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# Existence of closed geodesics on certain non-compact Riemannian manifolds

Akashdeep Dey Department of Mathematics, University of Toronto, Toronto,
Ontario M5S 2E4, Canada, Email<EMAIL_ADDRESS>
###### Abstract

Let $M$ be a complete Riemannian manifold. Suppose $M$ contains a bounded,
concave, open set $U$ with $C^{0}$ boundary. Moreover, we assume that
$M\setminus U$ is connected and either the relative homotopy set
$\pi_{1}(M,M\setminus U)=0$ or the image of the homomorphism
$\pi_{1}(M\setminus U)\rightarrow\pi_{1}(M)$ (induced by the inclusion
$M\setminus U\hookrightarrow M$) is a finite index subgroup or a normal
subgroup of $\pi_{1}(M)$ or the relative homology group $H_{1}(M,M\setminus
U)\neq 0$. Then there exists a non-trivial closed geodesic on $M$. This
partially proves a conjecture of Chambers, Liokumovich, Nabutovsky and Rotman
[CLNR]*Conjecture 1.1.

##  1 Introduction

In [Poi], Poincaré asked the question whether every closed Riemannian surface
$\Sigma$ admits a closed geodesic. If the fundamental group of $\Sigma$ (more
generally, any closed Riemannian manifold $N$) is non-zero, then one can
construct a non-trivial closed geodesic on $\Sigma$ (resp. on $N$) by
minimizing the length functional in a non-trivial (free) homotopy class of
maps $S^{1}\rightarrow\Sigma$ (resp. $S^{1}\rightarrow N$). In [Bir], Birkhoff
developed a min-max method and proved the existence of a closed geodesic on
every Riemannian two-sphere. This theorem was generalized in higher dimensions
by Fet and Lyusternik [LF]; they proved that every closed Riemannian manifold
admits a closed geodesic.

Minimal submanifolds are the higher dimensional generalizations of the
geodesics. In [alm], Almgren developed a min-max theory to prove the existence
of a closed (singular) minimal submanifold of dimension $l$ in any closed
Riemannian manifold $N^{n}$ and for every $1\leq l\leq n-1$. The regularity
theory in the co-dimension 1 case (i.e. when $l=n-1$) was further developed by
Pitts [pit] and Schoen–Simon [ss]. When $l=1$, the Almgren–Pitts min–max
theory produces stationary geodesic nets, which may not be smooth. (For a
precise statement, see [LS]*Proposition 3.2.) Using the phase-transition
regularization of the length functional, Chodosh and Mantoulidis [CM23] proved
that the geodesic nets, produced by the Almgren-Pitts min-max theory on
surfaces, are closed geodesics. In [LS], Liokumovich and Staffa proved that
for a generic Riemannian metric on a closed manifold, the union of all
stationary geodesic nets is dense. In [LiS], Li and Staffa proved the generic
equidistribution of closed geodesics on closed surfaces and stationary
geodesic nets on closed 3-manifolds.

In general, the above mentioned Fet-Lyusternik theorem does not hold on non-
compact manifolds. For example, $\mathbb{R}^{n}$, equipped with a Riemannian
metric of non-positive curvature, does not contain a closed geodesic. By the
works of Thorbergsson [Tho] and Bangert [Ban], every complete Riemannian
surface with locally convex ends admits a closed geodesic. Informally
speaking, a complete Riemannian manifold $M$ is said to have locally convex
ends if $M$ contains a bounded, concave (see Definition 1.1), open set. In
[CLNR], Chambers, Liokumovich, Nabutovsky and Rotman proved that every
complete Riemannian manifold with locally convex ends admits a stationary
geodesic net, which has at most one singular vertex. They also conjectured
that such a manifold contains a closed geodesic. In this paper, we prove this
conjecture (Theorem 1.3 and Corollary 1.4) under some additional topological
assumptions.

######  Definition 1.1.

Let $\mathscr{M}$ be a complete Riemannian manifold.
$\mathscr{C}\subset\mathscr{M}$ is called $\delta$-convex (where $\delta>0$)
if the following condition is satisfied. If $p,q\in\mathscr{C}$ so that
$d(p,q)\leq\delta$, then every minimizing geodesic from $p$ to $q$ is
contained in $\mathscr{C}$. $\mathscr{U}\subset\mathscr{M}$ is called concave
if $\mathscr{M}\setminus\mathscr{U}$ is $\delta$-convex for some $\delta>0$.

For a topological space $X$ and $A\subset X$, the inclusion $A\hookrightarrow
X$ induces a map $\pi_{0}(\Lambda A)\rightarrow\pi_{0}(\Lambda X)$ (here
$\pi_{0}(\cdot)$ denotes the set of path-components and $\Lambda\cdot$ denotes
the free loop space) and a homomorphism $\pi_{i}(\Lambda
A,c)\rightarrow\pi_{i}(\Lambda X,c)$ (where $\pi_{i}(\cdot)$ denotes the
$i$-th homotopy group) for all $i\geq 1$ and basepoints $c\in\Lambda A$. We
say that $(X,A)$ satisfies condition $(*)$ if $\pi_{1}(X,A)\neq 0$, the map
$\pi_{0}(\Lambda A)\rightarrow\pi_{0}(\Lambda X)$ is bijective and the
homomorphism $\pi_{i}(\Lambda A,c)\rightarrow\pi_{i}(\Lambda X,c)$ is an
isomorphism for all $i\geq 1$ and $c\in\Lambda A$ which are non-contractible
in $A$.

######  Theorem 1.2.

Let $M$ be a complete Riemannian manifold. Suppose $M$ contains a bounded,
concave, open set $U$ with $C^{0}$ boundary and $M\setminus U$ is connected.
Then either there exists a non-trivial closed geodesic on $M$ or
$(M,M\setminus U)$ satisfies condition $(*)$.

As a consequence of this theorem, we obtain the following.

######  Theorem 1.3.

Let $M$ be a complete Riemannian manifold. Suppose $M$ contains a bounded,
concave, open set $U$ with $C^{0}$ boundary. Moreover, we assume that
$M\setminus U$ is connected and either the relative homotopy set
$\pi_{1}(M,M\setminus U)=0$ or the image of the homomorphism
$\pi_{1}(M\setminus U)\rightarrow\pi_{1}(M)$ (induced by the inclusion
$M\setminus U\hookrightarrow M$) is a finite index subgroup or a normal
subgroup of $\pi_{1}(M)$ or the relative homology group $H_{1}(M,M\setminus
U)\neq 0$. Then there exists a non-trivial closed geodesic on $M$.

Theorem 1.3 has the following corollary.

######  Corollary 1.4.

Let $M$ be a complete Riemannian manifold such that $M$ has one end and
$\pi_{1}(M)$ is a finite group or an Abelian group. Suppose $M$ contains a
bounded, concave, open set $U$ with $C^{0}$ boundary. Then there exists a non-
trivial closed geodesic on $M$.

Let us briefly discuss the proof of Theorem 1.2 and elaborate on the condition
$(*)$ (see also Remark 4.9). When $\pi_{1}(M,M\setminus U)=0$, the proof for
the existence of a closed geodesic on $M$ is similar to the proof of the Fet-
Lyusternik theorem [Moore]*Proof of Theorem 2.3.2. If $\pi_{1}(M,M\setminus
U)=0$, we choose $k\geq 1$ such that $\pi_{i}(M,M\setminus U)=0$ for $1\leq
i\leq k$ and $\pi_{k+1}(M,M\setminus U)\neq 0$. This implies $\pi_{k}(\Lambda
M,\Lambda(M\setminus U),c_{p})\neq 0$ (here $c_{p}$ is the constant loop
defined by $p$; see Section 2.1). We choose $F:(I^{k},\partial
I^{k})\rightarrow(\Lambda M,\Lambda(M\setminus U))$ such that $[F]\neq 0$ in
$\pi_{k}(\Lambda M,\Lambda(M\setminus U),c_{p})$. We prove the existence of a
closed geodesic on $M$ by applying the Birkhoff curve shortening flow to the
map $F$. The Birkhoff curve shortening flow is a discrete gradient flow for
the length functional and (for the appropriately chosen parameters) the flow
maps $\Lambda(M\setminus U)$ to itself (since $U$ is concave).

If the map $\pi_{0}(\Lambda(M\setminus U))\rightarrow\pi_{0}(\Lambda M)$ is
not injective or if $\pi_{m}(\Lambda M,\Lambda(M\setminus U),c^{*})\neq 0$ for
some $m\geq 1$ and some $c^{*}\in\Lambda(M\setminus U)$ which is non-
contractible in $M\setminus U$, one can prove the existence of a closed
geodesic on $M$ using a min-max argument (similar to the above case when
$\pi_{1}(M,M\setminus U)=0$).

On the other hand, when $\pi_{1}(M,M\setminus U)\neq 0$, we aim to prove the
existence of a closed geodesic on $M$ using a minimization method. However, on
a non-compact manifold, it may not be possible to obtain a closed geodesic by
minimization since under the Birkhoff curve shortening flow, a loop might
escape to infinity. So, for our proof method to work, we need to additionally
assume that $M\setminus U$ is connected (so that a non-trivial element in
$\pi_{1}(M,M\setminus U)$ can be represented by a loop in $M$) and the map
$\pi_{0}(\Lambda(M\setminus U))\rightarrow\pi_{0}(\Lambda M)$ is not
surjective. (We note that if $M\setminus U$ is not connected,
$\pi_{1}(M,M\setminus U)$ is necessarily non-zero.) The map
$\pi_{0}(\Lambda(M\setminus U))\rightarrow\pi_{0}(\Lambda M)$ fails to be
surjective if and only if there exists a loop $\gamma$ in $M$ which is not
freely homotopic to a loop in $M\setminus U$. As a consequence, under the
Birkhoff curve shortening flow, $\gamma$ can not escape to infinity and hence
must converge to a non-trivial closed geodesic.

The condition $(*)$ for $(M,M\setminus U)$ has formal similarities with the
topology of the free loop space $\Lambda N$, where $N$ is a closed Riemannian
manifold with negative sectional curvature. $(M,M\setminus U)$ satisfies
condition $(*)$ if and only if $\pi_{1}(M,M\setminus U)\neq 0$, for each
$\tilde{\Pi}\in\pi_{0}(\Lambda M)$ there exists a unique
$\Pi\in\pi_{0}(\Lambda(M\setminus U))$ such that $\Pi\subset\tilde{\Pi}$ and
for each non-standard path-component $\tilde{\Pi}\in\pi_{0}(\Lambda M)$ (i.e.
$\tilde{\Pi}$ does not contain the constant loops),
$\pi_{i}(\tilde{\Pi},\Pi)=0$ for all $i\geq 1$. (We note that, if $M\setminus
U$ is connected, then $\pi_{1}(M,M\setminus U)\neq 0$ implies $\pi_{1}(M)\neq
0$, i.e. $\Lambda M$ has more than one path-components.) On the other hand, if
$N$ is a closed Riemannian manifold with negative sectional curvature, then
$\pi_{1}(N)\neq 0$; the standard path-component of $\Lambda N$ is homotopy
equivalent to $N$ and does not contain any non-trivial closed geodesic; each
non-standard path-component $\Pi$ of $\Lambda N$ is homotopy equivalent to
$S^{1}$ and contains a unique (up to reparametrization) closed geodesic, which
is the minimizer of the length functional in $\Pi$.

Acknowledgements. I am very grateful to Professors Yevgeny Liokumovich,
Alexander Nabutovsky and Regina Rotman for many helpful discussions. I would
like to thank Prof. Liokumovich for suggesting this problem to me. The author
is partially supported by NSERC Discovery grant.

##  2 Notations and Preliminaries

###  2.1 Notations

Here we summarize the notations which will be frequently used later.

  * •

$[m]$ : The set $\\{1,2,\dots,m\\}$.

  * •

$I$ : The interval $[0,1]$.

  * •

$\text{conv}_{p}(N)$ : The convexity radius of $N$ at $p$.

  * •

$\text{conv}(N)$ : The convexity radius of $N$.

  * •

$\text{inj}(N)$ : The injectivity radius of $N$.

  * •

$B(p,r)$ : The geodesic ball centered at $p$ with radius $r$.

  * •

$\overline{A}$ : The closure of $A$ (in a topological space).

  * •

We will identify $S^{1}$ with the quotient of the unit interval $[0,1]$. In
particular, a continuous map $\gamma:S^{1}\rightarrow X$ is same as a
continuous map $\gamma:[0,1]\rightarrow X$ with $\gamma(0)=\gamma(1)$.

  * •

$\Lambda X$ : The space of continuous maps from $S^{1}$ to $X$ (where $X$ is a
topological space), endowed with the compact-open topology.

  * •

$\Omega_{p}X=\\{\gamma\in\Lambda X:\gamma(0)=\gamma(1)=p\\}.$

  * •

$c_{p}$ : For $p\in X$ (where $X$ is a topological space), $c_{p}\in\Lambda X$
is the constant curve defined by $p$, i.e. $c_{p}(s)=p$ for all $s\in S^{1}$.

  * •

$\left|\gamma\right|$ : For $\gamma\in\Lambda X$ (where $X$ is a topological
spaces), $\left|\gamma\right|$ denotes the image of $\gamma$.

  * •

$\mathscr{L}N$ : The Sobolev space of $W^{1,2}$ maps from $S^{1}$ to $N$
(where $N$ is a Riemannian manifold).

  * •

$\ell(\gamma)$, $\mathscr{E}(\gamma)$ : For $\gamma\in\mathscr{L}N$,
$\ell(\gamma)$ and $\mathscr{E}(\gamma)$ denote the length and energy of
$\gamma$, i.e.

$\ell(\gamma)=\int_{0}^{1}\left|\gamma^{\prime}\right|,\quad\mathscr{E}(\gamma)=\int_{0}^{1}\left|\gamma^{\prime}\right|^{2}.$

###  2.2 The Birkhoff curve shortening map

In this subsection, we briefly discuss the properties of the Birkhoff curve
shortening map following Colding-Minicozzi [CM_article, CM_book]. Let $N$ be a
closed Riemannian manifold. We fix $E,R>0$ and $L\in\mathbb{N}$ so that

$0<R<\min\left\\{\frac{1}{2}\text{conv}(N),\frac{1}{4}\text{inj}(N)\right\\}\quad\text{and}\quad
L\geq\max\left\\{\frac{E}{R^{2}},\sqrt{E}\right\\}.$ (2.1)

For fixed $(E,R,L)$ satisfying (2.1), let $\Lambda$ denote the set of all maps
$c:S^{1}\rightarrow N$ such that the following conditions are satisfied. There
exist $L$ points $s_{0},s_{1},s_{2},\dots,s_{L}=s_{0}\in S^{1}$ such that for
all $i\in[L]$, $d(c(s_{i-1}),c(s_{i}))\leq R$; $c$ restricted to
$[s_{i-1},s_{i}]$ is the unique minimizing constant speed geodesic from
$c(s_{i-1})$ to $c(s_{i})$ and $\ell(c)\leq L$. Then,
$\Lambda\subset\mathscr{L}N$ is compact (with respect to the subspace
topology).

Let us use the notation
$\mathscr{L}^{E}=\\{c\in\mathscr{L}N:\mathscr{E}(c)\leq E\\}$. Let
$\gamma\in\mathscr{L}^{E}$ and $0\leq x<y\leq 1$ with $(y-x)\leq 1/L$. Then,
using (2.1),

$d(\gamma(x),\gamma(y))\leq\int_{x}^{y}\left|\gamma^{\prime}\right|\leq\left(\int_{x}^{y}1\right)^{1/2}\left(\int_{x}^{y}\left|\gamma^{\prime}\right|^{2}\right)^{1/2}\leq\sqrt{\frac{E}{L}}\leq
R.$ (2.2)

######  Definition 2.1.

The maps
$\Psi_{e},\Psi_{o},\tilde{\Psi}:\mathscr{L}^{E}\rightarrow\mathscr{L}^{E}$ are
defined as follows (cf. [CM_book]*p. 166, Steps 1-3). For
$i\in\\{0,1,2,\dots,2L\\}$, we set $x_{i}=\frac{i}{2L}$.

  * •

Denoting $\gamma_{e}=\Psi_{e}(\gamma)$, $\gamma_{e}$ is defined by the
condition that for all $i\in[L]$, $\gamma_{e}(x_{2i})=\gamma(x_{2i})$ and
$\gamma_{e}$ restricted to the interval $[x_{2i-2},x_{2i}]$ is a minimizing
constant speed geodesic. (By (2.2) and (2.1), there exists a unique minimizing
geodesic from $\gamma(x_{2i-2})$ to $\gamma(x_{2i})$.)

  * •

Denoting $\gamma_{o}=\Psi_{o}(\gamma)$, $\gamma_{o}$ is defined by the
condition that for all $i\in[L]$, $\gamma_{o}(x_{2i-1})=\gamma(x_{2i-1})$ and
$\gamma_{o}$ restricted to the interval $[x_{2i-1},x_{2i+1}]$ (here
$x_{2L+1}=x_{1}$) is a minimizing constant speed geodesic.

  * •

Denoting $\tilde{\gamma}=\tilde{\Psi}(\gamma)$, $\tilde{\gamma}$ is defined by
the condition that $\tilde{\gamma}$ is a reparametrization of $\gamma$ so that
$\tilde{\gamma}(0)=\gamma(0)$ and $\tilde{\gamma}$ has constant speed.

The Birkhoff curve shortening map $\Psi:\mathscr{L}^{E}\rightarrow\Lambda$ is
defined by $\Psi(\gamma)=\tilde{\Psi}(\Psi_{o}(\Psi_{e}(\gamma)))$.

######  Proposition 2.2.

There exists a continuous map
$\Phi_{e}:\mathscr{L}^{E}\times[0,1]\rightarrow\mathscr{L}N$ such that
$\Phi_{e}(\gamma,0)=\gamma$, $\Phi_{e}(\gamma,1/2)=\Psi_{e}(\gamma)$ and
$\Phi_{e}(\gamma,1)=\tilde{\Psi}(\Psi_{e}(\gamma)).$ Similarly, there exists a
continuous map $\Phi_{o}:\mathscr{L}^{E}\times[0,1]\rightarrow\mathscr{L}N$
such that $\Phi_{o}(\gamma,0)=\gamma$, $\Phi_{o}(\gamma,1/2)=\Psi_{o}(\gamma)$
and $\Phi_{o}(\gamma,1)=\tilde{\Psi}(\Psi_{o}(\gamma))$. Therefore, there
exists a continuous map
$\Phi:\mathscr{L}^{E}\times[0,1]\rightarrow\mathscr{L}N$ such that
$\Phi(\gamma,0)=\gamma$ and $\Phi(\gamma,1)=\Psi(\gamma)$.

###### Proof.

The proof of this proposition follows from the proofs of Lemma 5.3 and Lemma
5.5 in [CM_book]*Chapter 5. Let us briefly present the argument here. Let

$\Gamma=\\{(p,q)\in N\times N:d(p,q)<\text{conv}(N)\\}.$

We define $\mathbf{H}:\Gamma\rightarrow C^{1}([0,1],N)$ as follows
([CM_book]*p. 170, item (C2)). $\mathbf{H}(p,q):[0,1]\rightarrow N$ is the
minimizing constant speed geodesic from $p=\mathbf{H}(p,q)(0)$ to
$q=\mathbf{H}(p,q)(1)$, i.e.

$\mathbf{H}(p,q)(t)=\exp_{p}(t\exp_{p}^{-1}(q)).$ (2.3)

Let us use the notation $\tilde{\gamma}_{e}=\tilde{\Psi}(\Psi_{e}(\gamma)).$
Then, $\gamma_{e}=\tilde{\gamma}_{e}\circ P_{\gamma}$ where
$P_{\gamma}:[0,1]\rightarrow[0,1]$ is a monotone reparametrization of $[0,1]$
that fixes $0$ and $1$. For $(\gamma,s)\in\mathscr{L}^{E}\times[0,1]$,
$\Phi_{e}(\gamma,s):S^{1}\rightarrow N$ is defined by ([CM_book]*p. 174, eq.
(5.24) and (5.25))

$\Phi_{e}(\gamma,s)(x)=\begin{cases}\mathbf{H}(\gamma(x),\gamma_{e}(x))(2s)&\text{
if }s\in[0,1/2];\\\
\tilde{\gamma}_{e}\left((2-2s)P_{\gamma}(x)+(2s-1)x\right)&\text{ if
}s\in[1/2,1].\end{cases}$

One can define $\Phi_{o}$ in a similar way. Finally,
$\Phi:\mathscr{L}^{E}\times[0,1]\rightarrow\mathscr{L}N$ is defined by

$\Phi(\gamma,s)=\begin{cases}\Phi_{e}(\gamma,s)&\text{ if }s\in[0,1/2];\\\
\Phi_{o}(\gamma_{e},2s-1)&\text{ if }s\in[1/2,1].\end{cases}$

∎

######  Remark 2.3.

From Definition 2.1 and Proposition 2.2, it follows that

$\left|\Psi(\gamma)\right|,\left|\Phi(\gamma,s)\right|\subset\mathcal{N}_{4R}(\left|\gamma\right|)\;\forall\;\gamma\in\mathscr{L}^{E}\text{
and }s\in[0,1].$

Moreover, if $C\subset N$ is $2R$-convex (Definition 1.1) and
$\left|\gamma\right|\subset C$, then

$\left|\Psi(\gamma)\right|,\left|\Phi(\gamma,s)\right|\subset
C\;\forall\;s\in[0,1].$

The space $\mathscr{L}N$ is metrizable. Let us fix a metric $\mathbf{d}$ on
$\mathscr{L}N$. Let $\mathcal{G}\subset\Lambda$ denote the set of all constant
speed immersed closed geodesics in $N$ with length at most $L$.

######  Proposition 2.4.

For all $\varepsilon>0$, there exists $\delta>0$ such that if
$\gamma\in\Lambda$ satisfies
$\mathbf{d}\left(\gamma,\mathcal{G}\right)\geq\varepsilon$ then
$\ell(\Psi(\gamma))\leq\ell(\gamma)-\delta.$

##  3 Relative homotopy groups of the free loop space

Let $X$ be a topological space, $p\in X$. Let
$\mathbf{j}:\Omega_{p}X\rightarrow\Lambda X$ be the inclusion map,
$\mathbf{e}:\Lambda X\rightarrow X$ be defined by
$\mathbf{e}(\gamma)=\gamma(0)$ and $\mathbf{c}:X\rightarrow\Lambda X$ be
defined by $\mathbf{c}(p)=c_{p}$. (For the notations, see Section 2.1.) Then
$\mathbf{e}:\Lambda X\rightarrow X$ is a fibration; $\Omega_{p}X$ is the fiber
over $p$. Since $\mathbf{e}\circ\mathbf{c}=\text{id}_{X}$, the long exact
sequence of homotopy groups, corresponding to the fibration $\mathbf{e}$,
splits, i.e. for all $m\geq 1$,

${0}$${\pi_{m}(\Omega_{p}X,c_{p})}$${\pi_{m}(\Lambda
X,c_{p})}$${\pi_{m}(X,p)}$${0}$$\scriptstyle{\mathbf{j}_{*}}$$\scriptstyle{\mathbf{e}_{*}}$$\scriptstyle{\mathbf{c}_{*}}$

is a split exact sequence (see [Moore]*Section 2.3). Hence, $\pi_{m}(\Lambda
X,c_{p})$ is isomorphic to the semi-direct product
$\pi_{m}(\Omega_{p}X,c_{p})\rtimes\pi_{m}(X,p)$. In this section, we will
prove a relative version of this theorem.

Let $\mathcal{C}$ denote the category of pointed sets. Then the objects of
$\mathcal{C}$ are pairs $(S,s)$ where $S$ is a set and $s\in S$. (We call $s$
the distinguished element of $S$.) The morphisms of $\mathcal{C}$ are maps
$\varphi:(S,s)\rightarrow(S^{\prime},s^{\prime})$; we define the kernel of
$\varphi$ $\ker(\varphi)=\varphi^{-1}(s^{\prime})$. If $G$ is a group and
$1_{G}\in G$ is the identity element, then $(G,1_{G})$ in an object of
$\mathcal{C}$. Moreover, a group homomorphism maps the identity element to the
identity element. Thus, the category of groups can be regarded as a
subcategory of $\mathcal{C}$. In $\mathcal{C}$, we have the notion of an exact
sequence;

${\cdots}$${(S_{i-1},s_{i-1})}$${(S_{i},s_{i})}$${(S_{i+1},s_{i+1})}$${\cdots}$$\scriptstyle{\varphi_{i-1}}$$\scriptstyle{\varphi_{i}}$$\scriptstyle{\varphi_{i-1}}$

is called exact if $\text{im}(\varphi_{i-1})=\ker(\varphi_{i})$ for all $i$.

For a topological space $X$, $\pi_{0}(X)$ denotes the set of path-components
of $X$. For $p\in X$, $\pi_{0}(X,p)$ denotes the pointed set
$(\pi_{0}(X),X_{p})$, where $X_{p}$ is the path-component of $X$ containing
$p$. Let $\mathbf{j}$ and $\mathbf{e}$ be as defined above. Then

${\\{0\\}}$${\pi_{0}(\Omega_{p}X,c_{p})}$${\pi_{0}(\Lambda
X,c_{p})}$${\pi_{0}(X,p)}$${\\{0\\}}$$\scriptstyle{\mathbf{j}_{*}}$$\scriptstyle{\mathbf{e}_{*}}$

is an exact sequence. Another exact sequence of pointed sets, which will be
useful for our purpose, is the homotopy exact sequence of a pair of pointed
topological spaces. Let $X$ be a topological space, $A\subset X$ and $p\in A$.
Then the homotopy exact sequence of $(X,A,p)$ is the following long exact
sequence [Hat]*Theorem 4.3

$\cdots\rightarrow\pi_{i}(A,p)\rightarrow\pi_{i}(X,p)\rightarrow\pi_{i}(X,A,p)\rightarrow\pi_{i-1}(A,p)\rightarrow\cdots\\\
\rightarrow\pi_{1}(X,A,p)\rightarrow\pi_{0}(A,p)\rightarrow\pi_{0}(X,p).$
(3.1)

All but the last three objects in this sequence are groups. The distinguished
element of $\pi_{1}(X,A,p)$ is the homotopy class of the constant map.

######  Lemma 3.1.

Let us consider the following commutative diagram of groups.

${1}$${C_{1}^{\prime}}$${C_{1}}$${C_{1}^{\prime\prime}}$${1}$${1}$${C_{2}^{\prime}}$${C_{2}}$${C_{2}^{\prime\prime}}$${1}$${C_{3}^{\prime}}$${C_{3}}$${C_{3}^{\prime\prime}}$${1}$${C_{4}^{\prime}}$${C_{4}}$${C_{4}^{\prime\prime}}$${1}$${1}$${C_{5}^{\prime}}$${C_{5}}$${C_{5}^{\prime\prime}}$${1}$$\scriptstyle{f_{1}^{\prime}}$$\scriptstyle{g_{1}}$$\scriptstyle{h_{1}}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{1}^{\prime\prime}}$$\scriptstyle{j_{1}}$$\scriptstyle{f_{2}^{\prime}}$$\scriptstyle{g_{2}}$$\scriptstyle{h_{2}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{2}^{\prime\prime}}$$\scriptstyle{j_{2}}$$\scriptstyle{f_{3}^{\prime}}$$\scriptstyle{g_{3}}$$\scriptstyle{h_{3}}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{3}^{\prime\prime}}$$\scriptstyle{j_{3}}$$\scriptstyle{f_{4}^{\prime}}$$\scriptstyle{g_{4}}$$\scriptstyle{h_{4}}$$\scriptstyle{f_{4}}$$\scriptstyle{f_{4}^{\prime\prime}}$$\scriptstyle{g_{5}}$$\scriptstyle{h_{5}}$

We assume that

  1. (i)

all the squares commute, i.e. for $1\leq i\leq 4$, $f_{i}\circ
g_{i}=g_{i+1}\circ f^{\prime}_{i},\;f^{\prime\prime}_{i}\circ
h_{i}=h_{i+1}\circ f_{i}$ and for $i=1,2$, $f_{i}\circ j_{i}=j_{i+1}\circ
f^{\prime\prime}_{i};$

  2. (ii)

each column is exact;

  3. (iii)

for $i\neq 3$, the $i$-th row is a short exact sequence;

  4. (iv)

for $i=1,2$, short exact sequence in the $i$-th row splits, i.e. $h_{i}\circ
j_{i}=\textup{id}_{C_{i}^{\prime\prime}}$;

  5. (v)

for the third row, we have $\textup{im}(g_{3})\subset\ker(h_{3})$ and
$h_{3}\circ j_{3}=\textup{id}_{C_{3}^{\prime\prime}}$.

Then the third row is a split exact sequence.

Before we prove this lemma, let us recall the following fact about the split
exact sequences.

######  Lemma 3.2.

Let

${1}$${C^{\prime}}$${C}$${C^{\prime\prime}}$${1}$$\scriptstyle{g}$$\scriptstyle{h}$$\scriptstyle{j}$

be a split exact sequence of groups. Then for all $c\in C$, there exist
$c^{\prime}\in C^{\prime}$ and $c^{\prime\prime}\in C^{\prime\prime}$ such
that $c=g(c^{\prime})j(c^{\prime\prime})$.

###### Proof of Lemma 3.1.

Since $h_{3}\circ j_{3}=\textup{id}_{C_{3}^{\prime\prime}}$, $h_{3}$ is
surjective. Hence we only need to show that $g_{3}$ is injective and
$\ker(h_{3})\subset\text{im}(g_{3})$.

Suppose $c_{3}^{\prime}\in\ker(g_{3})$, i.e.

$g_{3}(c^{\prime}_{3})=1.$ (3.2)

Then

$\displaystyle
g_{4}(f_{3}^{\prime}(c^{\prime}_{3}))=f_{3}(g_{3}(c_{3}^{\prime}))=1$ (3.3)
$\displaystyle\implies f^{\prime}_{3}(c^{\prime}_{3})=1$ (3.4)
$\displaystyle\implies\exists\;c^{\prime}_{2}\in C^{\prime}_{2}\text{ such
that\;}c^{\prime}_{3}=f^{\prime}_{2}(c^{\prime}_{2})$ (3.5)
$\displaystyle\implies
f_{2}(g_{2}(c^{\prime}_{2}))=g_{3}(f^{\prime}_{2}(c^{\prime}_{2}))=1\;(\text{by
}\eqref{e.c'_3})$ (3.6) $\displaystyle\implies\exists\;c_{1}\in C_{1}\text{
such that\;}g_{2}(c^{\prime}_{2})=f_{1}(c_{1}).$ (3.7)

Lemma 3.2 implies that there exist $c^{\prime}_{1}\in C^{\prime}_{1}$ and
$c^{\prime\prime}_{1}\in C_{1}^{\prime\prime}$ such that
$c_{1}=g_{1}(c^{\prime}_{1})j_{1}(c_{1}^{\prime\prime})$. Hence (3.7) implies

$\displaystyle
g_{2}(c_{2}^{\prime})=f_{1}(g_{1}(c^{\prime}_{1}))f_{1}(j_{1}(c_{1}^{\prime\prime}))=g_{2}(f_{1}^{\prime}(c_{1}^{\prime}))j_{2}(f_{1}^{\prime\prime}(c_{1}^{\prime\prime}))$
(3.8) $\displaystyle\implies
g_{2}(c_{2}^{\prime}f_{1}^{\prime}(c_{1}^{\prime})^{-1})=j_{2}(f_{1}^{\prime\prime}(c_{1}^{\prime\prime}))$
(3.9) $\displaystyle\implies
h_{2}(g_{2}(c_{2}^{\prime}f_{1}^{\prime}(c_{1}^{\prime})^{-1}))=h_{2}(j_{2}(f_{1}^{\prime\prime}(c_{1}^{\prime\prime})))$
(3.10) $\displaystyle\implies 1=f_{1}^{\prime\prime}(c_{1}^{\prime\prime})$
(3.11) $\displaystyle\implies
g_{2}(c_{2}^{\prime}f_{1}^{\prime}(c_{1}^{\prime})^{-1})=1\;(\text{by
}\eqref{e.5.2})$ (3.12) $\displaystyle\implies
c_{2}^{\prime}=f_{1}^{\prime}(c_{1}^{\prime})$ (3.13) $\displaystyle\implies
c_{3}^{\prime}=1\;(\text{by }\eqref{e.5.3}).$ (3.14)

Therefore, $g_{3}$ is injective.

Suppose $d_{3}\in\ker(h_{3})$, i.e.

$h_{3}(d_{3})=1$ (3.15)

Then

$\displaystyle h_{4}(f_{3}(d_{3}))=f_{3}^{\prime\prime}(h_{3}(d_{3}))=1$
(3.16) $\displaystyle\implies\exists\;d_{4}^{\prime}\in C_{4}^{\prime}\text{
such that\;}f_{3}(d_{3})=g_{4}(d_{4}^{\prime})$ (3.17) $\displaystyle\implies
g_{5}(f_{4}^{\prime}(d_{4}^{\prime}))=f_{4}(g_{4}(d_{4}^{\prime}))=1$ (3.18)
$\displaystyle\implies f_{4}^{\prime}(d_{4}^{\prime})=1$ (3.19)
$\displaystyle\implies\exists\;d_{3}^{\prime}\in C_{3}^{\prime}\text{ such
that\;}d_{4}^{\prime}=f_{3}^{\prime}(d_{3}^{\prime})$ (3.20)
$\displaystyle\implies
f_{3}(d_{3})=g_{4}(f_{3}^{\prime}(d_{3}^{\prime}))=f_{3}(g_{3}(d_{3}^{\prime}))\;(\text{by
}\eqref{e.f.1})$ (3.21) $\displaystyle\implies
f_{3}(d_{3}g_{3}(d_{3}^{\prime})^{-1})=1$ (3.22)
$\displaystyle\implies\exists\;d_{2}\in C_{2}\text{ such
that\;}d_{3}g_{3}(d_{3}^{\prime})^{-1}=f_{2}(d_{2}).$ (3.23)

Lemma 3.2 implies that there exist $d_{2}^{\prime}\in C_{2}^{\prime}$ and
$d_{2}^{\prime\prime}\in C_{2}^{\prime\prime}$ such that
$d_{2}=g_{2}(d_{2}^{\prime})j_{2}(d_{2}^{\prime\prime})$. Hence, (3.23)
implies

$\displaystyle
d_{3}g_{3}(d_{3}^{\prime})^{-1}=f_{2}(d_{2})=f_{2}(g_{2}(d_{2}^{\prime}))f_{2}(j_{2}(d_{2}^{\prime\prime}))$
(3.24) $\displaystyle\implies
d_{3}g_{3}(d_{3}^{\prime})^{-1}=g_{3}(f_{2}^{\prime}(d_{2}^{\prime}))j_{3}(f_{2}^{\prime\prime}(d_{2}^{\prime\prime})).$
(3.25)

Applying $h_{3}$ on the both sides of (3.25) and using (3.15), we obtain

$1=f_{2}^{\prime\prime}(d_{2}^{\prime\prime});$

hence (3.25) implies

$d_{3}=g_{3}(f_{2}^{\prime}(d_{2}^{\prime})d_{3}^{\prime}).$

Therefore, $\ker(h_{3})\subset\text{im}(g_{3})$. ∎

######  Remark 3.3.

In Lemma 3.1, instead of assuming that all the objects in the diagram are
groups, let us assume that the objects in the 1st and 2nd row are groups; the
objects in the 3rd, 4th and 5th row are pointed sets and the exactness
conditions hold in the sense of pointed sets. Then, from the proof of Lemma
3.1, it follows that $\ker(g_{3})$ is a singleton set.

######  Proposition 3.4.

Let $X$ be a topological space, $A\subset X$ and $p\in A$. Then, for all
$m\geq 2$, the relative homotopy group $\pi_{m}(\Lambda X,\Lambda A,c_{p})$ is
isomorphic to the semi-direct product $\pi_{m}(\Omega_{p}X,\\\
\Omega_{p}A,c_{p})\rtimes\pi_{m}(X,A,p)$. Moreover, for $m=1$, the map
$\pi_{1}(\Omega_{p}X,\Omega_{p}A,c_{p})\rightarrow\pi_{1}(\Lambda X,\Lambda
A,c_{p})$, which is induced by the inclusion
$(\Omega_{p}X,\Omega_{p}A)\hookrightarrow(\Lambda X,\Lambda A)$, is non-
constant.

###### Proof.

For $m\geq 1$, let us consider the following commutative diagram.

${0}$${\pi_{m}(\Omega_{p}A)}$${\pi_{m}(\Lambda
A)}$${\pi_{m}(A)}$${0}$${0}$${\pi_{m}(\Omega_{p}X)}$${\pi_{m}(\Lambda
X)}$${\pi_{m}(X)}$${0}$${\pi_{m}(\Omega_{p}X,\Omega_{p}A)}$${\pi_{m}(\Lambda
X,\Lambda
A)}$${\pi_{m}(X,A)}$${0}$${\pi_{m-1}(\Omega_{p}A)}$${\pi_{m-1}(\Lambda
A)}$${\pi_{m-1}(A)}$${0}$${0}$${\pi_{m-1}(\Omega_{p}X)}$${\pi_{m-1}(\Lambda
X)}$${\pi_{m-1}(X)}$${0}$

The columns correspond to the homotopy exact sequence for the pairs
$(\Omega_{p}X,\Omega_{p}A)$, $(\Lambda X,\Lambda A)$ and $(X,A)$ with the
basepoints $c_{p}$, $c_{p}$ and $p$. The rows (except the 3rd row) correspond
to the short exact sequence for the loop space fibration (as discussed at the
beginning of this section). The maps in the 3rd row too

${\pi_{m}(\Omega_{p}X,\Omega_{p}A)}$${\pi_{m}(\Lambda X,\Lambda
A)}$${\pi_{m}(X,A)}$$\scriptstyle{\mathbf{j}_{*}}$$\scriptstyle{\mathbf{e}_{*}}$$\scriptstyle{\mathbf{c}_{*}}$

are induced by the maps $\mathbf{j}$, $\mathbf{e}$ and $\mathbf{c}$; from the
definitions of these maps, it follows that
$\text{im}(\mathbf{j}_{*})\subset\ker(\mathbf{e}_{*})$ and
$\mathbf{e}_{*}\circ\mathbf{c}_{*}=\text{id}_{\pi_{m}(X,A)}$.

If $m\geq 2$, all the objects in this diagram are groups. Therefore, by Lemma
3.1, the 3rd row is a split exact sequence; hence $\pi_{m}(\Lambda X,\Lambda
A,c_{p})\cong\pi_{m}(\Omega_{p}X,\Omega_{p}A,c_{p})\rtimes\pi_{m}(X,A,p)$. If
$m=1$, the objects in the 1st and 2nd row are groups; however the objects in
the 3rd, 4th and 5th row are only pointed sets. Therefore, by Remark 3.3, the
kernel of the map $\mathbf{j}_{*}$ in the 3rd row is a singleton set. Since
$\mathbf{j}_{*}$ maps the homotopy class of the constant map in
$\pi_{1}(\Omega_{p}X,\Omega_{p}A,c_{p})$ to the homotopy class of the constant
map in $\pi_{1}(\Lambda X,\Lambda A,c_{p})$,
$\mathbf{j}_{*}:\pi_{1}(\Omega_{p}X,\Omega_{p}A,c_{p})\rightarrow\pi_{1}(\Lambda
X,\Lambda A,c_{p})$ must be a non-constant map.

∎

######  Remark 3.5.

In the above commutative diagram, which appears in the proof of Proposition
3.4, the 4th and 5th row are split exact for all $m\geq 1$ and all the objects
are Abelian groups when $m\geq 3$. In Lemma 3.1, if we additionally assume
that the 4th and 5th row are split exact and all the objects are Abelian
groups, then the lemma follows from the five lemma for the Abelian groups.

######  Remark 3.6.

For all $m\geq 1$, there exists a canonical bijection
$D:\pi_{m}(\Omega_{p}X,\Omega_{p}A,c_{p})\rightarrow\pi_{m+1}(X,A,p)$, which
is an isomorphism if $m\geq 2$. Let us briefly recall the definition of $D$.

Let $I$ denote the interval $[0,1]$. Let $\mathscr{I}^{0}=\\{1\\}$ and for
$j\geq 2$,

$\mathscr{I}^{j-1}=\overline{\partial I^{j}\setminus(I^{j-1}\times\\{0\\})}.$

Let

$\tilde{\mathscr{I}}^{1}=\overline{\partial I^{2}\setminus(\\{0\\}\times
I)}\text{ and for }j\geq 3,\;\tilde{\mathscr{I}}^{j-1}=\overline{\partial
I^{j}\setminus(I^{j-2}\times\\{0\\}\times I)}.$

Given $\Xi:(I^{m},\partial
I^{m},\mathscr{I}^{m-1})\rightarrow(\Omega_{p}X,\Omega_{p}A,\\{c_{p}\\})$, we
define $\xi:(I^{m+1},\partial
I^{m+1},\tilde{\mathscr{I}}^{m})\rightarrow(X,A,\\{p\\})$ by
$\xi(s_{1},\dots,s_{m},s_{m+1})=\Xi(s_{1},\dots,s_{m})(s_{m+1})$. Then
$D:\pi_{m}(\Omega_{p}X,\Omega_{p}A,c_{p})\rightarrow\pi_{m+1}(X,A,p)$, defined
by $D([\Xi])=[\xi]$, is a well-defined map, which is a group homomorphism if
$m\geq 2$. The inverse of the map $D$, which we denote by $\Delta$, is defined
as follows. Given $\xi^{\prime}:(I^{m+1},\partial
I^{m+1},\tilde{\mathscr{I}}^{m})\rightarrow(X,A,\\{p\\})$, we define
$\Xi^{\prime}:(I^{m},\partial
I^{m},\mathscr{I}^{m-1})\rightarrow(\Omega_{p}X,\Omega_{p}A,\\{c_{p}\\})$ by
$\Xi^{\prime}(s_{1},\dots s_{m})=\xi^{\prime}(s_{1},\dots,s_{m},-)$. Then
$\Delta:\pi_{m+1}(X,A,p)\rightarrow\pi_{m}(\Omega_{p}X,\Omega_{p}A,c_{p})$ is
defined by $\Delta([\xi^{\prime}])=[\Xi^{\prime}]$.

##  4 Proof of Theorems 1.2 and 1.3

To prove Theorem 1.2, we need to prove the existence of a closed geodesic on
$M$ when one of the following conditions are satisfied:

  * (C1)

$\pi_{1}(M,M\setminus U)=0$;

  * (C2)

$\pi_{0}(\Lambda(M\setminus U))\rightarrow\pi_{0}(\Lambda M)$ is not
injective;

  * (C3)

$\pi_{m}(\Lambda(M\setminus U),c^{*})\rightarrow\pi_{m}(\Lambda M,c^{*})$ is
not an isomorphism for some $m\geq 1$ and $c^{*}\in\Lambda(M\setminus U)$
which is non-contractible in $M\setminus U$;

  * (C4)

$\pi_{0}(\Lambda(M\setminus U))\rightarrow\pi_{0}(\Lambda M)$ is not
surjective.

######  Lemma 4.1.

Let $X$ be a connected topological manifold. Suppose $A$ is a connected open
subset of $X$ so that $\overline{A}$ is a compact manifold with boundary
$\partial A$ and $X\setminus A$ is connected. Then the relative homotopy group
$\pi_{i}(X,X\setminus A)\neq 0$ for some $i\geq 1.$

###### Proof.

Suppose $\pi_{i}(X,X\setminus A)=0$ for all $i\geq 1.$ Then by the Hurewicz
theorem [Hat]*Theorem 4.37, the relative homology groups

$H_{i}(X,X\setminus A)=0\quad\forall\;i\geq 1.$ (4.1)

By the excision theorem [Hat]*Theorem 2.20, $H_{i}(X,X\setminus A)\cong
H_{i}(\mathcal{N}_{r}(A),\mathcal{N}_{r}(A)\setminus A)$ (for all $r>0$). By
[Hat]*Proposition 3.42, if $r$ is sufficiently small,
$(\mathcal{N}_{r}(A),\mathcal{N}_{r}(A)\setminus A)$ deformation retracts onto
$(\overline{A},\partial A)$; hence
$H_{i}(\mathcal{N}_{r}(A),\mathcal{N}_{r}(A)\setminus A)\cong
H_{i}(\overline{A},\partial A)$. Therefore, (4.1) implies

$H_{i}(\overline{A},\partial A)=0\quad\forall\;i\geq 1.$ (4.2)

However, by the Lefschetz duality [Hat]*Theorem 3.43,
$H_{n}(\overline{A},\partial A,\mathbb{Z}_{2})\neq 0$ (where $n=\dim(A)$);
hence by the universal coefficient theorem for homology [Hat]*Theorem 3A.3,
either $H_{n-1}(\overline{A},\partial A)\neq 0$ or
$H_{n}(\overline{A},\partial A)\neq 0$. ∎

Let $\mathscr{M}$ be a complete Riemannian manifold and
$\gamma\in\mathscr{L}\mathscr{M}$. We define the Radon measure
$\mathcal{L}_{\gamma}$ on $[0,1]$ as follows. If $A$ is a Borel subset of
$[0,1]$,

$\mathcal{L}_{\gamma}(A)=\int_{A}\left|\gamma^{\prime}(s)\right|\;ds.$

The Radon measure $\mu_{\gamma}$ on $\mathscr{M}$ is defined by
$\mu_{\gamma}=\gamma_{*}\mathcal{L}_{\gamma}$, i.e. if $B$ is a Borel subset
of $\mathscr{M}$, then $\mu_{\gamma}(B)=\mathcal{L}_{\gamma}(\gamma^{-1}(B))$.

######  Lemma 4.2.

The map $\gamma\mapsto\mu_{\gamma}$ is a continuous map from
$\mathscr{L}\mathscr{M}$ to the space of Radon measures on $\mathscr{M}$.

###### Proof.

Let $\\{\gamma_{i}\\}_{i=1}^{\infty}$ be a sequence in
$\mathscr{L}\mathscr{M}$ which converge to $\gamma_{\infty}$ and $\varphi\in
C^{0}_{c}(\mathscr{M})$. Then

$\int_{\mathscr{M}}\varphi\;d\mu_{\gamma}=\int_{0}^{1}\varphi(\gamma(s))\left|\gamma^{\prime}(s)\right|\;ds.$

Thus,

$\displaystyle\left|\int_{\mathscr{M}}\varphi\;d\mu_{\gamma_{\infty}}-\int_{\mathscr{M}}\varphi\;d\mu_{\gamma_{i}}\right|$
(4.3)
$\displaystyle\leq\int_{0}^{1}\Big{|}\varphi(\gamma_{\infty}(s))\left|\gamma_{\infty}^{\prime}(s)\right|-\varphi(\gamma_{i}(s))\left|\gamma_{i}^{\prime}(s)\right|\Big{|}\;ds$
(4.4)
$\displaystyle\leq\int_{0}^{1}\left|\varphi(\gamma_{\infty}(s))-\varphi(\gamma_{i}(s))\right|\left|\gamma_{\infty}^{\prime}(s)\right|\;ds+\int_{0}^{1}\left|\varphi(\gamma_{i}(s))\right|\Big{|}\left|\gamma_{\infty}^{\prime}(s)\right|-\left|\gamma_{i}^{\prime}(s)\right|\Big{|}\;ds.$
(4.5)

Let us assume that $\mathscr{M}$ is isometrically embedded in the Euclidean
space $\mathbb{R}^{P}$. For $x\in\mathscr{M}$, the tangent space
$T_{x}\mathscr{M}$ is a subspace of $T_{x}\mathbb{R}^{P}$ and
$T_{x}\mathbb{R}^{P}$ can be canonically identified with $\mathbb{R}^{P}$.
Using this identification, let us set

$\delta_{i}=\left(\int_{0}^{1}\left|\gamma_{\infty}^{\prime}(s)-\gamma_{i}^{\prime}(s)\right|^{2}\;ds\right)^{1/2}.$

Let

$\varepsilon_{i}=\sup_{s\in[0,1]}\left|\varphi(\gamma_{\infty}(s))-\varphi(\gamma_{i}(s))\right|.$

Since $\gamma_{i}\rightarrow\gamma_{\infty}$ in $\mathscr{L}\mathscr{M}$,
there holds $\varepsilon_{i},\delta_{i}\rightarrow 0$. Hence, from (4.5), we
obtain

$\left|\int_{\mathscr{M}}\varphi\;d\mu_{\gamma_{\infty}}-\int_{\mathscr{M}}\varphi\;d\mu_{\gamma_{i}}\right|\leq\varepsilon_{i}\ell(\gamma_{\infty})+\delta_{i}\left\|\varphi\right\|_{C^{0}}\rightarrow
0.$

∎

###### Proof of Theorem 1.2 when (C1) holds.

The proof is divided into eight parts.

Part 1. Since we are assuming that $\pi_{1}(M,M\setminus U)=0$, by Lemma 4.1,
there exists $k\geq 1$ such that

$\pi_{i}(M,M\setminus U)=0\;\forall\;1\leq i\leq k$ (4.6)

and $\pi_{k+1}(M,M\setminus U)\neq 0$. Therefore, by Proposition 3.4 and
Remark 3.6, $\pi_{k}(\Lambda M,\Lambda(M\setminus U),c_{p})\neq 0$. Let us
choose $\tilde{F}:(I^{k},\partial I^{k})\rightarrow(\Lambda
M,\Lambda(M\setminus U))$ so that

$[\tilde{F}]\neq 0\in\pi_{k}(\Lambda M,\Lambda(M\setminus U),c_{p}).$ (4.7)

Let $\tilde{f}:I^{k}\times S^{1}\rightarrow M$ be defined by
$\tilde{f}(v,s)=\tilde{F}(v)(s)$.

Let $\rho>0$ such that $M\setminus U$ is $\rho$-convex (Definition 1.1). Let
us choose $K\in\mathbb{N}$ so that
$\left|v-v^{\prime}\right|+\left|s-s^{\prime}\right|\leq 1/K$ implies

$d\left(\tilde{f}(v,s),\tilde{f}(v^{\prime},s^{\prime})\right)\leq\frac{\rho}{2}$
(4.8)

and

$d\left(\tilde{f}(v,s),\tilde{f}(v^{\prime},s^{\prime})\right)<\frac{1}{2}\inf\\{\text{conv}_{p}(M):p\in\text{image}(\tilde{f})\\}.$

We define $F:(I^{k},\partial
I^{k})\rightarrow(\mathscr{L}M,\mathscr{L}(M\setminus U))$ as follows. For
$v\in I^{k}$, $F(v):S^{1}\rightarrow M$ is defined by the condition that for
all $i\in\\{0,1,2,\dots K\\}$, $F(v)(i/K)=\tilde{F}(v)(i/K)$ and for all
$i\in[K]$, $F(v)$ restricted to $[\frac{i-1}{K},\frac{i}{K}]$ is a minimizing
constant speed geodesic. Then $F(v)\in\mathscr{L}M$ and
$F:I^{k}\rightarrow\mathscr{L}M$ is continuous. Moreover, by (4.8),
$F(\partial I^{k})\subset\mathscr{L}(M\setminus U)$. Let us define the
homotopy $G_{0}:(I^{k},\partial I^{k})\times I\rightarrow(\Lambda
M,\Lambda(M\setminus U))$ by

$G_{0}(v,t)(s)=\mathbf{H}(\tilde{F}(v)(s),F(v)(s))(t),$

where $\mathbf{H}$ is as in (2.3). Then $G_{0}(-,0)=\tilde{F}$ and
$G_{0}(-,1)=F$. Moreover, by (4.8), if $v\in\partial I^{k}$, then
$G_{0}(v,t)\in\Lambda(M\setminus U)$ for all $t\in I$.

Part 2. Let $f:I^{k}\times S^{1}\rightarrow M$ be defined by $f(v,s)=F(v)(s)$.
Let

$\lambda=\sup\\{\ell(F(v)):v\in I^{k}\\},\quad E=\sup\\{\mathscr{E}(F(v)):v\in
I^{k}\\},$ (4.9)

$V=\mathcal{N}_{2\lambda}(U)$,

$\alpha=\inf\\{\text{conv}_{p}(M):p\in V\\}$ (4.10)

and $W\subset M$ be a bounded open set which contains
$\mathcal{N}_{\alpha}(V)\>\cup\>\text{image}(f)$. Let us choose a closed
Riemannian manifold $N$ such that $N$ and $M$ have the same dimension and
there exists an isometric embedding $\iota:W\rightarrow N$. We will abuse
notation by identifying $W$ with $\iota(W)$.

Part 3. Let $\rho$ be as in Part 1 and $E$ be as in (4.9). We choose $R>0$ and
$L\in\mathbb{N}$ so that

$R<\min\left\\{\frac{\alpha}{4},\frac{\rho}{2}\right\\}$ (4.11)

and $(E,R,L)$ satisfies (2.1). For this fixed choice of $(E,R,L)$, let us
consider the maps $\Psi$ and $\Phi$ as defined in Definition 2.1 and
Proposition 2.2.

######  Lemma 4.3.

There exist sequences $\\{F_{i}:(I^{k},\partial
I^{k})\rightarrow(\mathscr{L}N,\mathscr{L}(N\setminus U))\\}_{i=0}^{\infty}$
and $\\{G_{i}:(I^{k},\partial I^{k})\times
I\rightarrow(\mathscr{L}N,\mathscr{L}(N\setminus U))\\}_{i=1}^{\infty}$ with
the following properties. $F_{0}=F$ and for all $i\geq 1$

  1. (i)

$G_{i}(-,0)=F_{i-1},\;G_{i}(-,1)=F_{i}$;

  2. (ii)

$\left|G_{i}(v,t)\right|\subset W\;\forall\;(v,t)\in I^{k}\times I$;

  3. (iii)

for $v\in I^{k}$, if $\left|F_{i-1}(v)\right|\cap U\neq\emptyset$, then
$F_{i}(v)=\Psi\left(F_{i-1}(v)\right)$;

  4. (iv)

for $v\in I^{k}$, if $\left|F_{i-1}(v)\right|\cap U=\emptyset$, then
$\left|F_{i}(v)\right|\cap U=\emptyset$ as well;

  5. (v)

for $v\in I^{k}$, if $\left|F_{i}(v)\right|\cap U\neq\emptyset$, then
$\ell(F_{i}(v))\leq\lambda$.

###### Proof.

We set $F_{0}=F$. By the choice of $W$, $\left|F(v)\right|\subset W$ for all
$v\in I^{k}$. Moreover, by the definition of $\lambda$,
$\ell(F(v))\leq\lambda$ for all $v\in I^{k}$. Let us assume that
$F_{j}:(I^{k},\partial I^{k})\rightarrow(\mathscr{L}N,\mathscr{L}(N\setminus
U))$ has already been defined for some $j\geq 0$ and $F_{j}$ has the following
properties:

  * •

$\left|F_{j}(v)\right|\subset W\;\forall v\in I^{k}$;

  * •

for $v\in I^{k}$, if $\left|F_{j}(v)\right|\cap U\neq\emptyset$, then
$\ell(F_{j}(v))\leq\lambda$.

We define $F_{j+1}:(I^{k},\partial
I^{k})\rightarrow(\mathscr{L}N,\mathscr{L}(N\setminus U))$ and
$G_{j}:(I^{k},\partial I^{k})\times
I\rightarrow(\mathscr{L}N,\mathscr{L}(N\setminus U))$ as follows. Let

$D^{0}_{j}=\\{v\in I^{k}:\left|F_{j}(v)\right|\cap(N\setminus
V)\neq\emptyset\\};\quad D^{1}_{j}=\overline{\\{v\in
I^{k}:\left|F_{j}(v)\right|\cap U\neq\emptyset\\}}.$

Then, $D^{0}_{j},\;D^{1}_{j}$ are closed subsets of $I^{k}$. Moreover, by our
assumption, if $v\in D^{1}_{j}$, then

$\left|F_{j}(v)\right|\subset\overline{\mathcal{N}_{\lambda}(U)};$

Hence, $D^{0}_{j}\>\cap\>D^{1}_{j}=\emptyset$. Let
$\chi_{j}:I^{k}\rightarrow[0,1]$ be a continuous function such that

$\chi_{j}=\begin{cases}0&\text{ on }D^{0}_{j};\\\ 1&\text{ on
}D^{1}_{j}.\end{cases}$ (4.12)

For $(v,t)\in I^{k}\times I$, we define

$G_{j+1}(v,t)=\Phi(F_{j}(v),\chi_{j}(v)t),\quad
F_{j+1}(v)=\Phi(F_{j}(v),\chi_{j}(v)).$

Then $G_{j+1}(-,0)=F_{j}$, $G_{j+1}(-,1)=F_{j+1}$. Remark 2.3 and (4.11) imply
that $G_{j+1}(v,t)\in\mathscr{L}(N\setminus U)$ for all $(v,t)\in\partial
I^{k}\times I$ and for $v\in I^{k}$, if $\left|F_{j}(v)\right|\cap
U=\emptyset$, then $\left|F_{j+1}(v)\right|\cap U=\emptyset$. Therefore, if
$\left|F_{j+1}(v)\right|\cap U\neq\emptyset$ then $\left|F_{j}(v)\right|\cap
U\neq\emptyset$. From the definition of $F_{j+1}$, it follows that if
$\left|F_{j}(v)\right|\cap U\neq\emptyset$, then $F_{j+1}(v)=\Psi(F_{j}(v))$;
in that case by Proposition 2.4,
$\ell(F_{j+1}(v))\leq\ell(F_{j}(v))\leq\lambda$.

It remains to show that $\left|G_{j+1}(v,t)\right|\subset W$ for all $(v,t)\in
I^{k}\times I$. If $v\in D^{0}_{j}$, then $G_{j+1}(v,t)=F_{j}(v)$ for all
$t\in I$; hence $\left|G_{j+1}(v,t)\right|=\left|F_{j}(v)\right|\subset W$. If
$v\notin D^{0}_{j}$, then $\left|F_{j}(v)\right|\subset V$. Therefore, by
Remark 2.3 and (4.11),

$\left|G_{j+1}(v,t)\right|\subset\mathcal{N}_{4R}\left(\left|F_{j}(v)\right|\right)\subset\mathcal{N}_{\alpha}(V)\subset
W.$

∎

Part 4. Since $U$ is an open set with $C^{0}$ boundary $\partial U$, there
exists an open set $V^{\prime}\subset\mathcal{N}_{\lambda}(U)$ and a
homeomorphism $\mathcal{F}:\partial U\times(-1,1)\rightarrow V^{\prime}$ such
that $\mathcal{F}(x,0)=x$ for all $x\in\partial U$;

$\mathcal{F}(\partial U\times(-1,0))\subset U;\quad\mathcal{F}(\partial
U\times(0,1))\subset V^{\prime}\setminus\overline{U}.$ (4.13)

Let $d_{\partial U}$ denote the signed distance from $\partial U$ such that
$d_{\partial U}<0$ on $U$. For $r\in\mathbb{R}$, let

$U(r)=\\{p\in N:d_{\partial U}(p)<r\\}.$

We choose $0<\eta\leq\alpha$ (where $\alpha$ is as in (4.10)) so that

$\\{-3\eta\leq d_{\partial U}\leq\eta\\}\subset V^{\prime}.$ (4.14)

For $\gamma\in\mathscr{L}N$, let $\mu_{\gamma}$ be the Radon measure as
defined before Lemma 4.2; if $B$ is a Borel subset of $N$, we use the notation

$\ell(\gamma,\;B)=\mu_{\gamma}(B).$

######  Claim 4.4.

There exists a closed geodesic $\sigma$ in $N$ such that
$\ell(\sigma)\leq\lambda$ and
$\ell\left(\sigma,\;\overline{U(\eta)}\right)\geq\eta$ .

The proof of this claim will be presented in Parts 5 – 8. Assuming the claim,
we obtain

$\left|\sigma\right|\subset\mathcal{N}_{\lambda}(U(\eta))\subset V.$

Viewing $V$ as a subset of $M$ we conclude that $\sigma$ is a non-trivial
closed geodesic in $M$.

Part 5. Let $\mathscr{G}\subset\mathscr{L}N$ denote the set of all constant
speed immersed closed geodesics in $N$ with length at most $\lambda$. We
assume that for all $\beta\in\mathscr{G}$, there holds

$\ell\left(\beta,\;\overline{U(\eta)}\right)<\eta$ (4.15)

$\mathscr{G}\subset\mathscr{L}N$ is compact; by Lemma 4.2,

$\mathcal{S}=\left\\{c\in\mathscr{L}N:\ell(c)\leq\lambda,\;\ell\left(c,\;\overline{U(\eta)}\right)\geq\eta\right\\}$

is a closed subset of $\mathscr{L}N$. Hence, (4.15) implies that there exists
$\tau>0$ such that for all $\gamma\in\mathcal{S}$,
$\mathbf{d}(\gamma,\mathscr{G})\geq\tau$ (where $\mathbf{d}$ is as in
Proposition 2.4). Therefore, by Proposition 2.4, there exists $\zeta>0$ such
that if $\gamma\in\Lambda\;\cap\>\mathcal{S}$, then

$\ell(\Psi(\gamma))\leq\ell(\gamma)-\zeta.$ (4.16)

We choose $Q\in\mathbb{N}$ such that $Q\geq\frac{\lambda}{\zeta}+1.$

######  Lemma 4.5.

For all $v\in I^{k}$, one of the following two alternatives hold.

  1. (i)

$\left|F_{Q}(v)\right|\cap U=\emptyset$.

  2. (ii)

$\left|F_{Q}(v)\right|\subset U(\eta)$ and $\ell(F_{Q}(v))<\eta$.

###### Proof.

Suppose there exists $v\in I^{k}$ such that for all $0\leq i\leq Q$,

$\left|F_{i}(v)\right|\cap U\neq\emptyset\text{ and
}\ell\left(F_{i}(v),\;\overline{U(\eta)}\right)\geq\eta.$ (4.17)

Then, by Lemma 4.3(iii), $F_{i+1}(v)=\Psi(F_{i}(v))$ for all $0\leq i\leq
Q-1$. Hence, by (4.16),

$\ell(F_{i+1}(v))\leq\ell(F_{i}(v))-\zeta\quad\forall\;1\leq i\leq Q-1.$
(4.18)

This implies

$\ell(F_{Q}(v))\leq\ell(F_{1}(v))-(Q-1)\zeta\leq\lambda-(Q-1)\zeta\leq 0,$

which contradicts (4.17). Therefore, for all $v\in I^{k}$, either

$\left|F_{j}(v)\right|\cap U=\emptyset\text{ for some }0\leq j\leq Q$ (4.19)

or

$\left|F_{i}(v)\right|\cap U\neq\emptyset\;\forall\;0\leq i\leq Q\text{ and
}\ell\left(F_{j}(v),\overline{U(\eta)}\right)<\eta\text{ for some }0\leq j\leq
Q.$ (4.20)

If (4.19) holds, then by Lemma 4.3(iv), $\left|F_{Q}(v)\right|\cap
U=\emptyset.$ If (4.20) holds, then $\left|F_{j}(v)\right|\subset U(\eta)$,
which implies $\ell(F_{j}(v))<\eta$. Therefore, by Lemma 4.3(iii),
$\ell(F_{Q}(v))<\eta$ which implies $\left|F_{Q}(v)\right|\subset U(\eta)$. ∎

Part 6. Let
$\Theta:\left\\{c\in\mathscr{L}N:\ell(c)<\text{conv}_{c(0)}(N)\right\\}\times[0,1]\rightarrow\mathscr{L}N$
be defined by

$\Theta(\gamma,t)(s)=\mathbf{H}(\gamma(s),\gamma(0))(t),$

where $\mathbf{H}$ is as in (2.3). Then
$\Theta(\gamma,0)=\gamma,\;\Theta(\gamma,1)=c_{\gamma(0)}$ and

$\left|\Theta(\gamma,t)\right|\subset
B(\gamma(0),\ell(\gamma))\quad\forall\;t\in[0,1].$ (4.21)

Let

$D^{0}=\left\\{v\in I^{k}:\left|F_{Q}(v)\right|\subset N\setminus
U(-\eta)\right\\},\;D^{1}=\left\\{v\in
I^{k}:\left|F_{Q}(v)\right|\cap\overline{U(-2\eta)}\neq\emptyset\right\\}.$

Then $D^{0}$, $D^{1}$ are disjoint closed subsets of $I^{k}$. Let
$\chi:I^{k}\rightarrow[0,1]$ be a continuous function such that

$\chi(v)=\begin{cases}0&\text{ if }v\in D^{0};\\\ 1&\text{ if }v\in
D^{1}.\end{cases}$ (4.22)

By Lemma 4.5, if $\left|F_{Q}(v)\right|\cap\overline{U(-\eta)}\neq\emptyset$
then $\ell(F_{Q}(v))<\eta$; hence $\left|F_{Q}(v)\right|\subset U$. We define
$G^{\prime}:(I^{k},\partial I^{k})\times
I\rightarrow(\mathscr{L}N,\mathscr{L}(N\setminus U))$ by

$G^{\prime}(v,t)=\begin{cases}\Theta\left(F_{Q}(v),\chi(v)t\right)&\text{ if
}\left|F_{Q}(v)\right|\cap\overline{U(-\eta)}\neq\emptyset;\\\ F_{Q}(v)&\text{
if }\left|F_{Q}(v)\right|\subset N\setminus U(-\eta).\end{cases}$ (4.23)

Then $G^{\prime}(v,0)=F_{Q}(v)$ and we define $F^{\prime}:(I^{k},\partial
I^{k})\rightarrow(\mathscr{L}N,\mathscr{L}(N\setminus U))$ by
$F^{\prime}(v)=G^{\prime}(v,1)$. Moreover, if $G^{\prime}(v,t)\neq F_{Q}(v)$,
then (4.21) implies $\left|G^{\prime}(v,t)\right|\subset U(\eta)$; therefore,
by Lemma 4.3(ii),

$\left|G^{\prime}(v,t)\right|\subset W\;\forall\;(v,t)\in I^{k}\times I.$
(4.24)

We also note that, if $F^{\prime}(v)$ is non-constant, then
$\left|F_{Q}(v)\right|\subset N\setminus\overline{U(-2\eta)}$, which implies

$\left|F^{\prime}(v)\right|\subset N\setminus U(-3\eta).$ (4.25)

Part 7. Given a continuous map $\phi:N\rightarrow N$ and $\gamma\in\Lambda N$,
let $\phi_{\\#}\gamma\in\Lambda N$ be defined by
$\left(\phi_{\\#}\gamma\right)(s)=\phi(\gamma(s)).$ (4.14) implies that there
exists $\theta\in(0,1)$ such that

$\\{-3\eta\leq d_{\partial U}\leq\eta\\}\subset\mathcal{F}(\partial
U\times[-1+\theta,1-\theta]).$ (4.26)

Let $\psi:\mathbb{R}\rightarrow[0,1]$ be a compactly supported $C^{1}$
function so that $\text{spt}(\psi)\subset(-1,1)$ and $\psi\equiv 1$ on
$[-1+\theta,1-\theta]$. Let $\\{\hat{\phi}^{t}\\}_{t\geq 0}$ denote the flow
of the vector field $\psi(\xi)\partial_{\xi}$ on $\mathbb{R}$. Then there
exists $T>0$ such that

$\hat{\phi}^{T}([-1+\theta,\infty))\subset[1-\theta,\infty).$ (4.27)

For $t\geq 0$, we define $\phi^{t}:N\rightarrow N$ as follows. If $p\in
V^{\prime}$ and $p=\mathcal{F}(x,\xi)$, we set
$\phi^{t}(p)=\mathcal{F}(x,\hat{\phi}^{t}(\xi))$; if $p\notin V^{\prime}$, we
set $\phi^{t}(p)=p$. Then $\\{\phi^{t}\\}_{t\geq 0}$ is a continuous family of
homeomorphisms of $N$. (4.26) and (4.27) imply

$\phi^{T}\left(N\setminus U(-3\eta)\right)\subset N\setminus U(\eta).$ (4.28)

We define $F^{\prime\prime}:(I^{k},\partial I^{k})\rightarrow(\Lambda
N,\Lambda(N\setminus U))$ and $G^{\prime\prime}:(I^{k},\partial I^{k})\times
I\rightarrow(\Lambda N,\Lambda(N\setminus U))$ by
$F^{\prime\prime}(v)=\phi^{T}_{\\#}F^{\prime}(v)$ and
$G^{\prime\prime}(v,t)=\phi^{Tt}_{\\#}F^{\prime}(v)$. Then
$G^{\prime\prime}(-,0)=F^{\prime}$ and
$G^{\prime\prime}(-,1)=F^{\prime\prime}$. Moreover, since $\phi^{t}(p)=p$ for
all $t\geq 0$ and $p\in N\setminus V^{\prime}$, by (4.24),

$\left|G^{\prime\prime}(v,t)\right|\subset W\;\forall\;(v,t)\in I^{k}\times
I.$ (4.29)

Furthermore, by (4.25) and (4.28), if $F^{\prime\prime}(v)$ is non-constant,
then

$\left|F^{\prime\prime}(v)\right|\subset N\setminus U(\eta).$ (4.30)

Concatenating the homotopies $\\{G_{i}\\}_{i=0}^{Q}$, $G^{\prime}$ and
$G^{\prime\prime}$, we conclude that there exists a homotopy
$G:(I^{k},\partial I^{k})\times I\rightarrow(\Lambda N,\Lambda(N\setminus U))$
such that $G(-,0)=\tilde{F}$ and $G(-,1)=F^{\prime\prime}$; Lemma 4.3(ii),
(4.24) and (4.29) imply that

$\left|G(v,t)\right|\subset W\;\forall\;(v,t)\in I^{k}\times I.$ (4.31)

Viewing $W$ as a subset of $M$, we can regard $G$ as a map from
$(I^{k},\partial I^{k})\times I$ to $(\Lambda M,\Lambda(M\setminus U))$ and
$F^{\prime\prime}$ as a map from $(I^{k},\partial I^{k})$ to $(\Lambda
M,\Lambda(M\setminus U))$. Therefore, using (4.7), we obtain

$[F^{\prime\prime}]\neq 0\in\pi_{k}(\Lambda M,\Lambda(M\setminus U),c_{p}).$
(4.32)

Part 8. For $j\in\mathbb{N}$, let $I(j)$ be the cell complex on $I$ whose
$0$-cells are

$[0],[j^{-1}],\dots,[1-j^{-1}],[1]$

and $1$-cells are

$[0,j^{-1}],[j^{-1},2j^{-1}],\dots,[1-j^{-1},1].$

Let $I^{k}(j)$ denote the cell complex on $I^{k}$ whose cells are
$c_{1}\otimes c_{2}\otimes\dots\otimes c_{k}$, where each $c_{i}\in I(j)$. By
abuse of notation, a cell $c_{1}\otimes c_{2}\otimes\dots\otimes c_{k}$ will
be identified with its support $c_{1}\times c_{2}\times\dots\times
c_{k}\subset I^{k}$.

We recall that $\mathbf{e}:\Lambda M\rightarrow M$ is defined by
$\mathbf{e}(\gamma)=\gamma(0)$. Let $f^{\prime\prime}:(I^{k},\partial
I^{k})\rightarrow(M,M\setminus U)$ be defined by
$f^{\prime\prime}(v)=\mathbf{e}(F^{\prime\prime}(v)).$ Let

$\left(f^{\prime\prime}\right)^{-1}(U)=\mathcal{U},\quad\left(f^{\prime\prime}\right)^{-1}(U(\eta))=\tilde{\mathcal{U}}.$

We choose $l\in\mathbb{N}$ so that if $c\in I^{k}(l)$ and
$c\cap\mathcal{U}\neq\emptyset$ then $c\subset\tilde{\mathcal{U}}$. Let
$\mathbf{X}\subset I^{k}(l)$ be the union of all top dimensional cells $c\in
I^{k}(l)$ (i.e. $\dim(c)=k$) such that $c\cap\mathcal{U}\neq\emptyset$. Let
$\mathbf{A}$ be the union of all cells $e\in\mathbf{X}$ such that
$e\cap\mathcal{U}=\emptyset$. Then

$\overline{(I^{k}(l)\setminus\mathbf{X})}\cap\mathbf{X}\subset\mathbf{A}.$
(4.33)

In the statement of Theorem 1.2, it is assumed that $M\setminus U$ is
connected. Using (4.6) and applying [Hat]*Lemma 4.6 to the map
$f^{\prime\prime}:(\mathbf{X},\mathbf{A})\rightarrow(M,M\setminus U)$, we
conclude that there exist $\hat{h}:(\mathbf{X},\mathbf{A})\times
I\rightarrow(M,M\setminus U)$ and $\hat{f}:\mathbf{X}\rightarrow M\setminus U$
such that $\hat{h}(-,0)=f^{\prime\prime}$, $\hat{h}(-,1)=\hat{f}$ and
$\hat{h}(v,t)=f^{\prime\prime}(v)$ for all $(v,t)\in\mathbf{A}\times I$. We
define $\hat{H}:(\mathbf{X},\mathbf{A})\times I\rightarrow(\Lambda
M,\Lambda(M\setminus U))$ and $\hat{F}:\mathbf{X}\rightarrow\Lambda(M\setminus
U)$ by

$\hat{H}(v,t)=c_{\hat{h}(v,t)}\;\text{ and }\;\hat{F}(v)=c_{\hat{f}(v)}.$

(4.30) implies $F^{\prime\prime}(v)=c_{f^{\prime\prime}(v)}$ for all
$v\in\tilde{\mathcal{U}}$. Hence, $\hat{H}(v,0)=F^{\prime\prime}(v)$ and
$\hat{H}(v,1)=\hat{F}(v)$ for all $v\in\mathbf{X}$ and
$\hat{H}(v,t)=F^{\prime\prime}(v)$ for all $(v,t)\in\mathbf{A}\times I$. We
extend the definitions of $\hat{H}$ and $\hat{F}$ on all of $I^{k}\times I$
and $I^{k}$ by setting $\hat{H}(v,t)=F^{\prime\prime}(v)$ and
$\hat{F}(v)=F^{\prime\prime}(v)$ for all
$v\in\overline{I^{k}(l)\setminus\mathbf{X}}$ and $t\in I$. Then (4.33) implies
$\hat{H}:(I^{k},\partial I^{k})\times I\rightarrow(\Lambda
M,\Lambda(M\setminus U))$ and $\hat{F}:I^{k}\rightarrow\Lambda(M\setminus U)$
are well-defined, continuous maps with $\hat{H}(v,0)=F^{\prime\prime}(v)$ and
$\hat{H}(v,1)=\hat{F}(v)$ for all $v\in I^{k}$. Hence
$[F^{\prime\prime}]=0\in\pi_{k}(\Lambda M,\Lambda(M\setminus U))$, which
contradicts (4.32). This finishes the proof of Claim 4.4. ∎

###### Proof of Theorem 1.2 when either (C2) or (C3) holds.

If (C2) holds, there exist $\Pi_{0},\;\Pi_{1}\in\pi_{0}(\Lambda(M\setminus
U))$ and $\tilde{\Pi}\in\pi_{0}(\Lambda M)$ such that
$\Pi_{0},\;\Pi_{1}\subset\tilde{\Pi}$. Since $M\setminus U$ is connected,
without loss of generality we can assume that $\Pi_{0}$ does not contain the
constant loops in $M\setminus U$. Let $P:I\rightarrow\tilde{\Pi}$ be a
continuous map such that $P(0)\in\Pi_{0}$ and $P(1)\in\Pi_{1}$. Then $[P]\neq
0\in\pi_{1}(\Lambda M,\Lambda(M\setminus U),c^{*})$, where $c^{*}\in\Pi_{0}$.
On the other hand, if (C3) holds, then the homotopy exact sequence of
$(\Lambda M,\Lambda(M\setminus U),c^{*})$ implies either $\pi_{m}(\Lambda
M,\Lambda(M\setminus U),c^{*})\neq 0$ or $\pi_{m+1}(\Lambda
M,\Lambda(M\setminus U),c^{*})\neq 0$. So let us assume that $\pi_{k}(\Lambda
M,\Lambda(M\setminus U),c^{*})\neq 0$ for some $k\geq 1$ and
$c^{*}\in\Lambda(M\setminus U)$ which is non-contractible in $M\setminus U$.

We now proceed as in the proof of Theorem 1.2 when (C1) holds. We choose
$\tilde{F}:(I^{k},\partial I^{k})\rightarrow(\Lambda M,\Lambda(M\setminus U))$
so that

$[\tilde{F}]\neq 0\in\pi_{k}(\Lambda M,\Lambda(M\setminus U),c^{*});$ (4.34)

repeating the argument of Parts 1 – 7 we construct
$F^{\prime\prime}:(I^{k},\partial I^{k})\rightarrow(\Lambda
M,\Lambda(M\setminus U))$ such that for $v\in I^{k}$, if
$F^{\prime\prime}(v)\notin\Lambda(M\setminus U)$, then (cf. (4.30))

$F^{\prime\prime}(v)\text{ is constant.}$ (4.35)

Moreover, there exists $G:(I^{k},\partial I^{k})\times I\rightarrow(\Lambda
M,\Lambda(M\setminus U))$ with $G(-,0)=\tilde{F}$ and
$G(-,1)=F^{\prime\prime}$. Therefore, if $\Pi$ denotes the path-component of
$\Lambda(M\setminus U)$ which contains $c^{*}$, then
$(F^{\prime\prime})^{-1}(\Pi)\neq I^{k}$ (by (4.34)); moreover,
$(F^{\prime\prime})^{-1}(\Pi)$ is a closed subset of $I^{k}$. Therefore, if
$u$ belongs to the (topological) boundary of $(F^{\prime\prime})^{-1}(\Pi)$,
then (4.35) implies $F^{\prime\prime}(u)$ is constant. This contradicts the
fact that $c^{*}$ is non-contractible in $M\setminus U$.

∎

###### Proof of Theorem 1.2 when (C4) holds.

The proof is divided into five parts.

Part 1. Condition (C4) implies that there exists $\tilde{\gamma}\in\Lambda M$
such that

$\tilde{\gamma}\text{ is not freely homotopic to any loop
}\hat{\gamma}\in\Lambda(M\setminus U)\text{ on }M.$ (4.36)

Let us choose $K\in\mathbb{N}$ so that $\left|s-s^{\prime}\right|\leq 1/K$
implies

$d(\tilde{\gamma}(s),\tilde{\gamma}(s^{\prime}))<\frac{1}{2}\inf\\{\text{conv}_{p}(M):p\in\left|\tilde{\gamma}\right|\\}.$

Let $\gamma:S^{1}\rightarrow M$ be defined by the condition that for all
$i\in\\{0,1,2,\dots K\\}$, $\gamma(i/K)=\tilde{\gamma}(i/K)$ and for all
$i\in[K]$, $\gamma$ restricted to $[\frac{i-1}{K},\frac{i}{K}]$ is a
minimizing constant speed geodesic. Then $\gamma\in\mathscr{L}M$. Let us
define the homotopy $J_{0}:S^{1}\times[0,1]\rightarrow M$ by
$J_{0}(s,t)=\mathbf{H}(\tilde{\gamma}(s),\gamma(s))(t)$, where $\mathbf{H}$ is
as in (2.3). Then $J_{0}(-,0)=\tilde{\gamma}$ and $J_{0}(-,1)=\gamma$.

Part 2. Let $\ell(\gamma)=\lambda$, $V=\mathcal{N}_{\lambda}(U)$,

$\alpha=\inf\\{\text{conv}_{p}(M):p\in V\\},$

$W=\mathcal{N}_{2\alpha}(V)$. Let us choose a closed Riemannian manifold $N$
such that $N$ and $M$ have the same dimension and there exists an isometric
embedding $\iota:W\rightarrow N$. We will abuse notation by identifying $W$
with $\iota(W)$. Let $\mathscr{E}(\gamma)=E$. We choose $R>0$ and
$L\in\mathbb{N}$ so that $4R<\alpha$ and $(E,R,L)$ satisfies (2.1). For this
fixed choice of $(E,R,L)$, let us consider the maps $\Psi$ and $\Phi$ as
defined in Definition 2.1 and Proposition 2.2.

Part 3. Let $\gamma_{0}=\gamma$ and for $i\in\mathbb{N}$, let
$\gamma_{i}=\Psi^{i}(\gamma)$. By Proposition 2.2 and Remark 2.3, for all
$i\in\mathbb{N}$, there exists $J_{i}:S^{1}\times[0,1]\rightarrow N$ such that

$J_{i}(-,0)=\gamma_{i-1},\;J_{i}(-,1)=\gamma_{i}\text{ and
}\left|J_{i}(-,t)\right|\subset\mathcal{N}_{4R}(\left|\gamma_{i}\right|)\;\forall\;t\in[0,1].$
(4.37)

For $j\in\mathbb{N}$, concatenating all the $J_{i}$’s for $1\leq i\leq j$, we
obtain $\tilde{J}_{j}:S^{1}\times[0,1]\rightarrow N$ such that
$\tilde{J}_{j}(-,0)=\gamma$ and $\tilde{J}_{j}(-,1)=\gamma_{j}$.

Part 4. We claim that $\left|\gamma_{i}\right|\cap U\neq\emptyset$ for all
$i$. To justify this claim, let us assume that there exists $m\in\mathbb{N}$
such that $\left|\gamma_{i}\right|\cap U\neq\emptyset$ for $0\leq i\leq m-1$
and $\left|\gamma_{m}\right|\cap U=\emptyset$. By Proposition 2.4,

$\ell(\gamma_{i})\leq\lambda\;\forall\;i\in\mathbb{N}.$ (4.38)

Hence, $\left|\gamma_{i}\right|\subset\mathcal{N}_{\lambda}(U)$ for all $0\leq
i\leq m-1.$ Therefore, by (4.37),
$\left|J_{i}(-,t)\right|\subset\mathcal{N}_{\alpha}(V)$ for all $i\in[m]$ and
$t\in[0,1]$. Thus, $\tilde{J}_{m}:S^{1}\times[0,1]\rightarrow W$ with
$\tilde{J}_{m}(-,0)=\gamma$ and $\tilde{J}_{m}(-,1)=\gamma_{m}$. Therefore,
viewing $W$ as a subset of $M$, $\tilde{\gamma}$ is freely homotopic to
$\gamma_{m}\in\Lambda(M\setminus U)$ on $M$, which contradicts (4.36).

Part 5. By Proposition 2.4, there exists a subsequence
$\\{\gamma_{i_{k}}\\}_{k=1}^{\infty}$ and a closed geodesic $\sigma$ in $N$
such that $\gamma_{i_{k}}$ converges to $\sigma$ in $\mathscr{L}N$. By (4.38),
$\ell(\sigma)\leq\lambda$. We claim that $\ell(\sigma)>0$ and
$\left|\sigma\right|\cap U\neq\emptyset$. Assuming the claim, we obtain
$\left|\sigma\right|\subset\mathcal{N}_{\lambda}(U)$. Viewing $W$ as a subset
of $M$, we conclude that $\sigma$ is a non-trivial closed geodesic in $M$.

To justify the claim, we first note that convergence in $\mathscr{L}N$ implies
convergence in $\Lambda N$. Therefore, there exists $l\in\mathbb{N}$ such that
$d(\sigma(s),\gamma_{i_{l}}(s))<\alpha$ for all $s\in S^{1}$. Moreover, by
Part 4, $\left|\sigma\right|\cap\overline{U}\neq\emptyset$; so
$\left|\sigma\right|\subset\mathcal{N}_{\lambda}(U)$. We define the homotopy
$\tilde{J}:S^{1}\times[0,1]\rightarrow W$ by

$\tilde{J}(s,t)=\mathbf{H}(\sigma(s),\gamma_{i_{l}}(s))(t),$

where $\mathbf{H}$ is as in (2.3). Further, by (4.37) and Part 4,
$\text{image}(\tilde{J}_{j})\subset W$, for all $j\in\mathbb{N}$. Thus,
concatenating $J_{0},\;\tilde{J}_{i_{l}}$ and $\tilde{J}$, we obtain
$J:S^{1}\times[0,1]\rightarrow W$ such that $J(-,0)=\tilde{\gamma}$ and
$J(-,1)=\sigma$. Therefore, viewing $W$ as a subset of $M$, $\tilde{\gamma}$
is freely homotopic to $\sigma$ on $M$. Hence, if $\left|\sigma\right|$ is a
point or $\left|\sigma\right|\cap U=\emptyset$, then that will be a
contradiction of (4.36).

∎

######  Lemma 4.6.

Let $G$ be an arbitrary group and $H$ be a proper finite index or normal
subgroup of $G$. Then

$G\neq\bigcup_{g\in G}gHg^{-1}.$ (4.39)

###### Proof.

If $H$ is a proper normal subgroup of $G$ then $gHg^{-1}=H$ for all $g\in G$;
hence (4.39) holds.

If $H$ is a finite index subgroup of $G$, we consider the set $S$ of the left
cosets of $H$ in $G$. Let $\mathfrak{S}$ denote the permutation group of $S$
and $\pi:G\rightarrow\mathfrak{S}$ be the homomorphism so that
$\pi(x)\in\mathfrak{S}$ maps $gH$ to $xgH$. Let $G^{\prime}=\pi(G)$. By
Burnside’s lemma,

$1=\frac{1}{\left|G^{\prime}\right|}\sum_{\tau\in
G^{\prime}}\left|\text{Fix}(\tau)\right|,$ (4.40)

where for $\tau\in\mathfrak{S}$, $\text{Fix}(\tau)$ is the set of the fixed
points of $\tau$ in $S$. Since the identity element of $G^{\prime}$ has more
than one fixed points, (4.40) implies there exists $\tau_{0}\in G^{\prime}$
such that $\text{Fix}(\tau_{0})=\emptyset$. $x\in gHg^{-1}$ (where $x,\;g\in
G$) if and only if $xgH=gH$. Therefore, if $\tau_{0}=\pi(x_{0})$,
$x_{0}\notin\bigcup\limits_{g\in G}gHg^{-1}.$ ∎

######  Remark 4.7.

Let $\mathbb{F}$ be an algebraically closed field. Let $G$ be the group of all
$2\times 2$ invertible matrices whose entries are in $\mathbb{F}$ and $H$ be
the group of all $2\times 2$ invertible upper triangular matrices whose
entries are in $\mathbb{F}$. Then, there holds $G=\bigcup\limits_{g\in
G}gHg^{-1}.$

######  Lemma 4.8.

Let $X$ be a topological space and $A\subset X$. Suppose the image of the
homomorphism $\pi_{1}(A)\rightarrow\pi_{1}(X)$ (induced by the inclusion
$A\hookrightarrow X$) is a proper finite index or normal subgroup of
$\pi_{1}(X)$. Then the map $\pi_{0}(\Lambda A)\rightarrow\pi_{0}(\Lambda X)$
(induced by the inclusion $A\hookrightarrow X$) is not surjective.

###### Proof.

Given a group $G$, let $\text{conj}(G)$ denote the set of conjugacy classes in
$G$. A group homomorphism $\phi:H\rightarrow G$ induces a map
$\phi_{\\#}:\text{conj}(H)\rightarrow\text{conj}(G)$ which maps the conjugacy
class of $h$ to the conjugacy class of $\phi(h)$.

$\phi_{\\#}\text{ is surjective }\iff G=\bigcup_{g\in G}g\phi(H)g^{-1}.$
(4.41)

Thus the homomorphism $\pi_{1}(A)\rightarrow\pi_{1}(X)$ (induced by the
inclusion $A\hookrightarrow X$) induces a map
$\text{conj}(\pi_{1}(A))\rightarrow\text{conj}(\pi_{1}(X))$. For a topological
space $Y$, $\pi_{0}(\Lambda Y)$ can be canonically identified with
$\text{conj}(\pi_{1}(Y))$. Under this identification, the map $\pi_{0}(\Lambda
A)\rightarrow\pi_{0}(\Lambda X)$ (induced by the inclusion $A\hookrightarrow
X$) corresponds to the map
$\text{conj}(\pi_{1}(A))\rightarrow\text{conj}(\pi_{1}(X))$, described above.
Therefore, Lemma 4.8 follows from (4.41) and Lemma 4.6. ∎

######  Remark 4.9.

Let $X$ be a topological space and $A\subset X$ be connected. Let us compare
the three conditions $\pi_{1}(X,A)=0$, $\pi_{0}(\Lambda
A)\rightarrow\pi_{0}(\Lambda X)$ is surjective and $H_{1}(X,A)=0$, which
appear in Theorems 1.2 and 1.3, with $(X,A)=(M,M\setminus U)$. From Lemma 4.6,
Remark 4.7, the existence of finite perfect groups and the following
discussion, it follows that, in general, these three conditions are not
equivalent. The homotopy exact sequence for the pair $(X,A)$ implies
$\pi_{1}(X,A)=0$ if and only if $\pi_{1}(A)\rightarrow\pi_{1}(X)$ is
surjective. On the other hand, in the proof Lemma 4.8, we explained that
$\pi_{0}(\Lambda A)\rightarrow\pi_{0}(\Lambda X)$ is surjective if and only if
$\text{conj}(\pi_{1}(A))\rightarrow\text{conj}(\pi_{1}(X))$ is surjective.
Lastly, the homology exact sequence for the pair $(X,A)$ implies that
$H_{1}(X,A)=0$ if and only if $\pi_{1}(A)^{ab}\rightarrow\pi_{1}(X)^{ab}$ is
surjective (where $G^{ab}$ denotes the abelianization of the group $G$).

###### Proof of Theorem 1.3.

By Theorem 1.2, there exists a closed geodesic on $M$ if either
$\pi_{1}(M,M\setminus U)=0$ or the map $\pi_{0}(\Lambda(M\setminus
U))\rightarrow\pi_{0}(\Lambda M)$ is not surjective. Since $M\setminus U$ is
connected, the homotopy exact sequence for the pair $(M,M\setminus U)$ implies
that if $\pi_{1}(M\setminus U)\rightarrow\pi_{1}(M)$ is surjective, then
$\pi_{1}(M,M\setminus U)=0$. On the other hand, if the image of the
homomorphism $\pi_{1}(M\setminus U)\rightarrow\pi_{1}(M)$ is a proper finite
index or normal subgroup of $\pi_{1}(M)$, then by Lemma 4.8,
$\pi_{0}(\Lambda(M\setminus U))\rightarrow\pi_{0}(\Lambda M)$ is not
surjective.

Since $M\setminus U$ is connected, every relative homology class in
$H_{1}(M,M\setminus U)$ can be represented by a closed loop
$\gamma:[0,1]\rightarrow M$ such that $\gamma(0)=\gamma(1)\in M\setminus U$.
If $[\gamma]\neq 0\in H_{1}(M,M\setminus U)$, then $\gamma$ can not be freely
homotopic to a loop in $M\setminus U$. Therefore, $H_{1}(M,M\setminus U)\neq
0$ implies $\pi_{0}(\Lambda(M\setminus U))\rightarrow\pi_{0}(\Lambda M)$ is
not surjective.

∎

## References


# Interpreting Criminal Charge Prediction and Its Algorithmic Bias via
Quantum-Inspired Complex Valued Networks

Abdul Rafae Khan  Jia Xu  Peter Varsanyi  Rachit Pabreja

###### Abstract

While predictive policing has become increasingly common in assisting with
decisions in the criminal justice system, the use of these results is still
controversial. Some software based on deep learning lacks accuracy (e.g., in
F-1), and importantly many decision processes are not transparent, causing
doubt about decision bias, such as perceived racial and age disparities. This
paper addresses bias issues with post-hoc explanations to provide a trustable
prediction of whether a person will receive future criminal charges given
one’s previous criminal records by learning temporal behavior patterns over
twenty years. Bi-LSTM relieves the vanishing gradient problem, attentional
mechanisms allow learning and interpretation of feature importance, and
complex-valued networks inspired quantum physics to facilitate a certain level
of transparency in modeling the decision process. Our approach shows a
consistent and reliable prediction precision and recall on a real-life
dataset. Our analysis of the importance of each input feature shows the
critical causal impact on decision-making, suggesting that criminal histories
are statistically significant factors, while identifiers, such as race and
age, are not. Finally, our algorithm indicates that a suspect tends to rather
than suddenly increase crime severity level over time gradually.

Deep Learning, Crime Prediction, Interpretability, Bias, Fairness, Quantum
Physics, Complex Valued Networks

##  1 Introduction

Automatic crime prediction using deep learning algorithms has played an
increasingly critical role in criminal justice systems and crime prevention
efforts. Previous work, such as (Luo et al., 2017), shows that deep learning
can assist decision-making by processing a very high volume of data that are
difficult for human researchers to analyze efficiently. Despite the success of
deep learning, it is still challenging to convince the general public to trust
our algorithms to assist a judge’s decisions due to a lack of interpretability
and sufficient accuracy. In particular, criticism that bias exists for
identifiers such as race and age points (Skeem & Lowenkamp, 2016) toward the
need for a fair, transparent, and explainable crime prediction method.

Our work presents a criminal charge prediction problem and introduces a
framework that leads to an accurate decision with post-hoc interpretability. A
criminal charge is a formal accusation made by a governmental authority
asserting that an individual has committed a crime. There are three primary
classifications of criminal offenses: felonies (we call these level $1$
crimes), including terrorism, murder, kidnapping, treason, elder abuse;
misdemeanors (level $2$), such as arson, extortion, threat, bribery, larceny;
and infractions (level $3$), the least severe crimes, including damaging
property, burglary, smuggling, obscenity.

We aim to understand individuals’ criminal behavior over twenty years and
focus our study on whether there are patterns in a criminal’s charge record
over time. These charge records include a person’s prior arrests, type of
crime or crimes committed, prior convictions, and personal attributes of race
and age. We look specifically at whether a person will receive a charge in the
next two years and if yes, its level given the criminal history in the
previous eighteen years and personal information. We also analyze each input
feature’s importance for prediction results as well as the changes in an
individual’s crime levels over time.

The first technical challenge in our study is the heavily imbalanced data. A
fully connected feed-forward neural network can give very low recall and
F-measure scores despite high precision. The negative samples are much fewer
than the positive samples with the predictor tending to predict everything
positive to minimize error. We apply an RNN along with the Bi-LSTM to capture
temporal patterns of a suspect’s criminal behavior over time, an approach that
brings both high precision and high recall.

The second challenge is the explanation for the causality inference of our
predictions. We analyze feature importance using values from attention weights
learned from the attentional Bi-LSTMs. The weights learned while training and
the node connections activated while predicting will indicate the importance
of the input features concerning the decision reached during classification.
Our results show that, contrary to the belief that a machine learning
algorithm is biased, our algorithms learned that criminal charge histories are
essential for prediction and that personal data such as race and age are
minimally useful. Our algorithms therefore do not need to explicitly rely on
personal information such as race and age that can cause biased decisions. Our
result, however, does not indicate whether there is bias in the human raw
dataset, a question that is outside the scope of this study. We summarize our
three major contributions below:

  1. 1.

We study four research questions on an individual’s future criminal charge
prediction and introduce Bi-LSTM with attention and Quantum-Inspired Complex
Valued Networks (QICVN) to alleviate the imbalanced data problem and to
interpret feature importance for decision-making.

  2. 2.

Interestingly, our results show that criminal history plays a significant role
in predicting a person’s future crime and crime level, while personal
identifiers, race and age, do not.

  3. 3.

Our models indicate that a person is more likely to commit the same or a
similar level of crime in the near future than a very different one.

In the following context, we will first describe our task and dataset,
introduce our models, then demonstrate experimental results and interpret our
models. Finally, we will discuss related work and conclude.

##  2 Task Description and Dataset

### Dataset

The real-life crime prediction data we use contains the criminal charge record
in Newark, New Jersey, from $1997$ to $2017$, with $17,335$ suspect
individuals. Each suspect is provided with her/his Personal ID, race, and a
list of Bookings. Each booking has a list of sentences, where each sentence
has the age of committing the crime and the individual criminal charge
history, such as the NCIC crime code, NCIC category code, and the crime level.
The NCIC category code includes the list of [”ASSL”,”TO”, ”DAD”, ”LARC”, ”FO”,
”PP”, ”BURG”, ”SV”, ”DP”, ”OP”]. The input features also include the number of
bookings, age average, number of crimes at level $1$, $2$, $3$, and so on. We
take the first 18 years as the training set and the last two years as the test
set. A suspect with at least two bookings is one sample. The label for each
sample is the crime level (or whether it is a specific crime level) during the
last booking.

### Task

Our goal is to predict a person’s crime or crime level ($1$ for most severe
and $3$ least severe) given her/his personal information and previous criminal
records. We propose a multitude of tasks addressed by Question
(Q)$1$/$2$/$3$/$4$:

Is a suspect going to commit a crime at level 1/2/3/any?

Age/Race | Crime level $1$ | Crime level $2$ | Crime level $3$
---|---|---|---
| Yes | No | Yes | No | Yes | No
All | 6.7 | 93.3 | 3.5 | 96.5 | 8.7 | 91.3
$\leq$ 20 | 16.9 | 83.1 | 7.2 | 92.8 | 12.8 | 87.2
21-30 | 7.8 | 92.2 | 4.3 | 95.7 | 8.1 | 91.9
31-50 | 5.9 | 94.1 | 2.9 | 97.1 | 9.1 | 90.9
$>$50 | 2.8 | 97.2 | 1.4 | 98.6 | 7.2 | 92.8
W | 6.8 | 93.2 | 3.6 | 96.4 | 9.2 | 90.8
A | 9.8 | 90.2 | 2.1 | 97.9 | 8.1 | 91.9
U | 2.6 | 97.4 | 2.1 | 97.9 | 5.1 | 94.9
B | 8.6 | 91.4 | 3.6 | 96.4 | 6.9 | 93.1
I | 3.1 | 96.9 | 0.8 | 99.2 | 6.2 | 93.8

Table 1: Highly imbalanced data: most bookings are labeled as non-crime. We
measure the percentage of the crime- and -non-crime labeled sample number [in
%], respectively. The statistics is calculated on the whole dataset, and on
three different data classifications of the age and race, respectively.

Essentially we train four different systems for each of the above four
questions, viewed as a binary classification task. We observe from Table 1
that this data is highly imbalanced, with most of the instances having the “No
crime” label. This resembles a real-life scenario, where a suspect stops
committing a crime more often than continuing to commit crimes. It is
technically challenging to handle imbalanced data. A fully connected feed-
forward neural network does not work in our experiments since it may classify
all samples as negative. For example, suppose there are only $2$ positive
samples among $100$ samples. In that case, the precision is still $98\%$,
although nothing is classified.

We will first describe our task and dataset in the following context,
introduce our models, then demonstrate experimental results and interpret our
models. Finally, we will discuss related work and conclude.

##  3 Methods

### Learning Temporal Patterns with Sequential Models

To solve the imbalanced data problem, we observe that the criminal charges of
an individual follow temporal patterns. A suspect likely repeats a similar
behavior, such as committing a crime with the same level or after an equal
amount of time (i.e. somewhat periodically). This kind of temporary pattern
can be learned with the whole dataset’s statistics and shared among different
individuals. We introduce a Recurrent Neural Network (RNN) to capture the
sequential patterns of individual criminal charge records along one’s
lifetime. To reduce the vanishing gradient problem, we add LSTMs (Hochreiter &
Schmidhuber, 1997) in the activation functions, which greatly reduced the data
imbalance problem. As a data preparation step, we had a list of booking for
each person. Each booking has an age that shows the person’s age when he/she
gets the sentence. We find that the maximum number of bookings for a person of
different ages (sorted ages from young ages to old ones) is 13. We decided to
create sequential data with 12 parts, and we use the last booking as a label
of the dataset.

### Interpreting Feature Importance with Attentional Mechanisms

The second challenge is that our models lack interpretability.  We provide a
certain degree of post-interpretability using the attention mechanism (Vaswani
et al., 2017). In particular, the attention mechanism allows to focus on, or
pay attention to, different parts of the input sequence to generate a
prediction. The attention scores obtained while predicting will indicate the
importance of the input features towards the decision and can provide post-hoc
interpretability. To obtain the feature importance: First, we check the impact
of all the input features on our model predictions by setting the window size
to be the total number of input features. After calculating the embedding of
each feature, we compute their relative weights. Then, we apply softmax
overall features to obtain probabilities of the predictions. Each input is
associated with a probability value, and the sum of all probabilities should
be $1$.

### Increasing Model Transparency with Quantum-inspired Complex Valued
Networks

Figure 1: Architecture of Quantum-Inspired Complex Valued Networks (QICVN).
Inputs: features; Outputs: whether a person will conduct a criminal charge at
a given level.

Attention mechanisms only interpret the input feature weights, while the deep
learning hidden layers and decision process are hard to explain. To increase
the transparency of our prediction model, we introduce complex-valued networks
based on the mathematical formulation of quantum physics. As a realization, we
adopt the model of (Li et al., 2019) from the matching task to a binary
classification task. The criminal history and other attributes are the feature
inputs to the model, and the output is yes or no to the future crime level.

Specifically, we first convert each feature value to a complex embedding of
size $d$. An embedding layer with a ReLU activation (Nair & Hinton, 2010)
takes as input the features one-by-one and converts them into two
$d$-dimensional vectors, one for representing amplitude and the other for
phase. Each number in the phase embedding is normalized to the range
$[-\pi,\pi]$. Given the complex embedding, we compute the relative weight of
the $i$-th feature as the $\ell_{2}$ norm of its amplitude. We then normalize
the amplitude by dividing by the $\ell_{2}$ norm. Using the normalized
amplitude $A_{i}$ and the phase $\varphi_{i}$, we compute the real ($R$) and
imaginary ($G$) parts of $i$-th feature as $R_{i}=A_{i}\cos(\varphi_{i})$ and
$G_{i}=A_{i}\sin(\varphi_{i})$. As in Figure 1, we compute the mixture ($M$)
for each feature in the n-gram as complex matrix multiplication, and then each
mixture is multiplied by a softmax probability of the feature’s relative
weight in the n-gram. After that, all the mixtures in the n-gram are summed up
to obtain the local mixture density matrix. The local mixtures are projected
using $K$ measurement matrices as in (Li et al., 2019). The resulting vectors
are max-pooled along each dimension, followed by a linear layer with softmax
activation.

##  4 Experiments

### Setup

Our model (Winata et al., 2018) is based on long short-term memory (LSTM)
network, and Bi-LSTM with attentions (Hochreiter & Schmidhuber, 1997) with two
hidden layers, where each hidden layer has ten nodes with or without attention
layers. The inputs are the features, and the outputs are fed to a linear layer
mapping to the target classes. The dropout parameter is set to 0.1. The
parameters were updated using the Adam optimizer (Kingma & Ba, 2014) with a
learning rate of 0.001 and random initialization.

### Prediction Results

Model | Crime level | Acc. | Prec. | Recall | F1
---|---|---|---|---|---
LSTM | 1 | 94.1 | 92.0 | 94.1 | 92.3
| 2 | 96.4 | 93.2 | 96.4 | 94.8
| 3 | 92.3 | 91.2 | 92.3 | 91.5
| Any | 87.5 | 86.8 | 87.5 | 87.0
Bi-LSTM | 1 | 94.5 | 93.0 | 94.5 | 93.1
with | 2 | 96.5 | 93.2 | 96.5 | 94.8
attention | 3 | 91.4 | 87.0 | 91.4 | 87.6
| Any | 90.6 | 90.5 | 90.6 | 90.5
QICVN | 1 | 85.2 | 93.9 | 85.2 | 88.5
| 2 | 85.0 | 95.7 | 85.0 | 89.3
| 3 | 81.2 | 90.2 | 81.2 | 84.5
| Any | 86.4 | 90.1 | 86.4 | 87.4

Table 2: LSTM, Bi-LSTM with attention, and QICVN results for crime level
prediction.

Table 2 shows the accuracy scores on the test data using different models on
four research questions in Section 2. We introduce sequential models,
including LSTM, Bi-LSTM with and without attention. We achieve both high
precision and high recall overcoming the data imbalance problem by considering
the criminal charge records along the previous eighteen years and learning the
common patterns across samples.

### Interpreting Results

Our attention model provides some interpretations for its predictions,
suggesting that the prediction relies on historical criminal records rather
than personal data, such as race and age. Feature weights are learned as a
means to measure the “causality” of model decisions in our algorithm. We
visualize softmax probabilities for randomly selected test samples. See Figure
2. We see that surprisingly, although there are many input features, only a
few of them have the highest impact. Importantly, Figure 2 shows that personal
information such as race and age contribute very little towards the
prediction. Instead, an individual’s criminal history features greatly impact
the prediction, such as “time since last crime” and “variance of the time gap
between successive crimes”. This indicates the personal information features
do not explicitly contribute to our algorithmic decisions. Note that our work
only focuses on the feature importance explicitly used in our model.
Discussions on any bias made by humans in the previous charge decisions
(Brantingham et al., 2018; Arnold et al., 2018) are out of the scope of this
paper.

Figure 2: Each column shows the relative weights of different features for one
random test sample or the average values across all years. The higher value,
the more important is the weight.

The correlation is computed as the probability (relative frequency) of the
lowest level (most severe one) of the crimes committed in the individual
history given a certain crime level $L$ in the test set. The algorithmic
“causality” of level an on predicting level b is computed as follows: For each
test sample, we sum up the weights of all features of level a, such as the
number of level a, yearly level a, on the model to predict whether a suspect
is going to commit a crime at level b. Then we normalize it with all level-
related weights and average the relative weights over all test samples. This
average weight shows how much level a influences the decision on predicting
level b.

The correlation and causality are different. The former computes the
probability of a future crime level given a previous crime level in our data.
The latter measures how much the features of a prior crime level contribute to
the decision on whether a person will commit a certain level of crime in the
future. Thus, the correlation is data-dependent, and the causality is model-
dependent. In our study on the transitions of a suspect from one crime level
to another, we find that crime level features are essential to predict a new
crime level. Table 3 shows whether a lower level crime record indicates a
higher level of crime or vice versa. Both our algorithmic causality (from our
model of Bi-LSTM with attention mechanisms) and correlation results indicate
that a suspect is likely to commit a crime with the same or similar level
again than a much higher or lower level.

| Crime level (L) | $L=1$ | $L=2$ | $L=3$
---|---|---|---|---
Correlation | $P(\textrm{level }1|L)$ | 0.53 | 0.37 | 0.25
| $P(\textrm{level }2|L)$ | 0.15 | 0.33 | 0.19
| $P(\textrm{level }3|L)$ | 0.32 | 0.30 | 0.56
Causality | level 1 to predict $L$ | 0.60 | 0.12 | 0.11
(Normalized | level 2 to predict $L$ | 0.30 | 0.09 | 0.12
feature weights) | level 3 to predict $L$ | 0.10 | 0.79 | 0.77

Table 3: Crime level transitions. Given L as the label, the
probability/weights of each lowest (most severe crime) level in history.

##  5 Related Work

There has been an increasing trend in criminal justice to leverage machine
learning for high-stakes prediction applications that deeply impact human
lives. (Yu et al., 2011) uses Support Vectors Machine (SVM) model, 2-layered
feed forward neural networkand Naive Bayes model for crime forecasting. (Stec
& Klabjan, 2018) and (Stalidis et al., 2018) use feed forward, convolutional
and recurrent-convolutional networks, (Luo et al., 2017) uses recurrent based
networks with attention mechanism to analyze law articles. Our model directly
predicts the criminal charge based on the criminal history records and
interprets our algorithms’ decisive factors. (Dressel & Farid, 2021) courts
across the United States are using computer software to predict whether a
person will commit a crime, the results of which are incorporated into bail
and sentencing decisions. It is imperative that such tools be accurate and
fair, but critics have charged that the software can be racially biased,
favoring white defendants over black defendants. We evaluate the claim that
computer software is more accurate and fairer than people tasked with making
similar decisions. We also evaluate and explain the presence of racial bias in
these predictive algorithms.

##  6 Conclusion

Our work introduces a trustable criminal charge prediction method with high
precision and high recall and post-interpretability. We show that deep
learning and quantum-inspired complex-valued networks can be a part of the
criminal justice assistant system as long as model transparency and accuracy
are taken care of. Perhaps most importantly, we draw attention to the
erroneous assumption that social features such as race and age are
statistically insignificant to influence our model prediction, even though
data may introduce bias.

## Acknowledgement

We appreciate the National Science Foundation (NSF) Award No. 1747728 and the
National Science Foundation of China (NSFC) Award No. 61672524 to fund this
research. We thank Kovid Inc. to provide the dataset within the NSF project,
the initial input of Subhadarshi Panda and the comments of Mengyan Dai,
Periklis Papakonstantinou, and Feng Gu. Finally, we sincerely thank the
support of the Google Cloud Research Program.

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