# A Generalized Isoperimetric Inequality via
Thick Embeddings of Graphs

Elia Portnoy

###### Abstract

We prove a generalized isoperimetric inequality for a domain diffeomorphic to
a sphere that replaces filling volume with $k$-dilation. Suppose $U$ is an
open set in $\mathbb{R}^{n}$ diffeomorphic to a Euclidean $n$-ball. We show
that in dimensions at least 4 there is a map from a standard Euclidean ball of
radius about $vol(\partial U)^{1/(n-1)}$ to $U$, with degree 1 on the
boundary, and $(n-1)$-dilation bounded by some constant only depending on $n$.
We also give an example in dimension 3 of an open set where no such map with
small $(n-1)$-dilation can be found.

The generalized isoperimetric inequality is reduced to a theorem about thick
embeddings of graphs which is proved using the Kolmogorov-Barzdin theorem and
the max-flow min-cut theorem. The proof of the counterexample in dimension 3
relies on the coarea inequality and a short winding number computation.

##  1 Introduction

In this paper we will prove a generalized isoperimetric inequality for a
domain diffeomorphic to a sphere. For reference, we recall the classic
isoperimetric inequality.

######  Theorem 1.

If $U$ is a smooth open set in $\mathbb{R}^{n}$ then,

$vol(U)\lesssim_{n}vol(\partial U)^{n/(n-1)}$

where $x\lesssim_{n}y$ means there is some constant $C$, only depending on
$n$, so that $x\leq Cy$.

Applying Banyaga’s generalization of Moser’s Theorem to manifolds with
boundary [B] we can deduce a slightly stronger theorem when $U$ is a ball.

######  Theorem 2.

Let $U\subset\mathbb{R}^{n}$ be a smooth bounded open set that is
diffeomorphic to a ball in $\mathbb{R}^{n}$. Let $B^{n}$ be a ball in
$\mathbb{R}^{n}$ with the same volume as $U$. Then there exists a smooth map,

$F:(B^{n},\partial B^{n})\to(U,\partial U)$

which has degree 1 on the boundary and $|det(DF)|=1$.

In other words, Theorem 2 says we can find a map from $B^{n}$ to $U$ that does
not expand $n$-dimensional volume. We can also try to control how much $F$
stretches the volume of $k$-dimensional subsets of the domain.

######  Definition 1.

Suppose $F:M\to N$ is a Lipschitz map between Riemannian manifolds that is
differentiable on a set of full measure $\Omega$. We define the k-dilation of
$F$ as

$Dil_{k}(F)=\sup_{p\in\Omega}|\Lambda^{k}DF(p)|$

Where for a symmetric matrix $A$, $|\Lambda^{k}A|$ denotes the product of its
$k$ largest eigenvalues. The geometric interpretation of this definition is as
follows. If $\Sigma$ is a $k$-dimensional submanifold of $M$ then,

$vol(F(\Sigma))\leq Dil_{k}(F)vol(\Sigma)$

where $vol$ stands for the $k$-dimensional volume.

The first theorem in this paper improves Theorem 2 as follows,

######  Theorem 3.

Let $U$ be a smooth bounded open set in $\mathbb{R}^{n}$ that admits a
diffeomorphism to a ball in $\mathbb{R}^{n}$ and let $B^{n}_{R}$ stand for a
standard ball of radius $R$ in $\mathbb{R}^{n}$. For $R=vol(\partial
U)^{1/(n-1)}$ and $n\geq 4$, there exists a Lipschitz map,

$F:(B^{n}_{R},\partial B^{n}_{R})\to(U,\partial U)$

which has degree 1 on the boundary and with $Dil_{n-1}(F)\lesssim_{n}1$.

For an example of this theorem we can consider the case when $U$ is the open
ellipse in $\mathbb{R}^{n}$ with axes lengths $(L,1,1\ldots 1)$, where $L\geq
1$. Then for $R=L^{1/(n-1)}$, we can take $F:B_{R}\to U$ to be the
diffeomorphism given be a linear map. The eigenvalues of $DF$ are
$(L^{(n-2)/(n-1)},L^{-1/(n-1)}\ldots L^{-1/(n-1)})$. Multiplying the $(n-1)$
largest eigenvalues of $DF$ we get $Dil_{n-1}(F)=1$.

Now we give an outline of the proof of Theorem 3. The first step is to
construct an expanding embedding of a neighborhood of a 1-dimensional subset
of $U$ into $B_{R}$, which will act as a skeleton for $F^{-1}$. We give some
definitions that will help us work with such embeddings.

######  Definition 2.

For some number $\omega>0$, and some open set $U\subset\mathbb{R}^{n}$, define
$G(U,\omega)$, to be the graph defined by intersecting $U$ with a
1-dimensional grid with side-length $\omega$. The grid’s vertices are
$\omega\mathbb{Z}^{n}$ and its edges are all the straight segments between
vertices of distance of $\omega$ from each other. We can assume this grid
transversely intersects $\partial U$, and we let the points where the grid
intersects $\partial U$ also be verticies of $G(U,\omega)$. Sometimes we will
treat $G(U,\omega)$ as a 1-dimensional subset of $U$ and sometimes we will
treat it as a graph.

For an example, take $U=[-3/2,3/2]^{n}$. Then $G(U,1)$ is the $n$-hypercube
graph along with $n2^{n}$ edges between the $n$-hypercube and $\partial U$.

######  Definition 3.

Fix some dimension $n>0$. Call a map between two graphs,
$\psi:\Gamma_{1}\to\Gamma_{2}$, thick, if $\psi$ maps vertices to vertices,
edges to non-trivial paths, and the pre-image of each edge in $\Gamma_{2}$
intersects $\lesssim_{n}1$ edges of $\Gamma_{1}$.

For $U$ an open set in $\mathbb{R}^{n}$, we say that a map,
$\psi:\Gamma_{1}\to U$, is $\omega$-thick if $\psi$ has image in
$G(U,\omega)\subset U$ and $\psi$ is a thick map from $\Gamma_{1}$ to
$G(U,\omega)$.

The following lemma says that we can construct a map that functions as a
skeleton for $F^{-1}$.

######  Lemma 1.

Let $n\geq 3$ and suppose we are given a smooth bounded open set
$U\subset\mathbb{R}^{n}$. For some small $\omega>0$, that depends on $U$, let
$\Gamma=G(U,\omega)$ and $\partial\Gamma=\Gamma\cap\partial U$. Then for
$R=vol(\partial U)^{1/(n-1)}$, there is an $\omega$-thick embedding,

$\psi:(\Gamma,\partial\Gamma)\to(B_{R},\partial B_{R})$

The technique of considering a skeleton for an inverse map by using a thick
embedding was previously used by Guth in [GC] to construct maps with small
$k$-dilation between $S^{m}$ and $S^{n}$ for $m>n$ and $k>(m+1)/2$. Since maps
between spaces of different dimensions were considered, Guth was able to
construct the thick embedding using a quantitative h-principle. We will rely
on a different method for maps between spaces of the same dimension. First we
cut our graph $\Gamma$ into pieces that roughly look like balls. We will then
rearrange these pieces into a larger ball and reconnect the edges between the
pieces using two lemmas. The first lemma was proven in dimension three by
Kolmogorov and Barzdin and generalized to higher dimensions by Guth.

######  Lemma 2.

[KB, GQ] Let $\Gamma$ be a graph which is a disjoint union of edges and let
$V\Gamma$ stand for the vertices of $\Gamma$. Suppose that for
$R=|V\Gamma|^{1/(n-1)}$ we are given an embedding,

$\psi:V\Gamma\to\partial B^{n}_{R}$

Then we can construct a 1-thick map,

$\Psi:\Gamma\to B^{n}_{R}$

which extends $\psi$.

This lemma roughly says that if want to map a graph of bounded degree,
$\Gamma$, into $B_{R}$, without too many edges intersecting any unit ball,
then we are free to chose the positions of verticies on $\partial B_{R}$, as
long as they are sufficiently spaced out. Kolmogorov and Barzdin proved this
lemma using a probabilistic method, and their bounds are sharp for expander
graphs. For the proof of Lemma 1 we will sometimes be forced to map vertices
into the interior of $B_{R}$, which presents an obstacle for their methods.
The second lemma gives us more flexibility in mapping half of the vertices of
$\Gamma$ at the cost of having less flexibility over where the other half of
the vertices gets mapped to.

######  Lemma 3.

Let $\Gamma$ be the bipartite graph on two sets of vertices $S$ and $T$ with
each vertex having degree 1. That is, $\Gamma$ is a set of $|S|=|T|$ disjoint
edges. Suppose we are given a 1-thick map $\psi:S\to B^{n}_{R}$ so that,

$|\psi(S)\cap B_{r}|\lesssim_{n}r^{n-1}$

for each $B_{r}\subset B_{R}$ with $r>1/2$. Then we can extend $\psi$ to a
1-thick map,

$\Psi:(\Gamma,T)\to(B_{R},\partial B_{R})$

The proof of Lemma 3 combines the min-cut max-flow theorem with a good ball
type argument used in a proof of the isoperimetric inequality (see [W]). We
use Lemma 2 to reconnect the edges between the balls, which we cut our graph
into, and we will use Lemma 3 to ensure that the edges adjacent to
$\partial\Gamma$ get mapped to edges adjacent to $\partial B_{R}$. This
completes the outline of Lemma 1.

Let’s now sketch how Lemma 1 implies Theorem 3. Let $W\subset B_{R}^{n}$ be a
$\omega/10$ neighborhood of $\psi(G(U,\omega))$ where $\psi$ is as in Lemma 1.
We can construct a $C_{1}\lesssim_{n}1$ bilipschitz embedding from $W$ to $U$.
Because we are in dimensions $\geq 4$, any two maps from a graph into
$B_{R}^{n}$ are isotopic. This means we can extend $\psi^{-1}$ to a Lipschitz
map $F_{1}:B^{n}_{R}\to U$. Let $\Phi:B^{n}_{R}\to B^{n}_{R}$ be a
$C_{2}\lesssim_{n}1$ Lipschitz map that contracts the complement of $F_{1}(W)$
to an $(n-2)$-dimensional set and let $F=\Phi\circ F_{1}$. Observe that
$F(B^{n}_{R}-W)$ is a subset of an $(n-2)$-dimensional set and that $F$ is
$C_{1}C_{2}$ Lipschitz on $W$. Putting these observations together we see that
$Dil_{n-1}(F)\lesssim_{n}1$.

In [GC], Guth showed that if $k$ is small compared to the dimension of the
domain, there are some topological restrictions to the existence of maps with
small $k$-dilation. This suggests that Theorem 3 might fail in lower
dimensions. We show that there are indeed 3-dimensional sets for which Theorem
3 fails to hold.

######  Theorem 4.

For any $N\in\mathbb{N}$, there is an open set $U_{N}\subset\mathbb{R}^{3}$,
diffeomorphic to a ball and with $vol_{2}(\partial U_{N})\lesssim 1$, so that
any $F:(B^{3}_{1},\partial B^{3}_{1})\to(U_{N},\partial U_{N})$, with degree 1
on the boundary, must have $Dil_{2}(F)\gtrsim N^{1/2}$.

Our $U_{N}$ will be an open set whose boundary winds around some axis $N$
times. Using the coarea formula we will show that any map that does not
increase area from $B^{3}$ to $U_{N}$ must map a disc with small area to a
disc which winds $N$ times around some axis. Then a winding number computation
will imply that the area of this image of this disc is $\gtrsim N^{1/2}$.

We end this section with some open questions,

######  Question 1.

The isoperimetric inequality holds for cycles of any dimension in
$\mathbb{R}^{n}$. Is there an analogue of Theorem 3 in higher codimensions?
More precisely, Let $W\subset\mathbb{R}^{n}$ be the image of an embedding (or
immersion) of $\partial B^{m}$ into $\mathbb{R}^{n}$. For
$R=vol_{m-1}(W)^{1/(m-1)}$ can we construct a map,

$F:(B^{m}_{R},\partial B^{m}_{R})\to(\mathbb{R}^{n},W)$

with degree 1 on the boundary and $Dil_{m-1}(F)\lesssim_{n}1$?

######  Question 2.

Are there similar isoperimetric inequalities for other $k$-dilation? For
instance, suppose $U$ is as in Theorem 3 and we are given a degree 1 map
$f:\partial B^{n}_{R}\to\partial U$ with $Dil_{k}(f)\leq 1$ for some $k>0$.
For which $k^{\prime}$ can we construct a $F:(B^{n}_{R},\partial
B^{n}_{R})\to(U,\partial U)$, with degree 1 on the boundary, and with
$Dil_{k^{\prime}}(F)\lesssim_{n}1$?

######  Question 3.

Suppose we are given a graph of bounded degree, $\Gamma$, and let
$Spec(\Gamma)$ be the spectrum of its laplacian. We wonder what is the
smallest $R(Spec(\Gamma))>0$ so that there is there a 1-thick embedding,

$\psi:\Gamma\to B^{3}_{R(Spec(\Gamma))}$

Are there some $\Gamma_{1},\Gamma_{2}$ with similar spectra, but
$R(Spec(\Gamma_{1})$ is much bigger than $R(Spec(\Gamma_{2})$?

This paper is organized as follows. In Section 2 we give a proof of Lemma 1
and show how it implies Theorem 3. In Section 3 we prove Lemma 3 using min-cut
max-flow. In Section 4 we present the example in dimension 3 from Theorem 4.

## Acknowledgements

The author would like to thank Larry Guth for suggesting the isoperimetric
inequality involving $k$-dilation and several other valuable discussions that
led to this paper.

##  2 Proof of Theorem 1

###### Proof of Lemma 1.

Our strategy is to partition $U$ by cutting it into well-behaved subsets. Then
we will isometrically embed these subsets into disjoint balls near $\partial
B_{R}$. In partitioning $U$ we had to cut some edges. Using the Kolmogorov-
Barzdin theorem we can reconnect these edges inside $B_{R}$. Finally, we will
add edges connecting the image of $\partial\Gamma$ to $\partial B_{R}$ using
Lemma 3 from Section 3.

There is a technical issue we have to address before moving on with the
outline above. We will construct our partition of $U$ inductively by cutting
away pieces of $U$. The worry is that after cutting many pieces off from $U$
we may be left with a set that is much less regular than the one we started
with. In order to avoid this we will take our set $U$ and all of the pieces in
the partition to be unions of $n$-cells from a lattice.

Let $G_{\omega}$ be a uniform $n$-lattice of side length $\omega$ in
$\mathbb{R}^{n}$. For small enough $\omega>0$ there is a $C\lesssim_{n}1$
bilipschitz map from $\overline{U}$ to a union of closed $n$-cells from
$G_{\omega}$, with degree 1 on the boundary. Since our problem is scale
invariant and invariant under $C\lesssim_{n}1$ bilipschitz maps we can take
$\omega=1$ and assume that $\overline{U}$ is a union of closed $n$-cells from
$G_{1}$, without loss of generality.

######  Definition 4.

A $G$-ball, denoted $B_{r}^{G}(p)$, is the union of closed $n$-cells from
$G_{1}$ that have non-trivial intersection with $\overline{B_{r}(p)}$. For
ease of notation we will sometimes omit the explicit dependence on the center
of the ball and simply write $B_{r}^{G}$.

We introduce the following notion of a good $G$-ball,

######  Definition 5.

Later in the proof we will fix constants $A>1,\tilde{A}>1$ and
$0<\delta=\delta(A)<1$ that only depend on $n$. Call $B^{G}_{r}$ a good
$G$-ball for an open set $W\subset\mathbb{R}^{n}$ if the following conditions
hold:

$vol(\partial W\cap B^{G}_{r})>A\,vol(W\cap\partial B^{G}_{r})$ (1)

$vol(\partial W\cap B^{G}_{r})>\delta r^{n-1}$ (2)

$\text{For $s>1/2$ and every ball }B^{G}_{s}\subset B^{G}_{r}\text{,
}vol(\partial W\cap B^{G}_{s})<\tilde{A}As^{n-1}$ (3)

Condition 1 tells us that if we remove $W\cap B^{G}_{r}$ from $W$, then we
decrease its boundary. This will be important in inductively constructing our
partition. Condition 2 says that the volume of the $\partial B^{G}_{r}$ is
still comparable to the volume of $\partial W\cap B^{G}_{r}$. This will help
us fit all the balls we used to partition $W$ into disjoint balls near
$\partial B_{R}$. Condition 3 says that $\partial W$ is not too concentrated
anywhere inside our ball, which will become important when we want to use
Lemma 3 later on.

The main feature of our partition of $U$ will be that each piece lies inside a
good $G$-ball and the sum of the boundaries of these $G$-balls is not much
bigger than $vol(\partial U)$. We show that such a partition exists in the
following lemma.

######  Lemma 4.

Let $U$ be a union of $n$-cells from $G_{1}$. There exists a partition of $U$,
denoted by $\mathcal{P}=\\{U_{i}\\}$, and a set of good $G$-balls for $U$,
denoted by $\\{B^{G}_{r_{i}}\\}$, so that,

$U_{i}\subset B^{G}_{r_{i}}$ $|\partial\mathcal{P}|:=\sum
r_{i}^{n-1}<A^{\prime}\,vol(\partial U)$

where $A^{\prime}>1$ is some large constant that only depends on $A$ and
$\delta$.

###### Proof of Lemma 4.

The proof is by induction on $vol(U)$. For the base case we consider
$vol(U)=1$ for which we can take $\mathcal{P}=\\{U\\}$. Note that $U$ lies in
a good $G$-ball $B^{G}_{2}(p)$ for $p$ in the center of $U$. Condition 1 is
satisfied because $vol(U\cap\partial B^{G}_{r})=0$. Condition 2 is satisfied
for small enough $\delta$, and condition 3 is satisfied for large enough $A$.

Now lets assume the inductive hypothesis holds for open sets with volume less
than $vol(U)$ and show that we can find a $G$-good ball for $U$. To do this we
will consider three cases.

Case 1: Suppose first that there is a ball $B^{G}_{r}$ with $vol(\partial
U\cap B^{G}_{r})>\tilde{A}Ar^{n-1}$. Let $B^{G}_{t}$ be a minimal $G$-ball
satisfying $vol(\partial U\cap B^{G}_{t})>\tilde{A}At^{n-1}$. In other words,
there is no $B^{G}_{s}$ strictly inside $B^{G}_{t}$ that also satisfies the
same inequality. For $A$ large enough, we have $t>n$, and so
$B^{G}_{t-n}\subsetneq B^{G}_{t}$ satisfies condition 3. Set
$B^{\prime}=B^{G}_{t-n}$ and observe that for $\tilde{A}$ large enough,

$vol(\partial U\cap
B^{\prime})>\tilde{A}At^{n-1}-\tilde{A}t^{n-1}>Avol(\partial B^{\prime})$

and so $B^{\prime}$ satisfies conditions 1 and 2. Thus, $B^{\prime}$ is a good
$G$-ball for $U$. For all the following cases we assume that $U$ satisfies
condition 3.

Case 2: Suppose there exists a $p\in U$ and an $0<r<\delta^{-1/(n-1)}$ so that
$vol(\partial B^{G}_{r}(p)\cap U)=0$. Condition 1 and 2 are easily satisfied,
so $B^{G}_{r}(p)$ is a good $G$-ball for $U$.

Case 3: Thus, we can take any $p\in U$ and assume that $vol(\partial
B^{G}_{r}(p)\cap U)>0$ for all $0<r<\delta^{-1/(n-1)}$. Define,

$t=\inf\\{r\geq\delta^{-1/(n-1)}:vol(U\cap\partial B^{G}_{r}(p))<\delta
r^{n-1}\\}$

Let $B^{\prime}=B^{G}_{t}(p)$. Suppose for contradiction that
$vol(B^{\prime}\cap\partial U)<A\delta t^{n-1}$. Then by the isoperimetric
inequality we have,

$vol(B^{\prime}\cap U)\lesssim_{n}(vol(B^{\prime}\cap\partial U)+vol(\partial
B^{\prime}\cap U))^{n/(n-1)}\lesssim_{n}(A\delta)^{n/(n-1)}t^{n}$

However, by the minimality property of $t$ we have,

$vol(B^{\prime}\cap U)\geq\sum_{k=1}^{t/3-1}vol(\partial B^{G}_{3k}(p)\cap
U)\gtrsim_{n}\delta t^{n}$

Picking $\delta>0$ small enough so that
$(A\delta)^{n/(n-1)}\lesssim_{n}\delta$ we get a contradiction. Thus,
$vol(B^{\prime}\cap\partial U)\geq A\delta t^{n-1}$ and so $B^{\prime}$ is a
good ball.

We have now shown that we can always find a good $G$-ball
$B^{\prime}=B^{G}_{t}(p)$ for $U$. If $U\subset B^{\prime}$ we complete the
inductive step by taking $\mathcal{P}=\\{B^{\prime}\\}$, otherwise let
$U_{2}=U-B^{\prime}$. Since
$vol(U_{2})<vol(U)$ we can apply the inductive hypothesis to $U_{2}$ to get a
partition $\mathcal{P}_{2}$ of $U_{2}$, so that each element of the partition
is inside a good $G$-ball, with,

$|\partial\mathcal{P}_{2}|<A^{\prime}vol(\partial U_{2})\leq
A^{\prime}(vol(\partial U)-vol(\partial U\cap B^{\prime})+vol(\partial
B^{\prime}\cap U))$ $<A^{\prime}vol(\partial U)-A^{\prime}(1-1/A)vol(\partial
U\cap B^{\prime})<A^{\prime}vol(\partial U)-A^{\prime}(1-1/A)\delta t^{n-1}$

The first inequality comes from the inductive hypothesis, the third comes from
Condition 1, and the fourth from Condition 2. Then we can take our partition
of $U$ to be $\mathcal{P}=\mathcal{P}_{2}\cup\\{B^{\prime}\cap U\\}$ and
compute,

$|\partial\mathcal{P}|=|\partial\mathcal{P}_{2}|+t^{n-1}<A^{\prime}vol(\partial
U)-(A^{\prime}(1-1/A)\delta-1)t^{n-1}<A^{\prime}vol(\partial U)$

where the last inequality holds for $A^{\prime}=A^{\prime}(A,\delta)$ large
enough. This completes the inductive step and the proof of Lemma 4. ∎

We now begin describing the 1-thick map $\psi$ from the statement of Lemma 1.
This will be done in a few steps and Figure 1 below may be helpful for
visualizing the steps. Let $\mathcal{P}=\\{U_{i}\\}$ and $\\{B^{G}_{r_{i}}\\}$
be the partition and good $G$-balls for $U$, given by Lemma 4. According to
Lemma 4,

$\sum_{i}r_{i}^{n-1}\leq A^{\prime}vol(\partial U)$

So for some $C\lesssim_{n}1$ large enough, and $R=Cvol(\partial U)^{1/(n-1)}$,
there exists points $\\{p_{i}\\}\subset B_{R}$ so that
$\\{B_{2r_{i}}(p_{i})\\}$ are disjoint balls inside $B_{R}$ with disjoint
radial projections to $\partial B_{R}$. We also denote,

$\Sigma=\bigcup_{i}\partial U_{i}$

Step 1: We begin by describing our map $\psi$ on $\Gamma\cap U_{i}$. Let
$\Gamma_{i}\subset\Gamma$ be the set of edges in $U_{i}$, that don’t intersect
$\partial U_{i}$. Since $\Gamma_{i}\subset B^{G}_{r_{i}}$ we can let $\psi$
map $\Gamma_{i}$ into $B_{2r_{i}}(p_{i})$ isometrically, with some
translation.

Step 2: Let $E_{i}$ be the set of edges in $\Gamma$ that intersect $\partial
U_{i}$. Each edge in $E_{i}$ will be mapped to a piece-wise linear path in
$B_{R}$ composed of several parts. To do this lets recall the statement of
Lemma 3,

######  Lemma.

Let $\Gamma^{\prime}$ be the bipartite graph on two sets of vertices $S$ and
$T$ with each vertex having degree 1. Suppose we are given a 1-thick map
$\psi:S\to B^{n}_{R^{\prime}}$ so that,

$|\psi(S)\cap B_{r}|\lesssim_{n}r^{n-1}$

for each $B_{r}\subset B_{R^{\prime}}$ with $r>1/2$. Then we can extend $\psi$
to a 1-thick map,

$\Psi:(\Gamma^{\prime},T)\to(B_{R^{\prime}},\partial B_{R^{\prime}})$

Let $S_{i}$ be the endpoints of $E_{i}$ in $U_{i}$. Notice that $S_{i}$ lies
in the 1-neighborhood of $\partial U_{i}$, and since $B_{r_{i}}^{G}$ is a good
G-ball for $U_{i}$ it satisfies condition 3. Thus, the assumption of Lemma 3
is satisfied for $S=S_{i}$, $B_{R^{\prime}}=B_{2r_{i}}(p_{i})$, and the $\psi$
we constructed in the previous step which is defined on $S_{i}$. We now use
Lemma 3 to construct the first part of these paths, which connect $S_{i}$ to
$\partial B_{2r_{i}}(p_{i})$.

The next part of these paths, will consist of a line segment connecting
$\partial B_{2r_{i}}(p_{i})$ with $\partial B_{R}$, in the radial direction.
We finish Step 2 by applying this procedure for every $i$. Notice that this
procedure gives us a 1-thick embedding.

Finally, notice that since $\partial\Gamma\subset\Sigma$, we have mapped every
edge adjacent to $\partial\Gamma$ to a path with an endpoint in $\partial
B_{R}$.

Step 3: Since we can make a $C\lesssim_{n}1$ bilpschitz perturbation of
$\Gamma$ we can assume that $V\Gamma\cap\Sigma=\partial\Gamma$. Let
$E_{\Sigma}$ be the set of edges that intersect $\Sigma$ but are not adjacent
to $\partial\Gamma$. In this step, we’ll finish defining $\psi$ on
$E_{\Sigma}$.

Each edge $e\in E_{\Sigma}$, has one endpoint $s_{i}\in U_{i}$ and another
endpoint $s_{j}\in U_{j}$, for some $j\neq i$. In the previous step we started
defining the image of $e$ to be two paths, one path going from $s_{i}$ to some
$t_{i}\in\partial B_{R}$ and another path going from $s_{j}$ to some
$t_{j}\in\partial B_{R}$. In order to finish defining our map on $e$ we need
to find a path from $t_{i}$ to $t_{j}$. To extend $\psi$ to $E_{\Sigma}$, lets
recall the statement of Lemma 2,

######  Lemma.

Let $\Gamma^{\prime}$ be a graph which is a disjoint union of edges. Suppose
that for $R^{n-1}\lesssim_{n}|V\Gamma|$ we are given an embedding,

$\psi:V\Gamma^{\prime}\to\partial B_{R}$

Then we can construct a 1-thick map,

$\Psi:\Gamma^{\prime}\to B_{R}$

which extends $\psi$.

Observe that we have,

$|\partial
E_{\Sigma}|\lesssim_{n}\sum_{i}|E_{i}|\lesssim_{n}\sum_{i}|U_{i}|\lesssim_{n}\sum_{i}r_{i}^{n-1}\lesssim_{n}R^{n-1}$

where the second to last inequality follows from condition 3 of a good G-ball,
and the last inequality comes from Lemma 4. Note that $\psi$ was defined on
$\partial E_{\Sigma}$ in the previous steps. Thus, applying Lemma 2 with
$\Gamma^{\prime}=E_{\Sigma}$, we can extend our $\psi$ to a 1-thick map on
$E_{\Sigma}$. This completes the construction of $\psi$.

∎

Figure 1: Sketch of $\psi$ for the case of a partition consisting of 2 sets.
G-good balls from Lemma 4 are shown in green. $E_{\Sigma}$ is shown in red.
The rest of the edges of $\Gamma$ are shown in blue.

We now give a technical lemma which allows us to modify a thick maps of graphs
to an expanding embeddings of its neighborhood.

######  Lemma 5.

Let $U,U^{\prime}\subset\mathbb{R}^{n}$ be open sets, which are unions of
$n$-cells from $G_{1}$, and suppose $\psi:G(U,1)\to U^{\prime}$ is a $1$-thick
map. For a graph $G$ embedded a subset in $\mathbb{R}^{n}$, denote its
$r$-neighborhood as $N_{r}(G)$. Then for some $\delta\gtrsim_{n}1$ and
$r=1/10$, there is a $\delta$-expanding embedding,

$\Psi:N_{r}(G(U,1))\to U^{\prime}$

In other words, $|D\Psi(v)|\geq\delta|v|$ for every tangent vector $v$. This
map satisfies,

$dist(\Psi(p),\psi(p))\lesssim_{n}1$

for every $p\in G(U,1)$.

###### Proof.

Denote by $K_{m}$ the complete graph on $m$ verticies. For some
$m\lesssim_{n}1$, let $T$ be the graph where we replace each vertex of
$G(U^{\prime},1)$ by $K_{m}$, and where we replace each edge in
$G(U^{\prime},1)$ by $m$ disjoint paths between the verticies of two complete
graphs, so that the degree of any vertex of $T$ is $\lesssim_{n}1$. We take
$m\lesssim_{n}1$ large enough, so that we can lift our thick map $\psi$ to a
genuine embedding,

$\psi^{\prime}:G(U,1)\to T$

Let $m^{\prime}U\subset\mathbb{R}^{n}$ denote the set $U$ scaled by
$m^{\prime}>0$. Then for $m^{\prime}\lesssim_{n}1$ large enough there is an
embedding,

$\phi:T\to m^{\prime}N_{r}(G(U^{\prime},1))$

so that for any vertices $v,v^{\prime}\in VT$,

$dist(\phi(v),\phi(v^{\prime})\geq 1$

and for any edges $e,e^{\prime}\in ET$ that do not share any endpoints,

$dist(\phi(e),\phi(e^{\prime}))\geq 1$

We can then construct a 1-expanding embedding $\Psi:N_{r}(G(U,1))\to
m^{\prime}N_{r}(G(U^{\prime},1))$ so that
$\Psi|_{G(U,1)}=\phi\circ\psi^{\prime}$. Shrinking the image of $\Psi$ by a
factor of $1/m^{\prime}$, we get a $\delta=1/m^{\prime}$ expanding embedding
as desired. ∎

We continue to the proof of Theorem 3.

###### Proof of Theorem 3.

Let $U\subset\mathbb{R}^{n}$ be our open set, which we can assume to be a
union of $n$-cells from $G_{1}$ and let $R=Cvol(\partial U)^{1/(n-1)}$ for
some large $C\lesssim_{n}1$. Let $W^{\prime}=N_{1/10}(G(U,1))$. Using Lemma 1
and Lemma 5 we can construct an $\delta\gtrsim_{n}1$ expanding embedding,

$\Psi:(W^{\prime},W^{\prime}\cap\partial U)\to(B_{R},\partial B_{R})$

Letting $F_{1}=\Psi^{-1}$, we can then extend $F_{1}$ to,

$F_{2}:(Im(\Psi)\cup\partial B_{R},\partial B_{R})\to(U,\partial U)$

so that $F_{1}$ is $C\lesssim_{n}1$ Lipschitz and has degree 1 on the
boundary. Next we extend $F_{1}$ to all of $B_{R}$ using an argument similar
to Lemma 11.2 from [GC].

If $n>3$ any two embeddings of a graph into $B^{n}_{R}$ are isotopic. If in
addition both embeddings map some vertices to $\partial B_{R}$, we can ensure
that those vertices are embedded in $\partial B_{R}$ throughout the isotopy.
By our hypothesis there is some diffeomorphism,

$F_{3}:(B_{R},\partial B_{R})\to(U,\partial U)$

Thus, since we assumed $n>3$ we can extend $F_{2}$ to a diffeomorphism
$F_{4}:(B_{R},\partial B_{R})\to(U,\partial U)$.

Let $\Phi:(U,\partial U)\to(U,\partial U)$ be a $C\lesssim_{n}1$ Lipschitz map
with degree 1 on the boundary which shrinks $U-N_{1/10}(G(U,1))$ to a
$(n-2)$-dimensional set $\Sigma$ (see Lemma 11.1 of [GC]). Now let
$F=\Phi\circ F_{4}$. On the the pre-image of $\Sigma$, the $(n-1)$-dilation of
$F$ is 0 because $dim(\Sigma)<n-1$ and on the pre-image of $U-\Sigma$, $F$ is
$C\lesssim_{n}1$ Lipschitz by the properties of $F_{4}$ and $\Phi$. Thus,
$Dil_{n-1}(F)\lesssim_{n}1$.

∎

##  3 Applications of Min-Cut Max-Flow to Thick Embeddings. Proof of Lemma 3.

The proof of Lemma 3 will rely on an integral max-flow min-cut theorem which
can be derived from the Ford-Fulkerson algorithm. The version used here is
stated as follows,

######  Theorem 5.

[E] Let $\Gamma$ be a directed graph with vertices $V\Gamma$ and edges
$E\Gamma$. Let $c:E\Gamma\to\mathbb{Z}_{\geq 0}$ be some given function, we
call the capacity. Let $S,T\subset V\Gamma$ be two disjoint subset of vertices
where each $v\in S$ has only edges directed out of it and each $v\in T$ has
only edges directed into it. A flow is a function
$f:E\Gamma\to\mathbb{Z}_{\geq 0}$ which satisfies the following conditions,

Capacity Constraint: For each $e\in E\Gamma$,

$f(e)\leq c(e)$ (4)

Conservation of Flow: For each $v\in V\Gamma-(S\cup T)$,

$\sum_{w\in V\Gamma,[w,v]\in E\Gamma}f([w,v])=\sum_{u\in V\Gamma,[v,u]\in
E\Gamma}f([v,u])$ (5)

Where $[v_{1},v_{2}]$ is an edge directed from $v_{1}$ to $v_{2}$.

A cut $\Sigma\subset E\Gamma$ for the sets $S$ and $T$ is a set of edges so
that the graph,

$\Gamma-\Sigma-\bigcup_{e\in E\Gamma,c(e)=0}e$

has no directed path from any vertex in $S$ to any vertex in $T$. Define the
capacity of a cut $\Sigma$ to be $c(\Sigma)=\sum_{e\in\Sigma}c(e)$ and the
capacity of a flow $f$ to be $\sum_{e:\partial e\cap S\neq\\{\\}}f(e)$. Then
the maximum capacity of a flow is equal to the minimum capacity of a cut.

We first prove a lemma that looks similar to Lemma 3.

######  Lemma 6.

Let $\Gamma$ be the bipartite graph on two sets of vertices $S$ and $T$ with
each vertex having degree 1. That is, $\Gamma$ is a set of $|S|=|T|$ disjoint
edges. Suppose we are given a 1-thick map $\psi:S\to B^{n}_{R}$ so that for
any open set $U\subset B_{R}$,

$|\psi(S)\cap U|\lesssim_{n}vol(\partial U)$

Then we can extend $\psi$ to a $1/2$-thick map,

$\Psi:(\Gamma,T)\to(B_{R},\partial B_{R})$

###### Proof.

We first construct an oriented graph with capacities. Let $G$ be the oriented
graph that has the same vertex set as $G(B_{R}^{n},1/2)$ but has an edge in
each direction between adjacent vertices in $G(B_{R}^{n},1/2)$. So
$|EG|=2|EG(B_{R}^{n},1/2)|$. If we treat edges in different directions as the
same segment in $B_{R}^{n}$, we can still think of $G$ as a subset of
$B_{R}^{n}$. Let $\partial G=\partial B_{R}^{n}\cap VG$. Define a capacity
function $c:EG\to\mathbb{Z}_{\geq 0}$ so that,

  1. 1.

For each $v\in\psi(S)$, let $\sum_{w\in VG:[v,w]\in
EG}c([v,w])=|\psi^{-1}(s)|$.

  2. 2.

For all $v\in\psi(S)$ and $[w,v]\in EG$, we have $c([w,v])=0$

  3. 3.

For all $v\in\partial G$ and $[v,w]\in EG$, we have $c([v,w])=0$.

  4. 4.

For all other edges in $EG$, we set the capacity be some large $M>0$, to be
defined later.

Let’s show that a minimal cut separating $\psi(S)$ from $\partial G$ has
capacity at least $|S|$. We apply an induction on $|S|$ to take care of the
cases when the cut passes near $\psi(S)$. The base case $|S|=1$ is easy to
see. We continue to proving the inductive hypothesis.

Let $ES$ denote the set of edges adjacent to $\psi(S)$ and let $\Sigma$ be a
minimal cut. If $\Sigma\cap ES\neq\emptyset$, there is an $s\in S$ and
$e=[\psi(s),v]\in\Sigma$ with $c(e)>0$. Consider a modified min-cut max-flow
problem where we change $S$ to $S^{\prime}=S-s$ and we change $c(e)$ to
$c^{\prime}(e)=c(e)-1$, but keep everything else the same. Notice that
$\Sigma$ is still a cut for the two sets $\psi(S^{\prime})$ and $\partial G$.
Also notice that the hypothesis of the lemma are satisfied since,

$|\psi(S^{\prime})\cap U|\leq|\psi(S)\cap U|$

for every open set $U$. Now we can apply our inductive hypothesis to get
$c^{\prime}(\Sigma)\geq|S^{\prime}|=|S|-1$. Since $e\in\Sigma$ we see that
$c(\Sigma)=c^{\prime}(\Sigma)+1\geq|S|$.

Thus, we can assume that $\Sigma\cap ES=\emptyset$. Let $H\subset G$ be the
component of $G-\Sigma$ containing $\psi(S)$. We define an open set $U$ to be
the $1/2$ neighborhood of $H$ and observe that,

$vol(\partial U)\lesssim_{n}|\Sigma|$

Since $\Sigma\cap ES=\emptyset$ we have $c(\Sigma)=M|\Sigma|$. Putting this
together with the assumption of the lemma we have, for some $C\lesssim_{n}1$
and $M=M(C)\lesssim_{n}1$, large enough,

$|S|\leq Cvol(\partial U)\leq C|\Sigma|=\frac{C}{M}c(\Sigma)\leq c(\Sigma)$

This completes the inductive hypothesis. By Theorem 5 there is a flow $f$ that
takes values in $\mathbb{Z}_{\geq 0}$ and has capacity $|S|$. Since $f$
satisfies the conservation property at each vertex, we can turn $f$ into a set
of paths that connect each vertex in $S$ to some vertex in $T$. The capacity
constraint on $f$ says that at most $M$ of these paths intersect at each edge.
This gives us a way to extend $\psi$ to a 1-thick embedding $\Psi$ and
completes the proof of the lemma.

∎

Now we show that the hypothesis in the previous lemma can be weakened to that
of Lemma 3. The proof uses a good ball type argument common in metric
geometry.

######  Lemma 7.

Let $S$ be a finite set of points in $B_{R}^{n}$ so that,

$|S\cap B_{r}|\lesssim_{n}r^{n-1}$

for each ball $B_{r}\subset B_{R}$ with $r>1/2$. Then for any open set
$U\subset\mathbb{R}^{n}$,

$|S\cap U|\lesssim_{n}vol(\partial U)$

###### Proof.

Suppose we are given our open set $U$. We begin by defining a cover for $U$.
For some small $1/4>\delta>0$ and each $p\in U$ let,

$t(p)=\inf\\{r>0:vol(\partial B_{r}(p)\cap U)<\delta r^{n-1}\\}$

and denote $B(p)=B_{t(p)}(p)$. From the definition of $t(p)$ we see that,

$vol(\partial B(p)\cap U)=\delta\,vol(\partial B(p))$ (6)

Since $\partial B(p)\cap\partial U$ is an embedded $(n-2)$-cycle in an
$S^{n-1}$ we can assume its minimal filling is a sum of connected components
of $\partial B(p)-(\partial B(p)\cap\partial U)$. There are only two such
possible fillings. Since $\delta<1/4$, equation 6 tells us that $\partial
B(p)\cap U$ is a minimal filling of $\partial B(p)\cap\partial U$ on $\partial
B(p)$.

We now use an observation of Federer-Fleming from [F]. Their observation is
that if we radially project a $(n-1)$-dimensional set $Z\subset B(p)$, from a
random point in $B(p)$ to $\partial B(p)$, then with high probability, the
volume of the image of $Z$ under the projection is $\lesssim_{n}vol(Z)$. We
can use such a projection, to project $B(p)\cap\partial U$ to a filling of
$\partial B(p)\cap\partial U$ on $\partial B(p)$ with volume
$\lesssim_{n}vol(B(p)\cap\partial U)$. Since $\partial B(p)\cap U$ was a
minimal filling for $\partial B(p)\cap\partial U$ we have,

$vol(\partial B(p)\cap U)\lesssim_{n}vol(B(p)\cap\partial U)$ (7)

We now use the Besicovitch Covering theorem to find a finite set of points
$\\{p_{i}\\}\subset U$ so that the set of balls $\\{B(p_{i})\\}$ have
multiplicity bounded by $C\lesssim_{n}1$ and cover all of $U$. Putting the
above observations together we obtain,

$vol(\partial U)\gtrsim_{n}\sum_{i}vol(\partial U\cap
B(p_{i}))\gtrsim_{n}\sum_{i}vol(U\cap\partial B(p_{i}))$ $=\sum_{i}\delta
r^{n-1}\gtrsim_{n}\sum_{i}\delta|B(p_{i})\cap
S^{\prime}|\gtrsim_{n}\delta|S^{\prime}|$

The second inequality comes from equation 7 and the second to last inequality
comes from the assumption on $S$ given in the hypothesis of the lemma. ∎

###### Proof of Lemma 3.

The proof follows by combining the two lemmas above. ∎

##  4 Proof of Theorem 4

Our example in dimension 3 will make use of the coarea formula. We state it
here for reference.

######  Lemma 8.

For $m\leq n$ and an open set $U\subset\mathbb{R}^{n}$, let
$f:U\to\mathbb{R}^{m}$ be a Lipschitz map. Then,

$\int_{U}|\Lambda^{m}DF(x)|dx\gtrsim_{n}\int_{\mathbb{R}^{m}}vol_{n-m}(f^{-1}(y)\cap
U)dy$

###### Proof of Theorem 4.

Given $N\in\mathbb{N}$, our goal is to construct a set $U_{N}\subset R^{3}$ of
$\textup{area}(\partial U_{N})\lesssim 1$ so that any Lipschitz map

$F:(B_{1}^{3},\partial B^{3})\to(U_{N},\partial U_{N})$

with degree 1 on the boundary and with $D=Dil_{2}(F)$, will have $D\gtrsim
N^{1/2}$.

We begin by defining two open sets inside $B_{1}^{3}$. For some small
$\delta>0$, let $T_{1}$ be an embedded tube which is a $\delta$-neighborhood
of an embedded curve that sits inside $\\{1/2\leq x^{2}+y^{2}\leq 3/4\\}$ and
spins $N$ times around the $z$-coordinate axis. Also ensure that $T_{1}$
intersects the upper hemisphere of $\partial B_{1}$ in one small disc. Let
$T_{2}$ be a $\delta$-neighborhood of an embedded $B_{1}^{2}$, say defined by
$\\{x^{2}+y^{2}\leq 1/2,\,z\in[1/2-2\delta,1/2]\\}$. We ensure that $T_{2}$
intersects $T_{1}$ in a small neighborhood of some point. For $T=T_{1}\cup
T_{2}$, we let $U_{N}=B_{1}^{3}-T$. Note that $\partial U_{N}$ is
diffeomorphic to $\partial B_{1}^{3}$ and $\textup{area}(\partial
U_{N})\lesssim 1$ as long as $\delta>0$ is small enough. The set $U_{2}$ is
illustrated in Figure 2 below.

We now show that $D\gtrsim N^{1/2}$. Define $\pi:B_{1}^{3}\to B_{1}^{2}$ to be
the projection in the $z$-direction and let
$U^{\prime}=U_{N}\cap\\{x^{2}+y^{2}\leq 1/3,\,z\geq 1/2\\}$, in other words,
the region above the middle of $T_{2}$. By the coarea formula applied to
$\pi\circ F|_{F^{-1}(U^{\prime})}$, there is a point in $B^{2}_{1}$ whose pre-
image is a segment $\gamma^{\prime}\subset B_{1}^{3}$ with
$\textup{length}(\gamma^{\prime})\lesssim D$ and endpoints on $\partial
B_{1}^{3}$. Note that one of the end points of $F(\gamma^{\prime})$ lies on
$T_{2}$ and the other lies in the upper hemisphere of $\partial B^{3}_{1}$.
Let $\gamma^{\prime\prime}$ be the geodesic segment in $\partial B_{1}^{3}$
between the endpoints of $\gamma^{\prime}$ and define a loop
$\gamma=\gamma^{\prime}\cup\gamma^{\prime\prime}$. Note that
$F(\gamma^{\prime\prime})$ must wind around the entire length of $T_{1}$. By
the cone inequality we can fill $\gamma$ by a disc $\Sigma$ of area $\lesssim
D$.

Consider what $(\pi\circ F)(\gamma)$ looks like in $B^{2}_{1}$. All points in
$B^{2}_{1}$ with radius in $[1/3,1/2]$ have winding number $\gtrsim N$ with
respect to $(\pi\circ F)(\gamma)$. Thus,

$area(F(\Sigma))\geq area(\pi\circ F(\Sigma))\gtrsim N$

Where area is counted with multiplicity here. Putting together this
observation with $area(F(\Sigma))\leq area(\Sigma)D\lesssim D^{2}$, we get
$D\gtrsim N^{1/2}$.

∎

Figure 2: Sketch of $U_{2}$

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The Dijkgraaf-Witten invariants from second Chern classes

Takefumi Nosaka 111 E-mail address<EMAIL_ADDRESS>
###### Abstract

Given a 3-cocycle $\psi$ in the cohomology of a finite group $G$, we can
define the Dijkgraaf-Witten invariant of closed 3-manifolds. In this paper, we
focus on the case where $\psi$ is a 3-cocycle canonically obtained from the
second Chern class of a representation of $G$, and show some procedures for
computing the invariant, and clarify its topological interpretation under some
conditions.

Keywords
Group cohomology, 3-manifold, Chern classes, splitting principle

Subject Codes
20J06, 57K31, 19L10, 58J28,

##  1 Introduction

Let $M$ be a connected closed 3-manifold with orientation 3-class $[M]\in
H_{3}(M;{\mathbb{Z}})\cong{\mathbb{Z}}$. Let $G$ be a group $G$, and $BG$ be
an Eilenberg-MacLane space of type $(G,1)$. Fix a group 3-cocycle $\psi\in
H^{3}(BG;A)=H^{3}(G;A)$ with trivial coefficient module $A$. Then, for a group
homomorphism $f:\pi_{1}(M)\rightarrow G$, we can consider the pairing $\langle
f^{*}(\psi),[M]\rangle\in A$. As an important example, if $G$ is a Lie group
and $\psi$ is a Chern-Simons 3-class, the pairing is called the Chern-Simons
invariant with relation to hyperbolic volume. Meanwhile, as another example,
the Dijkgraaf-Witten invariant [DW, §6] is a toy model of the Chern-Simons
invariant when $G$ is of finite order. Precisely, the invariant is defined as
the following formal sum in the group ring ${\mathbb{Z}}[A]$:

$\mathrm{DW}_{\psi}(M):=\sum_{f\in\mathop{\mathrm{Hom}}\nolimits(\pi_{1}(M),G)}1_{{\mathbb{Z}}}\langle
f^{*}(\psi),[M]\rangle\in{\mathbb{Z}}[A].$ (1)

Here, $1_{{\mathbb{Z}}}$ is the unit of ${\mathbb{Z}}.$ The DW invariant seems
simple; however, there are few examples of the resulting computations and
studies of the invariant, unless $G$ is abelian or nilpotent.

However, to further explore the invariant (1) and to compute the pairing, it
is a problem to examine what kind of $\psi$ we should choose, and how we find
an explicit expression of $\psi$ when $G$ is not abelian. Since the Chern-
Simons 3-class is, roughly speaking, considered to be an inverse of a
transgression of the second Chern-class of the Lie group (see, e.g., [CS] or
[DW, §§3–5]), it is sensible to focus on a class of group 3-cocycles as such
inverses from second Chern-classes of $G$ with $|G|<\infty$. For this, if
$|G|<\infty,$ notice from the universal coefficient theorem the following
isomorphisms among the group (co)-homology:

$H^{n+1}(G;{\mathbb{Z}})\stackrel{{\scriptstyle\beta^{-1}}}{{\cong}}H^{n}(G;{\mathbb{Z}}/|G|{\mathbb{Z}})\cong
H^{n}(G;{\mathbb{Q}}/{\mathbb{Z}})\cong H_{n}(G;{\mathbb{Z}}),$ (2)

where the first isomorphism is given by the inverse of the Bockstein map from

$0\longrightarrow{\mathbb{Z}}\xrightarrow{\ |G|\textrm{-times}\
}{\mathbb{Z}}\longrightarrow{\mathbb{Z}}/|G|{\mathbb{Z}}\longrightarrow
0\quad\mathrm{(exact)}.$ (3)

In addition, as seen in [Ev2, Th], any complex representation
$\rho:G\rightarrow\mathrm{GL}_{n}({\mathbb{C}})$ uniquely admits the Chern-
class $c_{i}(\rho)\in H^{2i}(G;{\mathbb{Z}})$. Thus, we have a 3-cocycle of
the form

$\beta^{-1}(c_{2}(\rho))\in H^{3}(G;{\mathbb{Z}}/|G|{\mathbb{Z}}).$ (4)

In this paper, as a finite model of the Chern-Simons 3-class, we consider the
Dijkgraaf-Witten invariant (1) with $\psi=\beta^{-1}(c_{2}(\rho))$. First, as
a result of a Riemann-Roch type theorem, we develop a formula for describing
$12\beta^{-1}(c_{2}(\rho))$ (see Proposition 2.2), and show (Section 3) that,
under a certain condition, the twelvefold invariant can be computed from the
cohomology rings of some covering spaces of $M$. For example, the condition
holds if $G={\rm SL}_{2}(\mathop{\mathbb{F}_{q}}\nolimits)$ with $q>10$; see
Corollary 3.7. In Section 4, we argue a relation to the Chern-Simons
invariants and suggest a procedure for computing the Dijkgraaf-Witten
invariants of some Seifert 3-manifolds. Finally, Section 5 observes other
approaches to the invariants with $\psi=\beta^{-1}(c_{2}(\rho))$, as a
normalization of the 12–multiple.

Conventional notation. Let $G$ be a finite group, and $M$ be a closed
3-manifold with an orientation class $[M]\in
H_{3}(M;{\mathbb{Z}})\cong{\mathbb{Z}}$. By $BG$ we mean an Eilenberg-MacLane
space of type $(G,1)$. We denote the cyclic group
${\mathbb{Z}}/|G|{\mathbb{Z}}$ by ${\mathbb{Z}}_{|G|}$ for simplicity. Let
$\mu_{m}\in{\mathbb{C}}$ be $\mathrm{exp}(2\pi\sqrt{-1}/m)$ as a root of
unity.

##  2 Twelvefold second Chern classes

Let $\rho:G\rightarrow\mathrm{GL}_{n}({\mathbb{C}})$ be any representation,
and $c_{i}(\rho)\in H^{2i}(G;{\mathbb{Z}})$ be the Chern class of $\rho$; see,
e.g., [Ev2, Th] for the definitions. The purpose of this section is to develop
an algorithm to describe a representative 3-cocycle of the twelvefold
$12\beta^{-1}(c_{2}(\rho)).$

Let $c(\rho)\in H^{*}(G;{\mathbb{Z}})$ be the total Chern class, that is,
$c(\rho)=1+c_{1}(\rho)+c_{2}(\rho)+\cdots.$ Let $R_{\mathbb{C}}(G)$ be the
representation ring of $G$. As is known as Brauer theorem, there are finitely
many $n_{i}(\rho)\in{\mathbb{Z}}$ and subgroups $H_{i}\subset G$ with
one–dimensional representation
$\phi_{i}:H_{i}\rightarrow\mathrm{GL}_{1}({\mathbb{C}})$ such that

$\rho=\sum_{i}n_{i}(\rho)\mathrm{Ind}_{H_{i}}^{G}(\phi_{i})\in
R_{{\mathbb{C}}}(G).$ (5)

Here, $\mathrm{Ind}_{H_{i}}^{G}(\phi_{i})$ is the induced representation. By
the Whitney sum formula of $c(\rho)$, we have

$c(\rho)=\prod(1+c_{1}(\mathrm{Ind}_{H_{i}}^{G}(\phi_{i}))+c_{2}(\mathrm{Ind}_{H_{i}}^{G}(\phi_{i}))+\cdots)^{n_{i}(\rho)}\in
H^{*}(G;{\mathbb{Z}})$ (6)

To summarize, to compute $c_{2}(\rho)$, we may focus only
$c_{1}(\mathrm{Ind}_{H_{i}}^{G}(\phi_{i}))$ and
$c_{2}(\mathrm{Ind}_{H_{i}}^{G}(\phi_{i}))$.

Thus, we shall restrict ourselves to a subgroup $H\subset G$ and a
one–dimensional representation
$\phi:H\rightarrow\mathrm{GL}_{1}({\mathbb{C}})$, and show Lemma 2.1 below.
Identify ${\mathbb{Z}}_{|G|}$ with the multiplicative subgroup
$\\{\mu_{|G|}^{m}\mid m\in{\mathbb{Z}}\\}\subset{\mathbb{C}}^{\times}$. Thus
$\phi$ factors through a homomorphism
$\hat{\phi}:H\rightarrow{\mathbb{Z}}_{|G|}$ as a 1-cocycle, where
$\phi(h)=\mu_{|G|}^{\hat{\phi}(h)}$ for any $h\in H$. Denote the transfer map
by $\mathrm{Tr}^{G}_{H}:H^{n}(H;A)\rightarrow H^{n}(G;A)$; see, e.g., [Br,
§III.9] for the definition. Then, the following claims that the 3-cocycle
$12\beta^{-1}\bigl{(}c_{2}(\mathrm{Ind}_{H}^{G}(\phi))\bigr{)}$ can be
computed from $\beta$ and the transfers.

######  Lemma 2.1.

Take another subgroup $H^{\prime}\subset G$ with a representation
$\phi^{\prime}:H^{\prime}\rightarrow\mathrm{GL}_{1}({\mathbb{C}})$. Then, the
following two formulas hold:

$2\beta^{-1}\bigl{(}c_{1}(\mathrm{Ind}_{H}^{G}(\hat{\phi}))\smile
c_{1}(\mathrm{Ind}_{H^{\prime}}^{G}(\hat{\phi}^{\prime}))\bigr{)}=2\mathrm{Tr}^{G}_{H}(\hat{\phi})\smile\beta\circ\mathrm{Tr}^{G}_{H^{\prime}}(\hat{\phi}^{\prime}),$
(7)
$12\beta^{-1}\bigl{(}c_{2}(\mathrm{Ind}_{H}^{G}(\phi))\bigr{)}=6\bigl{(}\mathrm{Tr}^{G}_{H}(\hat{\phi})\smile\beta\circ\mathrm{Tr}^{G}_{H}(\hat{\phi})-\mathrm{Tr}^{G}_{H}(\hat{\phi}\smile\beta(\hat{\phi}))\bigr{)}\in
H^{3}(G;{\mathbb{Z}}_{|G|}).$ (8)

Furthermore, if $\hat{\phi}$ extends to a 1-cocycle
$G\rightarrow{\mathbb{Z}}_{|G|}$, then (8) vanishes.

###### Proof.

Recall the integral Riemann-Roch theorem in group cohomology (see, e.g., [Th,
Theorem 6.3]), which states

$6c_{2}(\mathrm{Ind}_{H}^{G}(\phi))=3c_{1}(\mathrm{Ind}_{H}^{G}(\phi)^{2})-3\mathrm{Tr}^{G}_{H}(c_{1}(\phi)\smile
c_{1}(\phi))\in H^{4}(G;{\mathbb{Z}}),$ (9)
$2c_{1}(\mathrm{Ind}_{H}^{G}(\phi))=2\mathrm{Tr}^{G}_{H}(c_{1}(\phi))\in
H^{2}(G;{\mathbb{Z}}).$ (10)

Notice from [Th, p. 68] that $c_{1}(\phi)=\beta(\hat{\phi})\in
H^{2}(H;{\mathbb{Z}})$, and from [Br, V.3.(3.3)] that

$\beta(u\smile v)=\beta(u)\smile v,\quad\quad\mathrm{for\ any\ }u\in
H^{n}(G;{\mathbb{Z}}_{|G|}),v\in H^{m}(G;{\mathbb{Z}}).$ (11)

Since $\mathrm{Ind}_{H}^{G}\circ\beta=\beta\circ\mathrm{Ind}_{H}^{G}$ (see
[Ev1, Exercise 4.2.4]), the twice of the right hand side of (9) is reduced to
$6\beta\bigl{(}\hat{\phi}\smile
c_{1}(\mathrm{Ind}_{H}^{G}(\phi))-\mathrm{Tr}^{G}_{H}(\hat{\phi}\smile
c_{1}(\phi))\bigr{)}$. By applying this to (11), we readily obtain (8) as
desired. In addition, we immediately check (7) from (10) and (11). Finally,
the last claim follows directly from the projection formula; see, e.g., [Br,
V.(3.8)]. ∎

To summarize, we readily reach the following conclusion:

######  Proposition 2.2.

For a representation $\rho:G\rightarrow\mathrm{GL}_{n}({\mathbb{C}})$, we
suppose (5). Then,

$12(\beta^{-1}(c_{2}(\rho))=6\sum_{k=1}^{m}n_{k}(\rho)\Bigl{(}\bigl{(}\sum_{j=1}^{m}n_{j}(\rho)\mathrm{Tr}^{G}_{H_{j}}(\hat{\phi}_{j})\smile\beta\circ\mathrm{Tr}^{G}_{H_{k}}(\hat{\phi}_{k})\bigr{)}-\mathrm{Tr}^{G}_{H_{k}}(\hat{\phi_{k}}\smile\beta(\hat{\phi_{k}}))\Bigr{)}.$
(12)

######  Remark 2.3.

We here remark a topological meaning of the first term in (12). In general,
for any $1$-cocycles $\phi_{j}:G\rightarrow{\mathbb{Z}}_{|G|}$ as in
$\phi=\mathrm{Tr}^{G}_{H_{j}}(\hat{\phi}_{j})$ and any homomorphism
$f:\pi_{1}(M)\rightarrow{\mathbb{Z}}_{|G|}$, we regard $\phi_{j}\circ f$ as
first cohomology classes via
$H^{1}(M;{\mathbb{Z}}_{|G|})=\mathop{\mathrm{Hom}}\nolimits(\pi_{1}(M),{\mathbb{Z}}_{|G|})$.
Hence, the pairing $\langle f^{*}(\psi),[M]\rangle$ with
$\psi=\hat{\phi_{j}}\smile\beta(\hat{\phi_{k}})$ can be computed from the
cohomology ring $H^{*}(M;{\mathbb{Z}}_{|G|})$ or the linking form of $M$. In
addition, the pairing can be computed in several ways: for example, see [MOO]
for such computations from some quasitriangular Hopf algebras, branched
coverings, or Kirby diagrams. Meanwhile, the second term in (12) will be
topologically considered in Section 3.

By Lemma 2.1, we should describe explicit formula of $\beta$ and
$\mathrm{Tr}^{G}_{H}$ in the non-homogeneous complex of $G$. First, from the
definition of the Bockstein map, $\beta(\hat{\phi}):H\times
H\rightarrow{\mathbb{Z}}$ is represented by

$\beta(\hat{\phi})(g,h):=\left\\{\begin{array}[]{ll}1,&\mathrm{if}\quad\overline{\hat{\phi}(g)}+\overline{\hat{\phi}(h)}\geq|G|,\\\
0,&\mathrm{if}\quad\overline{\hat{\phi}(g)}+\overline{\hat{\phi}(h)}<|G|.\end{array}\right.$
(13)

Here, for $m\in{\mathbb{Z}}_{|G|}$, we denote the representative in
$\\{1,2,\dots,|G|\\}\subset{\mathbb{Z}}$ by $\overline{m}$. Next, let us
choose a left complete set of representatives,
$T:=\\{g_{i}\\}_{i=1}^{|G/H|}\subset G$, such that $T\cap H=\\{1_{G}\\}.$ For
$\sigma\in G$, we fix notation

$\bar{\sigma}:=H\sigma\cap
T,\quad\quad\widetilde{\sigma}:=\sigma(\bar{\sigma})\in H.$

Then, according to [Br, Exercise II.10.2],
$\mathrm{Tr}^{G}_{H}(\hat{\psi}):G\rightarrow{\mathbb{Z}}_{|G|}$ is
represented by a corresponding

$\sigma\longmapsto\sum_{i:1\leq
i\leq|G/H|}\hat{\phi}(g_{i}\sigma\overline{g_{i}\sigma}^{-1}).$ (14)

In addition, for a 3-cocycle $\psi:H^{3}\rightarrow A$, the transfer
${\mathrm{Tr}}_{H}^{G}(\hat{\psi})$ is known to be represented as a map
$G^{3}\rightarrow A$ defined by setting

$(\sigma_{1},\sigma_{2},\sigma_{3})\longmapsto\sum_{i:1\leq
i\leq|G/H|}\psi(\widetilde{\bar{g_{i}}\sigma_{1}},\widetilde{\overline{g_{i}\sigma_{1}}\sigma_{2}},\widetilde{\overline{g_{i}\sigma_{1}\sigma_{2}}\sigma_{3}}).$
(15)

In summary, combing the above formula (13)–(15) and Lemma 2.1, we can obtain
an explicit formula of $12\beta^{-1}(c_{2}({\mathrm{Ind}}_{H}^{G}\phi))$ as a
map $G^{3}\rightarrow{\mathbb{Z}}_{|G|}$. In particular, if every 4-cocycle
$C\in H^{4}(G;{\mathbb{Z}})$ admits a representation
$\rho:G\rightarrow\mathrm{GL}_{n}({\mathbb{C}})$ such that
$12C=12c_{2}(\rho)$, then we can obtain a representative
$G^{3}\rightarrow{\mathbb{Z}}_{|G|}$ of the 12–multiple of every 3-cocycle of
$G$. For example, if $G$ has a periodic cohomology, $G$ satisfies this
condition; see [Th] for the detail and other examples, with a comparison with
studies of the Chern subring $\subset H^{\rm even}(G;{\mathbb{Z}})$.

##  3 Finite groups of type $C_{n,m}$ and covering spaces

In this section, we will introduce a class of finite groups, and show that, if
$G$ lies in the class, some multiple of the Dijkgraaf-Witten invariant of $M$
can be computed from the cohomology rings of some finite covering spaces of
$M$. For this purpose, let us notice the following lemmas:

######  Lemma 3.1.

For a surjective homomorphism $f:\pi_{1}(M)\rightarrow G$ and a subgroup
$\iota:H\subset G$, let $\pi:M_{f,H}\rightarrow M$ be the finite covering
associated with the subgroup $f^{-1}(H)\subset\pi_{1}(M)$. Suppose
$m\in\mathbb{N}$, $\bar{\psi}\in H^{3}(G;{\mathbb{Z}}_{|G|})$ and $\psi\in
H^{3}(H;{\mathbb{Z}}_{|G|})$ such that $\iota^{*}(\bar{\psi})=m\psi$. Then,

$|G/H|m\langle\mathrm{Tr}^{G}_{H}(\psi),f_{*}([M])\rangle=|G/H|m\langle\psi,\mathrm{res}(f)_{*}([M_{f,H}])\rangle\in{\mathbb{Z}}_{|G|}.$
(16)

In particular, if $\psi=\hat{\phi_{k}}\smile\beta(\hat{\phi_{k}})$ as in the
second term in (12), then the pairing (16) is computed from the cup product of
the space $M_{f,H}$

###### Proof.

By $\pi_{*}([M_{f,H}])=|G/H|[M]$ and
$f\circ\pi=\iota_{*}\circ\mathrm{res}(f)$, we have

$m|G/H|\langle\mathrm{Tr}^{G}_{H}(\psi),f_{*}([M])\rangle=\langle\mathrm{Tr}^{G}_{H}(\iota^{*}(\bar{\psi})),f_{*}\circ(\pi)_{*}([M_{f,H}])\rangle=|G/H|\langle\bar{\psi},(\iota)_{*}\circ\mathrm{res}(f)_{*}([M_{f,H}])\rangle$
$=|G/H|\langle\iota^{*}(\bar{\psi}),\mathrm{res}(f)_{*}([M_{f,H}])\rangle=m|G/H|\langle\psi,\mathrm{res}(f)_{*}([M_{f,H}])\rangle.$
(17)

Both hand sides of (17) are exactly (16). ∎

We remark that the surjectivity of $f$ is not so important; indeed, if $f$ is
not surjective, we may replace $H\subset G$ as $H\cap\mathrm{Im}(f)\subset
G\cap\mathrm{Im}(f)$. We also give a sufficient condition for the assumption
in Lemma 3.1:

######  Lemma 3.2.

Suppose an inclusion $\iota:H\hookrightarrow G$ such that
$\mathrm{g.c.d}(|H|,|G/H|)$ is divisible by $m$. Then, any $\psi\in
H^{n}(H;{\mathbb{Z}}_{|G|})$ admits $\bar{\psi}\in
H^{n}(G;{\mathbb{Z}}_{|G|})$ such that $\iota^{*}(\bar{\psi})-m\psi\in{\rm
Ker}(\mathrm{Tr}^{G}_{H})$. In particular, if $\mathrm{Tr}^{G}_{H}$ is
injective, $\iota^{*}(\bar{\psi})=m\psi$.

###### Proof.

Choose $s\in{\mathbb{N}}$ such that $|G/H|=ms$ and $(s,|H|)=1$. Since
$H^{n}(H;{\mathbb{Z}}_{|G|})$ is annihilated by $|H|$, we can define
$\bar{\psi}$ to be $s^{-1}\mathrm{Tr}_{H}^{G}(\psi)$. By the familiar equality
$|G/H|x=\mathrm{Tr}_{H}^{G}(\iota^{*}(x))$ for any $x\in
H^{n}(G;{\mathbb{Z}}_{|G|})$, we have
$\mathrm{Tr}_{H}^{G}(\iota^{*}(\bar{\psi})-m\psi)=0$ as required. ∎

From these lemmas, it seems sensible to consider the following definition:

######  Definition 3.3.

For $n,m\in{\mathbb{Z}}$, we say a finite group $G$ to be of type $C_{n,m}$,
if there are finitely many $n_{i}\in{\mathbb{Z}}$ with $1\leq i\leq k$ and
subgroups $\iota_{i}:H_{i}\subset G$ with one–dimensional representation
$\phi_{i}:H_{i}\rightarrow\mathrm{GL}_{1}({\mathbb{C}})$ such that

  1. (i)

The subring of $H^{*}(G;{\mathbb{Z}})$ generated by the total Chern classes
$c(\mathrm{Ind}_{H_{j}}^{G}(\phi_{i}))$ with $1\leq i\leq k$ includes the
$m$–multiple of the $2n$-th cohomology $mH^{2n}(G;{\mathbb{Z}})$.

  2. (ii)

For any $i$, the greatest common divisor of $|H_{i}|$ and $|G/H_{i}|$ is
divisible by $m$, and the image of the restriction $\iota_{i}^{*}$ contains
$mH^{2n}(H_{i};{\mathbb{Z}})$.

######  Example 3.4.

For any $n,m\in{\mathbb{Z}}$, every cyclic group is of type $C_{n,m}$ by
definition.

If any Sylow group of $G$ is of $C_{n,m}$-type, so is $G$, since
$\oplus_{K\subset G:\textrm{Sylow
group}}H^{*}(K;{\mathbb{Z}}_{|G|})\rightarrow H^{*}(G;{\mathbb{Z}}_{|G|})$
induced by the inclusions is known to be surjective; see [Br, Theorem
III.10.3]. For example, for any odd prime power $q\in{\mathbb{Z}}$ with
$q>10$, the special linear group
$\mathrm{SL}_{2}(\mathop{\mathbb{F}}\nolimits_{q})$ is of type $C_{2,2}$. In
fact, it is classically known (see, e.g., [Hut]) that, if $p$ is odd, then any
$p$-Sylow group, $G_{p}$, of
$\mathrm{SL}_{2}(\mathop{\mathbb{F}}\nolimits_{q})$ is cyclic, and the induced
map $H_{3}(G_{p};{\mathbb{Z}})\rightarrow
H_{3}(\mathrm{SL}_{2}(\mathop{\mathbb{F}}\nolimits_{q});{\mathbb{Z}})\cong{\mathbb{Z}}/q^{2}-1$
is injective; the $2$-Sylow group $G_{2}$ is the generalized quaternion group
of order $2^{t}$, and contains a cyclic subgroup ${\mathbb{Z}}_{2^{t-1}}$
which induces an injection
$H_{3}({\mathbb{Z}}_{2^{t-1}};{\mathbb{Z}})\rightarrow
H_{3}(G_{2};{\mathbb{Z}})\cong{\mathbb{Z}}_{2^{t}}.$

Similarly, we can show that the projective one
$\mathrm{PSL}_{2}(\mathop{\mathbb{F}}\nolimits_{q})$ is also of type
$C_{2,2}$.

Let us state a theorem. If $\psi=\hat{\phi_{k}}\smile\beta(\hat{\phi_{k}})$,
the left hand side of (16) is equal to the $m|G/H|$–multiple of the second
term in (12). As mentioned in Remark 2.3, the right hand side of (16) can be
computed from the linking form of the covering space $M_{f,H_{i}}$. If $G$ is
of type $C_{2,m}$, the $m$–multiple of any 3-cocycle $\psi$ is equal to a
certain sum of the forms
$\beta^{-1}(c_{2}(\mathrm{Ind}_{H_{i}}^{G}(\phi_{i})))$, and satisfies the
assumption of Lemmas 3.1 and 3.2. In conclusion, we have a summary:

######  Theorem 3.5.

Let $G$ be a finite group of type $C_{2,m}$. Let $m^{\prime}\in\mathbb{N}$ be
$\mathrm{l.c.d}(12,m^{2})$. Then, for any 3-cocycle $\lambda\in
H^{3}(G;{\mathbb{Z}}_{|G|})$, the DW invariant (1) with
$\psi=m^{\prime}\lambda$ can be computed from the set
$\mathop{\mathrm{Hom}}\nolimits(\pi_{1}(M),G)$ and the linking forms of the
covering spaces $M_{f,H_{i}}$.

######  Remark 3.6.

The assumption of type $C_{2,m}$ seems necessary because of unnaturality of
the transfer. To be precise, for any homomorphism $f:K\rightarrow G$ and a
subgroup $H\subset G$ of finite index,
$\mathrm{res}(f)^{*}\circ{\mathrm{Tr}}_{H}^{G}$ and
${\mathrm{Tr}}_{f^{-1}(H)}^{K}\circ f^{*}$ are not always equal. For instance,
for any odd prime $p\in{\mathbb{Z}}$, if ${\mathbb{Z}}_{p}=:H\subset
G:=\mathfrak{S}_{p}$ and
$f:K:={\mathbb{Z}}_{p}\hookrightarrow\mathfrak{S}_{p}$, then
$f^{*}\circ{\mathrm{Tr}}_{H}^{G}=0$ and ${\mathrm{Tr}}_{f^{-1}(H)}^{K}\circ
f^{*}$ is the identity on $H^{1}(\bullet;{\mathbb{Z}}_{p}).$ Thus, in general,
(16) does not always hold without the assumption.

######  Corollary 3.7.

Let $G$ be either $\mathrm{SL}_{2}(\mathop{\mathbb{F}}\nolimits_{q})$ or
$\mathrm{PSL}_{2}(\mathop{\mathbb{F}}\nolimits_{q})$ with odd $q\geq 10$. Take
a generator $\phi$ of $H^{3}(G;{\mathbb{Z}}_{|G|})$. Then, the DW invariant
(1) with $\psi=12\phi$ can be computed from the set
$\mathop{\mathrm{Hom}}\nolimits(\pi_{1}(M),G)$ and the linking forms of some
finite covering spaces of $M$.

######  Example 3.8.

We now give a computation of DW invariants from the dihedral group. Let
$m\in\mathbb{N}$ be odd, and $G$ be the dihedral group
$D_{m}:={\mathbb{Z}}_{m}\rtimes{\mathbb{Z}}_{2}$. Take
$\phi:{\mathbb{Z}}_{m}\rightarrow\mathrm{GL}_{1}({\mathbb{C}})={\mathbb{C}}^{\times}$
which sends $n$ to $\mu_{m}^{n}.$ As is known (see, e.g., page 74 of [Th]),
the group cohomology $2H^{4}(D_{m};{\mathbb{Z}})\cong{\mathbb{Z}}_{m}$ is
generated by $c_{2}(\rho)$, where
$\rho={\mathrm{Ind}}_{{\mathbb{Z}}_{m}}^{D_{m}}\phi$. Namely, the group
$D_{m}$ is of type $C_{2,m}$. Hence, by (12) and (16), the $12$–multiple of
the DW invariant (1) with $\psi=\beta^{-1}(c_{2}(\rho))$ can be computed from
the cohomology rings of some double covering spaces of $M$.

For more concrete computations, let us fix some 3-manifolds. Consider the
generalized quaternion group $Q_{4m}$ of order $4m$, which has a presentation
$\langle x,y\mid x^{m}=y^{2},xyx=y\rangle$. It is known that

$H_{1}(Q_{4m};{\mathbb{Z}})\cong{\mathbb{Z}}_{4},\quad
H_{2}(Q_{4m};{\mathbb{Z}}),\quad
H_{3}(Q_{4m};{\mathbb{Z}})\cong{\mathbb{Z}}_{4m}.$

Since $Q_{4m}$ injects $SU(2)=S^{3}$, we have a closed 3-manifold
$M_{m}:=S^{3}/Q_{4m}$. Notice $H_{1}(M_{m};{\mathbb{Z}})={\mathbb{Z}}/4$ and
$H_{2}(M_{m};{\mathbb{Z}})=0$. Since $x^{m}$ lies in the center of $Q_{4m}$,
and the quotient $Q_{4m}/\langle x^{m}\rangle$ is isomorphic to $D_{m}$, we
have a bijection
$\mathop{\mathrm{Hom}}\nolimits(\pi_{1}(M_{m}),D_{m})\simeq\mathop{\mathrm{Hom}}\nolimits(D_{m},D_{m})$.
By the Lyndon-Hochschild-Serre spectral sequence, the projection
$Q_{4m}\rightarrow D_{m}$ induces a surjection
$H_{3}(Q_{4m};{\mathbb{Z}})\rightarrow
2H_{3}(D_{m};{\mathbb{Z}})\cong{\mathbb{Z}}_{m}.$ Meanwhile, the fourfold
covering space of $M_{m}$ by the abelianization is the lens space $L(m,q)$ for
some $q\in{\mathbb{Z}}_{m}$. Therefore, the cover $L(m,q)\rightarrow M_{m}$
induces a canonical injection

$\mathcal{B}:{\mathbb{Z}}_{m}\cong\mathop{\mathrm{Hom}}\nolimits(\pi_{1}(L(m,q)),{\mathbb{Z}}_{m})\hookrightarrow\mathop{\mathrm{Hom}}\nolimits(\pi_{1}(M_{m}),D_{m})$

such that the complement
$\mathop{\mathrm{Hom}}\nolimits(\pi_{1}(M_{m}),D_{m})\setminus\mathrm{Im}(\mathcal{B})$
consists of the trivial map. Since
$\mathrm{DW}_{\beta^{-1}(c_{1}(\phi)^{2})}(L(m,q))=\sum_{j=1}^{m}1_{{\mathbb{Z}}}(qj^{2})\in{\mathbb{Z}}[{\mathbb{Z}}_{m}]$
is known (see, e.g, [DW, MOO]), by (16), we conclude
$\mathrm{DW}_{\beta^{-1}c_{2}(\rho)}(M_{m})=\sum_{j=0}^{m}1_{{\mathbb{Z}}}(2qj^{2})\in{\mathbb{Z}}[{\mathbb{Z}}_{2m}]$.

More generally, it is sensible to examine more examples of groups that are
(not) of type $C_{2,m}$, and to describe the second Chern classes with
computations of the associated Dijkgraaf-Witten invariants. For example, if
$G=\mathrm{SL}_{2}(\mathbb{F}_{q})$ as in Corollary 3.7, we can compute the
invariants via the procedure in Remark 2.3; however, the resulting are
complicated. In this paper, we omit to add other examples of computation;
however, the computation of some Seifert homology spheres with
$G=\mathrm{PSL}_{2}(\mathbb{F}_{p})$ is seen in [Y].

##  4 Relation to the Chern-Simons invariant

We also consider the DW invariant from the viewpoint of the Chern-Simons
invariant.

As mentioned in [CS], the Chern-Simons 3-class can be considered as a group
3-cocycle $\mathrm{CCS}\in
H^{3}(\mathrm{GL}_{N}(\mathbb{C});{\mathbb{C}}/{\mathbb{Z}})$ with $N\geq 2$.
Thus, given a homomorphism
$F:\pi_{1}(M)\rightarrow\mathrm{GL}_{N}(\mathbb{C})$, the Chern-Simons
invariant, $\mathrm{CS}(F)$, is defined as
$\langle\mathrm{CCS},F_{*}([M])\rangle\in{\mathbb{C}}/{\mathbb{Z}}$.

######  Proposition 4.1.

Take a representation $\rho:G\rightarrow\mathrm{GL}_{N}(\mathbb{C})$. Define
$\kappa:{\mathbb{Z}}_{|G|}\hookrightarrow{\mathbb{Q}}/{\mathbb{Z}}$ by
$k\mapsto k/|G|.$ Then, the image of $\rho^{*}(\mathrm{CCS})$ is contained in
${\mathbb{Q}}/{\mathbb{Z}}$, and

$\rho^{*}(\mathrm{CCS})=\kappa(\beta^{-1}(c_{2}(\rho)))\in
H^{3}(G;{\mathbb{Q}}/{\mathbb{Z}}).$ (18)

In particular, the DW invariant with $\psi=\beta^{-1}(c_{2}(\rho))$ is equal
to $\sum_{f}1_{{\mathbb{Z}}}\kappa(\mathrm{CS}(\rho\circ
f))\in{\mathbb{Z}}[{\mathbb{Z}}_{|G|}]$ where $f$ runs over
$\mathop{\mathrm{Hom}}\nolimits(\pi_{1}(M),G)$.

There are many procedures to compute the Chern-Simons invariant. For example,
[JW] mentions relations from the $e$-invariant and $\eta$-invariant in
differential topology. If the image of $F$ is contained in
$\mathrm{SL}_{2}(\mathbb{C})$, the cocycle expression of $\mathrm{CCS}$ is
described in [DZ]. As a special case, if $M$ is a Seifert manifold over
$S^{2}$ and $H_{*}(M;{\mathbb{Z}})\cong H_{*}(S^{3};{\mathbb{Z}})$ and the
Seifertor is sent to trivial by $f$, Theorem C in [JW]222In the theorem C, the
image is required to be contained in $\mathrm{SL}_{N}({\mathbb{C}})$; however,
if we consider the inclusion
$\mathrm{GL}_{N}({\mathbb{C}})\hookrightarrow\mathrm{SL}_{N+1}({\mathbb{C}})$
which sends $A$ to $\begin{pmatrix}A&0\\\
0&\mathop{\mathrm{det}}\nolimits(A)^{-1}\\\ \end{pmatrix}$, we can apply the
theorem for the computation of $\mathrm{CS}(F).$ gives an explicit formula to
compute $2N$–multiple of the Chern-Simons invariant
$2N\rho^{*}(\mathrm{CCS})$; In conclusion, following Proposition 4.1, the DW
invariant with $\psi=2N\beta^{-1}(c_{2}(\rho))$ of Seifert homology spheres
can be sometimes easily computed with the help of a computer program; see also
[Ch, Y] for the computation of $\mathop{\mathrm{Hom}}\nolimits(\pi_{1}(M),G)$.

Now, we give the proof of Proposition 4.1. Let $K$ be a complex Lie group with
finite connected components, and $I_{k}(K,{\mathbb{C}})$ denote the ring
consisting of $K$-invariant polynomials on the Lie algebra $\mathfrak{k}$.
Recall from the classical Chern-Weil theory that there is a natural
homomorphism $W:I_{k}(K,{\mathbb{C}})\rightarrow H^{2k}(BK^{\rm
top},{\mathbb{C}})$, where $BK^{\rm top}$ is the topological classifying space
of $K$. Let $r$ denote the map $H^{*}(BK^{\rm top};{\mathbb{Z}})\rightarrow
H^{*}(BK^{\rm top};{\mathbb{C}})$ induced by the inclusion
${\mathbb{Z}}\hookrightarrow{\mathbb{C}}$. From [CS], consider

$J^{k}(K,{\mathbb{C}}):=\\{(P,u)\in I^{k}(K,{\mathbb{C}})\times H^{2k}(BK^{\rm
top};{\mathbb{Z}})\mid W(P)=r(u)\\}.$

Then, in [CS], there is a homomorphism
$\textrm{C-C-S}_{K}:J^{k}(K,{\mathbb{C}})\rightarrow
H^{2k-1}(BK;{\mathbb{C}}/{\mathbb{Z}})$ such that any homomorphism
$\phi:G\rightarrow K$ ensures the naturality
$\phi^{*}\circ\textrm{C-C-S}_{G}=\textrm{C-C-S}_{K}\circ\phi^{*}$; see also
Page 1 of [DZ]. As important examples, it is widely known that if
$K=\mathrm{GL}_{n}({\mathbb{C}})$ and $u=c_{2}\in H^{4}(BK^{\rm
top},{\mathbb{Z}})$ is the Chern class, and
$P(A)=((\mathrm{Tr}A)^{2}-\mathrm{Tr}(A^{2}))/8\pi^{2}$, then
$\textrm{C-C-S}_{K}(P,u)=\textrm{CCS}$. Furthermore, if $G$ is of finite order
and $u=c_{2}(\rho)$, then $BG^{\rm top}=BG$ and $I^{k}(G,{\mathbb{C}})$ is
zero; thus, $\textrm{C-C-S}_{G}$ is equal to
$\beta_{{\mathbb{C}}/{\mathbb{Z}}}^{-1}(c_{2}(\rho))$ by definition, where
$\beta_{{\mathbb{C}}/{\mathbb{Z}}}$ is the Bockstein map from

$0\longrightarrow{\mathbb{Z}}\longrightarrow{\mathbb{C}}\longrightarrow{\mathbb{C}}/{\mathbb{Z}}\longrightarrow
0\quad\mathrm{(exact)}.$ (19)

###### Proof of Proposition 4.1.

As mentioned above, $\beta_{{\mathbb{C}}/{\mathbb{Z}}}^{-1}(c_{2}(\rho))$ lies
in $H^{3}(G;{\mathbb{Q}}/{\mathbb{Z}})$. Since
$\beta^{-1}=\kappa\circ\beta_{{\mathbb{C}}/{\mathbb{Z}}}^{-1}$, the above
naturality deduces to

$\kappa(\beta^{-1}(c_{2}(\rho)))=\beta_{{\mathbb{C}}/{\mathbb{Z}}}^{-1}(c_{2}(\rho))=\textrm{C-C-S}_{G}(\rho^{*}(P),\rho^{*}(c_{2}))=\rho^{*}(\textrm{C-C-S}_{K}(P,c_{2}))=\rho^{*}(\textrm{CCS}),$

which is the required (18). ∎

##  5 Other cohomological approaches to second Chern classes

We end this paper by suggesting other three approaches to second Chern
classes, and examining renormalization of the 12–multiple in Proposition 2.2.

First, let us review [Ev2, Theorem 4]. Take a subgroup $H\subset G$ with a
representation $\phi:H\rightarrow\mathrm{GL}_{1}({\mathbb{C}})$. Let
$\pi={\mathrm{Ind}}_{H}^{G}(1)$. Then, the theorem says the equalities

$c_{1}({\mathrm{Ind}}_{H}^{G}(\phi))={\mathrm{Tr}}_{H}^{G}(c_{1}(\phi))+c_{1}(\pi)\in
H^{2}(G;{\mathbb{Z}}),$ (20)
$c_{2}({\mathrm{Ind}}_{H}^{G}(\phi))=\mathcal{N}_{H}^{G}(1+c_{1}(\phi))+c_{1}(\pi)\smile{\mathrm{Tr}}_{H}^{G}(c_{1}(\phi))+c_{2}(\pi)\in
H^{4}(G;{\mathbb{Z}}).$ (21)

Here, $\mathcal{N}_{H}^{G}$ is the multiplicative transfer of Evens (see [Ev1,
§6.1]), and $2c_{1}(\pi)=0$ and $6c_{2}(\pi)=0$ are known (compared with (9)).

As mentioned in [Q] as an unpublished note, a representative 4-cocycle of the
first term in (21) is described as follows. Let $\mathfrak{S}_{|G/H|}$ be the
symmetric group of order $|G/H|$ with a permutation action on the product
$H^{|G/H|}$. We can define the semidirect product
$H^{|G/H|}\rtimes\mathfrak{S}_{|G/H|}$. Take a group homomorphism
$\Phi:G\rightarrow H^{|G/H|}\rtimes\mathfrak{S}_{|G/H|}$ defined in [Ev1,
§5.2], which is called the monomial representation, and consider a non-
homogeneous 4-cocycle
$\tilde{c}_{4}:(H^{|G/H|}\rtimes\mathfrak{S}_{|G/H|})^{4}\rightarrow{\mathbb{Z}}$
in the non-homogeneous cochain group defined by setting

$[(h_{1},\dots,h_{|G/H|},\sigma)|(h_{1}^{\prime},\dots,h_{|G/H|}^{\prime},\sigma^{\prime})|(h_{1}^{\prime\prime},\dots,h_{|G/H|}^{\prime\prime},\sigma^{\prime\prime})|(h_{1}^{\prime\prime\prime},\dots,h_{|G/H|}^{\prime\prime\prime},\sigma^{\prime\prime\prime})]\mapsto$
$\sum_{1\leq i\neq
j\leq|G/H|}c_{1}(\phi)(h_{\sigma^{-1}(i)},h_{(\sigma\sigma^{\prime})^{-1}(i)}^{\prime})\cdot
c_{1}(\phi)(h_{(\sigma\sigma^{\prime}\sigma^{\prime\prime})^{-1}(j)}^{\prime\prime},h_{(\sigma\sigma^{\prime}\sigma^{\prime\prime}\sigma^{\prime\prime\prime})^{-1}(j)}^{\prime\prime\prime}),$
(22)

where the 2-cocycle $c_{1}(\phi)=\beta(\hat{\phi})$ is represented by a map
$H\times H\rightarrow{\mathbb{Z}}$ as in (13). Then,
$\mathcal{N}_{H}^{G}(1+c_{1}(\phi))=\Phi^{*}(\tilde{c}_{4})$ is shown [Q].
Moreover, in general, for a non-homogeneous 4-cocycle
$C_{4}:G^{4}\rightarrow{\mathbb{Z}}$, the inverse $\beta^{-1}(C_{4})$ is
represented by a map

$G\times G\times
G\longrightarrow{\mathbb{Z}}_{|G|};(g_{1},g_{2},g_{3})\longmapsto[\sum_{g\in
G}\frac{C_{4}(g,g_{1},g_{2},g_{3})}{|G|}]$ (23)

where we should notice that the sum $\sum_{g\in G}C_{4}(g,g_{1},g_{2},g_{3})$
is annihilated by $|G|$ for any $(g_{1},g_{2},g_{3})\in G^{3}.$ This formula
(23) of $\beta^{-1}(C_{4})$ can be easily proved from the definition of the
Bockstein map. In conclusion, by (21)–(23), we can find a representative
3-cocycle of $\beta^{-1}(c_{2}({\mathrm{Ind}}_{H}^{G}(\phi)))$ as a map
$G^{3}\rightarrow{\mathbb{Z}}_{|G|}$. Meanwhile, following an outline in [No,
§5], for some 3-manifolds $M$, we can sometimes describe a chain 3-class of
$[M]$; hence, for any homomorphism $f:\pi_{1}(M)\rightarrow G$, we can find a
3-chain of the pushforward $f_{*}([M])\in H_{3}(G;{\mathbb{Z}})$. In
conclusion, by combing the 3-chain with the presentation of
$c_{2}({\mathrm{Ind}}_{H}^{G}(\phi))$, we can compute the Dijkgraaf-Witten
invariant with $\psi=\beta^{-1}c_{2}({\mathrm{Ind}}_{H}^{G}(\phi))$.

As pointed out in [Ev2, Q], the 12–multiple is essentially derived from the
(stable) third homology
$H_{3}(\mathfrak{S}_{n};{\mathbb{Z}})\cong{\mathbb{Z}}_{12}\oplus({\mathbb{Z}}_{2})^{d_{n}}$
with relation to the stable homotopy group of spheres
$\pi_{3}^{S}\cong{\mathbb{Z}}_{24}$, where $d_{4}=d_{5}=1,$ and $d_{n}=2$ for
$n\geq 6$. The third framed bordism group is also isomorphic to $\pi_{3}^{S}$.
Thus, a refinement of the 12–multiple of the DW invariant should be
interpreted from the sphere spectrum or some (co)-homology operations.

As the second approach, let us observe a topological approach by the splitting
principle. Given a representation
$\rho:G\rightarrow\mathrm{GL}_{n}({\mathbb{C}})$, we can construct a cellular
map $\bar{\rho}:BG\rightarrow\mathrm{Gr}(n)$ such that
$c_{i}(\rho)=\bar{\rho}^{*}(c_{i})$, where $\mathrm{Gr}(n)$ is the infinite-
dimensional complex Grassmannian manifold and $c_{i}\in
H^{2i}(\mathrm{Gr}(n);{\mathbb{Z}})$ is the Chern class; see, e.g., [Th, §5]
for the construction. Let $\gamma$ be the tautological vector bundle on
$\mathrm{Gr}(n)$ such that $c_{i}=c_{i}(\gamma)$. For a homomorphism
$f:\pi_{1}(M)\rightarrow G$, the splitting principle admits a fibration
$p_{f}:M_{f}\rightarrow M$ such that
$p_{f}^{*}:H^{*}(M;{\mathbb{Z}})\rightarrow H^{*}(M_{f};{\mathbb{Z}})$ is
injective and the pullback $p_{f}^{*}\circ f^{*}(\gamma)$ is decomposed as a
Whiteny sum of some line bundles on $M_{f}$. As in (7), the 3-cocycle
$\beta^{-1}p_{f}^{*}\circ f^{*}(c_{2}(\rho))$ can be computed from the
cohomology ring of $M_{f}$. In summary, if we can construct $M_{f}$ as a
manifold and determine the cohomology ring $H^{*}(M_{f};{\mathbb{Z}}_{|G|})$,
we can compute the pairing $\langle
f^{*}(\beta^{-1}(c_{2}(\rho))),[M]\rangle\in{\mathbb{Z}}_{|G|}$.

Finally, we briefly mention modular representations. Choose distinct primes
$p,\ell\in{\mathbb{N}}$. While we considered only complex representations, we
can also obtain from any representation $\rho:G\rightarrow{\rm
GL}_{n}(\overline{\mathbb{F}}_{p})$ uniquely a Chern class $c_{i}(\rho)\in
H^{2i}(G;\widehat{{\mathbb{Z}}}_{\ell})$, where
$\widehat{{\mathbb{Z}}}_{\ell}$ is the $\ell$-adic integer ring; see [Kr] or
[Th, p.118] for the details. Therefore, similarly to (4), we have a 3-cocycle
$\beta^{-1}(c_{2}(\rho))$. If $p,\ell>3$, and a similar Riemann-Roch theorem
is true, we can easily obtain similar results in Sections 2 and 3. For
example, the cohomology $H^{*}({\rm
GL}_{n}(\mathop{\mathbb{F}}\nolimits_{p^{k}});{\mathbb{Z}}[1/p])$ is generated
by such Chern classes, as is stated in the famous result of Quillen, while
${\rm GL}_{n}(\mathop{\mathbb{F}}\nolimits_{p^{k}})$ is not of type $C_{2,m}$
for enough large $n,k$. Thus, if $G$ is a Chevalley group, to analyze the DW
invariant, we might employ Chern classes from some modular representations of
$G$.

### Acknowledgments

The work was partially supported by JSPS KAKENHI, Grant Number 00646903.

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Department of Mathematics, Tokyo Institute of Technology 2-12-1 Ookayama,
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