equation}
Thus the spectral sequence stabilizes at the $(r+1)$st page:
\begin{equation}
E^{r+1}_{p, q} = E^{r+2}_{p, q} = \cdots \eqcolon E^\infty_{p, q}.
\end{equation}
Extending the analysis in Secs.\,\ref{sec:d1} and \ref{subsubsec:higher_differentials} to higher $r$ and general values of $p$ and $q$, we get that
\begin{eqnarray}
&& E^\infty_{p, q} \nonumber\\[20pt]
&\isomorphic& \braces{\parbox{4.5in}{configurations of $(-q)$-dimensional SPT phases on $p$-cells (i.e.\,cells in $I_p$) that satisfy all gluing conditions along lower-dimensional cells (i.e.\,cells in $I_0 \sqcup I_1 \sqcup \ldots \sqcup I_{p-1}$) modulo all equivalence relations due to higher-dimensional cells (i.e.\,cells in $I_{p+1} \sqcup I_{p+2} \sqcup \ldots \sqcup I_d$)}}.
\end{eqnarray}
Alternatively,
\begin{eqnarray}
&& E^\infty_{p, n - p} \nonumber\\[20pt]
&\isomorphic& \braces{\parbox{4.5in}{configurations of degree-$n$ SPT phenomena on $p$-cells (i.e.\,cells in $I_p$) that satisfy all gluing conditions along lower-dimensional cells (i.e.\,cells in $I_0 \sqcup I_1 \sqcup \ldots \sqcup I_{p-1}$) modulo all equivalence relations due to higher-dimensional cells (i.e.\,cells in $I_{p+1} \sqcup I_{p+2} \sqcup \ldots \sqcup I_d$)}}.
\end{eqnarray}






\subsubsection{Filtration and abelian group extension}
\label{sec:Group extension}

$h^G_n\paren{X, Y}$ classifies degree-$n$ SPT phenomena on $X$ relative $Y$. Restricting to the $p$-skeleton, we have that $h^G_n\paren{X_p, X_p \cap Y}$ classifies degree-$n$ SPT phenomena on $X_p$ relative $X_p \cap Y$. By definition, this is the classification of degree $n$-SPT phenomena on $X_p$ that are glued along cells in $I_0 \sqcup I_1 \sqcup \ldots \sqcup I_{p-1}$ modulo all equivalence relations due to cells in $I_1 \sqcup I_2 \sqcup \ldots \sqcup I_p$.

For $h^G_n\paren{X_p, X_p \cap Y}$, there is no equivalence relation due to cells of dimension $> p$. However, if we consider the homomorphism $h^G_n(X_p,X_p \cap Y) \to h^G_n(X,Y)$ induced by the inclusion map $\paren{X_p, X_p \cap Y} \fromto \paren{X, Y}$, then its image
\begin{align}
F_p h^G_n(X,Y):= \image \brackets{h^G_n(X_p,X_p \cap Y) \to h^G_n(X,Y)}
\end{align}
will have equivalence relations due to cells of dimension $> p$ incorporated. More explicitly,
\begin{eqnarray}
&& F_p h^G_n(X,Y) \nonumber\\[20pt]
&\isomorphic& \braces{\parbox{4.5in}{degree-$n$ SPT phenomena on $X_p$ that are glued along cells in $I_0 \sqcup I_1 \sqcup \ldots \sqcup I_{p-1}$ modulo all equivalence relations due to cells in $I_1 \sqcup I_2 \sqcup \ldots \sqcup I_d$}}. \label{F_p_meaning}
\end{eqnarray}
This gives a filtration
\begin{equation}
0 \eqcolon F_{-1} h^G_n(X,Y) \subset F_0 h^G_n(X,Y) \subset F_1 h^G_n(X,Y) \subset \cdots \subset F_d h^G_n(X,Y) \coloneq h^G_n(X, Y).
\end{equation}
Furthermore, Eq.\,(\ref{F_p_meaning}) implies that the quotient
\begin{eqnarray}
&& F_p h^G_n(X,Y) / F_{p - 1} h^G_n(X,Y) \nonumber\\[20pt]
&\isomorphic& \braces{\parbox{4.5in}{configurations of degree-$n$ SPT phenomena on cells in $I_p$ that satisfy all gluing conditions along cells in $I_0 \sqcup I_1 \sqcup \ldots \sqcup I_{p-1}$ modulo all equivalence relations due to cells in $I_p \sqcup I_{p+1} \sqcup \ldots \sqcup I_d$}}.
\end{eqnarray}
This explains the isomorphism
\begin{equation}
F_p h^G_n(X,Y) / F_{p-1} h^G_n(X,Y) \isomorphic E^\infty_{p, n - p}. \label{Fp/Fp-1=Einf}
\end{equation}

We can express Eq.\,(\ref{Fp/Fp-1=Einf}) as an abelian group extension,
\begin{align}
0 \fromto F_{p-1} h^G_n(X,Y) \fromto F_p h^G_n(X,Y) \fromto E^\infty_{p, n - p} \fromto 0.
\label{eq:ahss_extension}
\end{align}
The fact that an abelian group extension can be nontrivial reflects the fact that configurations of degree-$n$ SPT phenomena on $p$-cells classified by $E^\infty_{p, n - p}$ can be nontrivially extended by degree-$n$ SPT phenomena on $X_{p-1}$ classified by $F_{p-1} h^G_n\paren{X, Y}$. For example, consider $n = 0$ and suppose an SPT phenomenon $a$ on $X_p$ represents an element of $E^\infty_{p, -p}$ of order $k > 1$. This means a stack of $k$ copies of $a$ can be trivialized on the $p$-cells. However, in general it can remain nontrivial on $X_{p-1}$. See Sec.~V A 1 of our work \cite{Shiozaki2018} for an example where the abelian group extension is nontrivial.







