Combinatorial Structure of Manifolds with

Poincaré Conjecture

Linfan MAO

(Chinese Academy of Mathematics and System Science, Beijing 100080, P.R.China)

E-mail<EMAIL_ADDRESS>

Abstract: A manifold $M^{n}$ inherits a labeled $n$-dimensional graph
$\widetilde{M}[G^{L}]$ structure consisting of its charts. This structure
enables one to characterize fundamental groups of manifolds, classify those of
locally compact manifolds with finite non-homotopic loops by that of labeled
graphs $G^{L}$. As a by-product, this approach also concludes that every
homotopy $n$-sphere is homeomorphic to the sphere $S^{n}$ for an integer
$n\geq 1$, particularly, the Perelman’s result for $n=3$.

Key Words: Combinatorial Euclidean space, manifold, fundamental group, labeled
graph, homotopic loop.

AMS(2010): 51H20.

§$1.$ Introduction

An $n$-manifold is a second countable Hausdorff space of locally Euclidean
$n$-space without boundary. Terminologies and notions used in this paper are
standard. We follow references [1], [13] for topology, [5], [6] for graphs or
topological graphs and [11] for combinatorial manifolds. Here, we only mention
conceptions appeared in references [8], [10]-[11] for combinatorial manifolds.

Definition $1.1$ A combinatorial Euclidean space
$\mathscr{E}_{G}(n_{\nu};\nu\in\Lambda)$ underlying a connected graph $G$ is a
topological spaces consisting of ${\bf R}^{n_{\nu}}$, $\nu\in\Lambda$ for an
index set $\Lambda$ such that

$V(G)=\\{{\bf R}^{n_{\nu}}|\nu\in\Lambda\\}$;

$E(G)=\\{\ ({\bf R}^{n_{\mu}},{\bf R}^{n_{\nu}})|\ {\bf R}^{n_{\mu}}\cap{\bf
R}^{n_{\nu}}\not=\emptyset,\mu,\nu\in\Lambda\\}$.

If $|\Lambda|=1$, i.e., the dimension of Euclidean spaces in
$\mathscr{E}_{G}(n_{\nu};\nu\in\Lambda)$ are all in the same $n$, the notation
$\mathscr{E}_{G}(n_{\nu};\nu\in\Lambda)$ is abbreviated to
$\mathscr{E}_{G}(n,\ \nu\in\Lambda)$ for simplicity.

Notice that a Euclidean space ${\bf R}^{n}$ is an $n$-dimensional vector space
with a normal basis $\overline{\epsilon}_{1}=(1,0,\cdots,0)$,
$\overline{\epsilon}_{2}=(0,1,0\cdots,0)$, $\cdots$,
$\overline{\epsilon}_{n}=(0,\cdots,0,1)$, namely, it has $n$ orthogonal
orientations. In Definition $1.1$, we do not assume ${\bf R}^{n_{\mu}}\cap{\bf
R}^{n_{\nu}}={\bf R}^{\min\\{n_{\mu},n_{\nu}\\}}$. In fact, let
$\mathcal{X}_{v_{\mu}}$ be the set of orthogonal orientations in ${\bf
R}^{n_{v_{\mu}}}$, $\mu\in\Lambda$, respectively ([12]). Then

${\bf R}^{n_{\mu}}\bigcap{\bf
R}^{n_{\nu}}=\mathcal{X}_{v_{\mu}}\bigcap\mathcal{X}_{v_{\nu}}.$

A combinatorial fan-space $\widetilde{\bf R}(n_{\nu};\nu\in\Lambda)$ is a
combinatorial Euclidean space
$\mathscr{E}_{K_{|\Lambda|}}(n_{\nu};\nu\in\Lambda)$ of ${\bf R}^{n_{\nu}},\
\nu\in\Lambda$ such that for any integers $\mu,\nu\in\Lambda,\ \mu\not=\nu$,

${\bf R}^{n_{\mu}}\bigcap{\bf
R}^{n_{\nu}}=\bigcap\limits_{\lambda\in\Lambda}{\bf R}^{n_{\lambda}}.$

If $|\Lambda|=m<+\infty$, for $\forall p\in\widetilde{{\bf
R}}(n_{\nu};\nu\in\Lambda)$ we can present it by an $m\times n_{m}$ coordinate
matrix $[\overline{x}]$ following with $x_{il}=\frac{x_{l}}{m}$ for $1\leq
i\leq m,1\leq l\leq\widehat{m}$, where $\widehat{m}$ is the number of
orthogonal orientations in
$\mathcal{X}_{v_{\mu}}\bigcap\mathcal{X}_{v_{\nu}}$,

$[\overline{x}]=\left[\begin{array}[]{cccccccc}x_{11}&\cdots&x_{1\widehat{m}}&x_{1(\widehat{m})+1)}&\cdots&x_{1n_{1}}&\cdots&0\\\
x_{21}&\cdots&x_{2\widehat{m}}&x_{2(\widehat{m}+1)}&\cdots&x_{2n_{2}}&\cdots&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
x_{m1}&\cdots&x_{m\widehat{m}}&x_{m(\widehat{m}+1)}&\cdots&\cdots&x_{mn_{m}-1}&x_{mn_{m}}\end{array}\right],$

which enables us to generalize the conception of manifold to combinatorial
manifold, a locally combinatorial Euclidean space.

Definition $1.2$ For a given integer sequence $0<n_{1}<n_{2}<\cdots<n_{m}$,
$m\geq 1$, a topological combinatorial manifold $\widetilde{M}$ is a Hausdorff
space such that for any point $p\in\widetilde{M}$, there is a local chart
$(U_{p},\varphi_{p})$ of $p$, i.e., an open neighborhood $U_{p}$ of $p$ in
$\widetilde{M}$ and a homoeomorphism
$\varphi_{p}:U_{p}\rightarrow\widetilde{\bf
R}(n_{1}(p),n_{2}(p),\cdots,n_{s(p)}(p))=\bigcup\limits_{i=1}^{s(p)}{\bf
R}^{n_{i}(p)}$ with

$\\{n_{1}(p),n_{2}(p),\cdots,n_{s(p)}(p)\\}\subseteq\\{n_{1},n_{2},\cdots,n_{m}\\}$
and

$\bigcup\limits_{p\in\widetilde{M}}\\{n_{1}(p),n_{2}(p),\cdots,n_{s(p)}(p)\\}=\\{n_{1},n_{2},\cdots,n_{m}\\}$,

denoted by $\widetilde{M}(n_{1},n_{2},\cdots,n_{m})$ or $\widetilde{M}$ on the
context and

$\widetilde{{\mathcal{A}}}=\\{(U_{p},\varphi_{p})|p\in\widetilde{M}(n_{1},n_{2},\cdots,n_{m}))\\}$

an atlas on $\widetilde{M}(n_{1},n_{2},\cdots,n_{m})$.

A topological combinatorial manifold $\widetilde{M}(n_{1},n_{2},\cdots,n_{m})$
is finite if it is just combined by finite manifolds without one manifold
contained in the union of others.

If $m=1$, then $\widetilde{M}(n_{1},n_{2},\cdots,n_{m})$ is exactly the
manifold $M^{n_{1}}$ by definition. Furthermore, if these manifolds $M_{i},\
1\leq i\leq m$ in $\widetilde{M}(n_{1},n_{2},\cdots,n_{m})$ are Euclidean
spaces ${\bf R}^{n_{i}},\ 1\leq i\leq m$, then
$\widetilde{M}(n_{1},n_{2},\cdots,n_{m})$ is nothing but the combinatorial
Euclidean space $\mathscr{E}_{G}(n_{\nu};\nu\in\Lambda)$ with
$\Lambda=\\{1,2,\cdots,m\\}$.

For a finitely combinatorial manifold
$\widetilde{M}(n_{1},n_{2},\cdots,n_{m})$ consisting of manifolds $M_{i},\
1\leq i\leq m$, we can construct a vertex-edge labeled graph
$G^{L}[\widetilde{M}]$ defined by

$V(G^{L}[\widetilde{M}])=\\{M_{1},M_{2},\cdots,M_{m}\\},$

$E(G^{L}[\widetilde{M})=\\{\ (M_{i},M_{j})\ |\ M_{i}\bigcap
M_{j}\not=\emptyset,1\leq i,j\leq n\\}$

with a labeling mapping $\Theta:V(G^{L}[\widetilde{M}])\cup
E(G^{L}[\widetilde{M}])\rightarrow{\bf Z}^{+}$ determined by

$\Theta(M_{i})={\rm dim}M_{i}\ \ {\rm and}\ \ \Theta(M_{i},M_{j})={\rm
dim}M_{i}\bigcap M_{j}$

for integers $1\leq i,j\leq m$, which is inherent structure of combinatorial
manifolds. Particularly, for a combinatorial Euclidean space
$\mathscr{E}_{G}(n_{1},\cdots,n_{m})$, this vertex-edge labeled graph is
defined on $G$ by labeling $\Theta:V(G)\cup E(G)\rightarrow{\bf Z}^{+}$
determined by

$\Theta({\bf R}^{n_{i}})=n_{i}\ \ {\rm and}\ \ \Theta({\bf R}^{n_{i}},{\bf
R}^{n_{j}})={\rm dim}{\bf R}^{n_{i}}\bigcap{\bf R}^{n_{j}}$

for integers $1\leq i,j\leq m$.

The objective of this paper is to characterize the inherent combinatorial
structure of $n$-manifolds with finite non-homotopic loops. For such
manifolds, there is a well-known Poincaré conjecture first proved by Perelman
([3]) following.

Theorem $1.3$(Perelman,[15]-[17]) Any closed simply connected $3$-manifold is
homeomorphic to $S^{3}$.

Notice that a homotopy sphere is an $n$-manifold homotopy equivalent to the
$n$-sphere. A generalized Poincaré conjecture says that any homotopy
$n$-sphere is homeomorphic to $S^{n}$. Combining works of Smale in 1961 for
$n\geq 5$ ([18]), Freedman’s in 1982 for $n=4$ ([2]), Theorem $1.3$ and
classical result for $n=1,2$, we conclude that

Theorem $1.4$ Any homotopy $n$-sphere is homeomorphic to $S^{n}$ for $n\geq
1$.

Then can we find a unified proof on Theorem $1.4$? By applying a combinatorial
notion [9], we find an inherent labeled graph $G^{L}[M^{n}]$ structure
consisting of charts homeomorphic to a Euclidean space ${\bf R}^{n}$ for a
topological manifold $M^{n}$ in this paper, which enables us to classify
locally compact $n$-manifolds with finite non-homotopic loops by labeled
graphs for integers $n\geq 1$, also simplifies the calculation of fundamental
groups of $n$-manifolds and get a combinatorial proof for Theorem $1.4$.

§$2.$ Dimensional Graphs

We discuss combinatorial Euclidean spaces $\mathscr{E}_{G}(n)$ in this section
whose importance is shown in the next result.

Theorem $2.1$ A locally compact $n$-manifold $M^{n}$ is a combinatorial
manifold $\widetilde{M}_{G}(n)$ homeomorphic to a Euclidean space
$\mathscr{E}_{G^{\prime}}(n)$ with countable graphs $G\cong G^{\prime}$
inherent in $M^{n}$, denoted by $G[M^{n}]$.

Proof Let $M^{n}$ be a locally compact $n$-manifold with an atlas

$\mathscr{A}[M^{n}]=\\{\ (U_{\lambda};\varphi_{\lambda})\ |\
\lambda\in\Lambda\\},$

where $\Lambda$ is a countable set. Then each $U_{\lambda},\
\lambda\in\Lambda$ is itself an $n$-manifold by definition. Define an
underlying combinatorial structure $G$ by

$V(G)=\\{U_{\lambda}|\lambda\in\Lambda\\},$

$E(G)=\\{\ (U_{\lambda},U_{\iota})_{i},1\leq i\leq\kappa_{\lambda\iota}+1|\
U_{\lambda}\bigcap U_{\iota}\not=\emptyset,\lambda,\iota\in\Lambda\\}$

where $\kappa_{\lambda\iota}$ is the number of non-homotopic loops in formed
between $U_{\lambda}$ and $U_{\iota}$. Then we get a combinatorial manifold
$\widetilde{M}_{G}(n)$ underlying a countable graph $G$.

Define a combinatorial Euclidean space
$\mathscr{E}_{G^{\prime}}(n,\lambda\in\Lambda)$ of spaces ${\bf R}^{n}$ by

$V(G^{\prime})=\\{\varphi_{\lambda}(U_{\lambda})|\lambda\in\Lambda\\},$

$E(G^{\prime})=\\{\
(\varphi_{\lambda}(U_{\lambda}),\varphi_{\iota}(U_{\iota}))_{i},1\leq
i\leq\kappa^{\prime}_{\lambda\iota}+1|\
\varphi_{\lambda}(U_{\lambda})\bigcap\varphi_{\iota}(U_{\iota})\not=\emptyset,\lambda,\iota\in\Lambda\\},$

where $\kappa^{\prime}_{\lambda\iota}$ is the number of non-homotopic loops in
formed between $\varphi_{\lambda}(U_{\lambda})$ and
$\varphi_{\iota}(U_{\iota})$. Notice that
$\varphi_{\lambda}(U_{\lambda})\bigcap\varphi_{\iota}(U_{\iota})\not=\emptyset$
if and only if $U_{\lambda}\bigcap U_{\iota}\not=\emptyset$ and
$\kappa_{\lambda\iota}=\kappa^{\prime}_{\lambda\iota}$ for
$\lambda,\iota\in\Lambda$. We know that $G\ \cong\ G^{\prime}$ by definition.

Now we prove that $\widetilde{M}_{G}(n)$ is homeomorphic to
$\mathscr{E}_{G^{\prime}}(n,\lambda\in\Lambda)$. By assumption, $M^{n}$ is
locally compact. Whence, there exists a partition of unity
$c_{\lambda}:U_{\lambda}\rightarrow{\bf R}^{n}$, $\lambda\in\Lambda$ on the
atlas $\mathscr{A}[M^{n}]$. Let $A_{\lambda}={\rm supp}(\varphi_{\lambda})$.
Define functions $h_{\lambda}:M^{n}\rightarrow{\bf R}^{n}$ and ${\bf
H}:M^{n}\rightarrow\mathscr{E}_{G^{\prime}}(n)$ by

$h_{\lambda}(x)=\left\\{\begin{array}[]{lr}c_{\lambda}(x)\varphi_{\lambda}(x)&{\rm
if}\ x\in U_{\lambda},\\\ {\bf 0}=(0,\cdots,0)&{\rm if}\ x\in
U_{\lambda}-A_{\lambda}.\end{array}\right.$

and

${\bf H}=\sum\limits_{\lambda\in\Lambda}\varphi_{\lambda}c_{\lambda},\ \ \
{\rm and}\ \ \ {\bf
J}=\sum\limits_{\lambda\in\Lambda}c_{\lambda}^{-1}\varphi_{\lambda}^{-1}.$

Then $h_{\lambda}$, ${\bf H}$ and ${\bf J}$ all are continuous by the
continuity of $\varphi_{\lambda}$ and $c_{\lambda}$ for
$\forall\lambda\in\Lambda$ on $M^{n}$. Notice that
$c_{\lambda}^{-1}\varphi_{\lambda}^{-1}\varphi_{\lambda}c_{\lambda}=$the unity
function on $M^{n}$. We get that ${\bf J}={\bf H}^{-1}$, i.e., ${\bf H}$ is a
homeomorphism from $M^{n}$ to $\mathscr{E}_{G^{\prime}}(n,\lambda\in\Lambda)$.
$\Box$

According to Theorem $2.1$, a combinatorial Euclidean space homeomorphic to a
$n$-manifold $M^{n}$ can be denoted by $\mathscr{E}_{G[M^{n}]}(n,\
\mu\in\Lambda)$. We classify such combinatorial Euclidean spaces
$\mathscr{E}_{G}(n,\mu\in\Lambda)$ into two classes by considering the
intersection $\varphi_{\mu}^{-1}({\bf R}^{n})\cap\varphi_{\nu}^{-1}({\bf
R}^{)}$ for $\mu,\nu\in\Lambda$ following:

Class 1. For $\forall\mu,\nu\in\Lambda$, $\varphi_{\mu}^{-1}({\bf
R}^{n})\cap\varphi_{\nu}^{-1}({\bf R}^{)}=\emptyset$ or homeomorphic to ${\bf
R}^{n}$.

Class 2. There are $\mu,\nu\in\Lambda$ such that $\varphi_{\mu}^{-1}({\bf
R}^{n})\cap\varphi_{\nu}^{-1}({\bf R}^{)}\not=\emptyset$ with more than $2$
arcwise connected components.

We respectively discuss Classes $1$ and $2$ by dimensional graphs following.

$2.1$ Dimensional Graphs $\widetilde{M}^{n}[G]$

Let $B^{n}=\\{(x_{1},x_{2},\cdots,x_{n})|\ (x_{1},x_{2},\cdots,x_{n})\in{\bf
R}^{n},\ x_{1}^{n}+x_{2}^{n}+\cdots+x_{n}^{n}<1\\}$,
$S^{n}=\\{(x_{1},x_{2},\cdots,x_{n})|\ (x_{1},x_{2},\cdots,x_{n})\in{\bf
R}^{n},\ x_{1}^{n}+x_{2}^{n}+\cdots+x_{n}^{n}=1\\}$ be a $n$-dimensional open
ball or $n$-sphere for an integer $n\geq 1$, respectively. The combinatorial
Euclidean spaces $\mathscr{E}_{G}(n,\lambda\in\Lambda)$ in Class $1$ is such a
topological space that each non-homotopic loops comes from the graph $G$,
which can be characterized by graphs in spaces, i.e., $n$-dimensional graphs
defined following.

Definition $2.2$ An $n$-dimensional graph $\widetilde{M}^{n}[G]$ is a
combinatorial Euclidean space $\mathscr{E}_{G}(n)$ of ${\bf
R}^{n}_{\mu},\mu\in\Lambda$ underlying a combinatorial structure $G$ such that

($1$) $V(G)$ is discrete consisting of $B^{n}$, i.e., $\forall v\in V(G)$ is
an open ball $B_{v}^{n}$;

($2$) $\widetilde{M}^{n}[G]\setminus V(\widetilde{M}^{n}[G])$ is a disjoint
union of open subsets $e_{1},e_{2},\cdots,e_{m}$, each of which is
homeomorphic to an open ball $B^{n}$;

($3$) the boundary $\overline{e}_{i}-e_{i}$ of $e_{i}$ consists of one or two
$B^{n}$ and each pair $(\overline{e}_{i},e_{i})$ is homeomorphic to the pair
$(\overline{B}^{n},B^{n})$;

($4$) a subset $A\subset\widetilde{M}^{n}[G]$ is open if and only if
$A\cap\overline{e}_{i}$ is open for $1\leq i\leq m$.

A topological graph $\mathscr{T}[G]$ of a graph $G$ is a $1$-dimensional graph
in a topological space $\mathscr{P}$. We restate it in the following.

Definition $2.3$ A topological graph $\mathscr{T}[G]$ is a pair $(X,X^{0})$ of
a Hausdorff space $X$ with its a subset $X^{0}$ such that

($1$) $X^{0}$ is discrete, closed subspaces of $X$;

($2$) $X-X^{0}$ is a disjoint union of open subsets
$e_{1},e_{2},\cdots,e_{m}$, each of which is homeomorphic to an open interval
$(0,1)$;

($3$) the boundary $\overline{e}_{i}-e_{i}$ of $e_{i}$ consists of one or two
points. If $\overline{e}_{i}-e_{i}$ consists of two points, then
$(\overline{e}_{i},e_{i})$ is homeomorphic to the pair $([0,1],(0,1))$; if
$\overline{e}_{i}-e_{i}$ consists of one point, then
$(\overline{e}_{i},e_{i})$ is homeomorphic to the pair
$(S^{1},S^{1}-\\{1\\})$;

($4$) a subset $A\subset\widetilde{T}[G]$ is open if and only if
$A\cap\overline{e}_{i}$ is open for $1\leq i\leq m$.

Observation shows that there is a natural relation between a $n$-dimensional
graph $\widetilde{M}^{n}[G]$ with that of its a topological graph
$\mathscr{T}_{0}[G]=(X_{0},X^{0}_{0})$ which is constructed from
$\widetilde{M}^{n}[G]_{1}$ by:

($1$) Let $X^{0}_{0}=\\{\ O_{v}\ |\ O_{v}\ {\rm is\ the\ center\ of}\
B^{n}_{v}\ {\rm for}\ v\in V(G)\\}$;

($2$) For $\forall uv\in E(G)$, let $uv$ be a line segment
$e_{uv}:[0,1]\rightarrow B^{n}_{u}\cup B^{n}_{v}$ such that $e_{uv}(0)=O_{u}$
and $e_{uv}(1)=O_{v}$.

Then, the following result shows that a $n$-dimensional graph
$\widetilde{M}^{n}[G]$ is in fact a blown up of a topological graph
$\mathscr{T}[G]$ to dimensional $n$.

Theorem $2.4$ For any integer $n\geq 1$, $\mathscr{T}_{0}[G]$ is a deformation
retract of $\widetilde{M}^{n}[G]$.

Proof If $n=1$, then $\widetilde{M}^{n}[G]=\mathscr{T}_{0}[G]$ is itself a
topological graph. So we assume $n\geq 2$. Define a mapping
$f:\widetilde{M}^{n}[G]\times I\rightarrow\widetilde{M}^{n}[G]$ by

$f(\overline{x},t)=(1-t)\overline{x}+t\overline{x}_{0},$

for $\forall\overline{x}\in\widetilde{M}^{n}[G]_{1},t\in I$, where
$\overline{x}_{0}=O_{v}$ if $\overline{x}\in B^{n}_{v}$, and
$\overline{x}_{0}=p(\overline{x})$ if $\overline{x}\in e_{i}$, where
$p:uv\rightarrow e_{uv}$ a projection for $1\leq i\leq m$, such as those shown
in Fig.$2.1$.

Fig.$2.1$

Then we know that $f$ is continuous by definition and for
$\forall\overline{x}\in\widetilde{M}^{n}[G]$,

$f(\overline{x},0)=\overline{x},$

$f(\overline{x},1)=p(\overline{x}),$

and $f(\overline{x},t)=\overline{x}$ for
$\forall\overline{x}\in\mathscr{T}_{0}[G]$ and $t\in I$. Therefore,
$\mathscr{T}_{0}[G]$ is a deformation retract of $\widetilde{M}^{n}[G]$ by
definition. $\Box$

Notice that the inclusion mapping
$i:\mathscr{T}_{0}[G]\rightarrow\widetilde{M}^{n}[G]$ is an isomorphism
between groups $\pi(\mathscr{T}_{0}[G],v_{0})$ and
$\pi(\widetilde{M}^{n}[G]_{1},v_{0})$. We get a conclusion by Theorem $2.4$
following.

Corollary $2.5$ Let $\widetilde{M}^{n}[G]$ be a $n$-dimensional graph. Then
for $v_{0}\in\mathscr{T}_{0}[G]$,

$\pi(\widetilde{M}^{n}[G],v_{0})\cong\pi(\mathscr{T}_{0}[G],v_{0}).$

We have known the structure of fundamental group of a topological graph
$\mathscr{T}[G]$ in [13]. Whence, we can characterize the fundamental group of
a $n$-dimensional graph $\widetilde{M}^{n}[G]$ by applying Corollary $2.5$
following.

Theorem $2.6$ Let $T_{span}$ be a spanning tree in the topological graph
$\mathscr{T}_{0}[G]$, $\\{e_{\lambda}:\ \lambda\in\Lambda\\}$ the set of edges
of $\mathscr{T}_{0}[G]$ not in $T_{span}$ and
$\alpha_{\lambda}=A_{\lambda}e_{\lambda}B_{\lambda}\in\pi(\mathscr{T}_{0}[G],v_{0})$
a loop associated with $e_{\lambda}=a_{\lambda}b_{\lambda}$ for
$\forall\lambda\in\Lambda$, where $v_{0}\in\mathscr{T}_{0}[G]$ and
$A_{\lambda}$, $B_{\lambda}$ are unique paths from $v_{0}$ to $a_{\lambda}$ or
from $b_{\lambda}$ to $v_{0}$ in $T_{span}$. Then

$\pi(\mathscr{T}_{0}[G],v_{0})=\left<\alpha_{\lambda}|\lambda\in\Lambda\right>.$

An $n$-dimensional tree is a $n$-dimensional graph $\widetilde{M}^{n}[G]$ with
a tree $G$, denoted by $\widetilde{M}^{n}[T]$. Applying Theorem $2.4$, we know
the next result.

Theorem $2.7$ An $n$-dimensional tree $\widetilde{M}^{n}[T]$ is contractible.

Proof By Theorem $2.4$, we know that there is a continuous mapping
$f:\widetilde{M}^{n}[T]\times I\rightarrow\widetilde{M}^{n}[T]$ such that
$\mathscr{T}_{0}[T]$ is a deformation retract of $\widetilde{M}^{n}[T]$, i.e.,
for $\forall\overline{x}\in\widetilde{M}^{n}[T]$,

$f(\overline{x},0)=\overline{x},$

$f(\overline{x},1)=p(\overline{x}),$

and $f(\overline{x},t)=\overline{x}$ for
$\forall\overline{x}\in\mathscr{T}_{0}[T]$ and $t\in I$. Notice that
$\mathscr{T}_{0}[T]$ is contractible ([13]), we have a continuous mapping
$g:\mathscr{T}_{0}[T]\times I\rightarrow\mathscr{T}_{0}[T]$ such that
$\\{v_{0}\\}$ is a deformation retract for $\forall v_{0}\in\mathscr{T}[T]$.
Whence, the composition mapping $g\circ f:\widetilde{M}^{n}[T]\times
I\rightarrow\widetilde{M}^{n}[T]$ is continuous such that for
$\forall\overline{x}\in\widetilde{M}^{n}[T]$,

$g\circ f(\overline{x},0)=\overline{x},$

$g\circ f(\overline{x},1)=p(\overline{x}),$

and $g\circ f(v_{0},t)=v_{0}$ for $\forall t\in I$, i.e., $\\{v_{0}\\}$ is a
deformation retract of $\widetilde{M}^{n}[T]_{1}$. This completes the proof.
$\Box$

$2.2$ Labeled Dimensional Graphs $\widetilde{M}^{n}[G^{L}]$

In Class $2$, non-homotopic loops come from both $\varphi_{\mu}^{-1}({\bf
R}^{n})\cap\varphi_{\nu}^{-1}({\bf R})$ for $\mu,\nu\in\Lambda$ and the
combinatorial structure $G$ of $\mathscr{E}_{G}(n,\lambda\in\Lambda)$, which
enables us to construct the labeled dimensional graph
$\widetilde{M}^{n}[G^{L}]$ following.

Definition $2.8$ Let $\mathscr{A}[M^{n}]=\\{\ (U_{\lambda};\varphi_{\lambda})\
|\ \lambda\in\Lambda\\}$ be an atlas of an orientable manifold $M^{n}$. A
labeled dimensional graph $\widetilde{M}^{n}[G^{L}]$ of a combinatorial
Euclidean space $\mathscr{E}_{G}(n,\lambda\in\Lambda)$ is such a
$n$-dimensional graph $\widetilde{M}^{n}[G]$ with labeling $L:\
(U_{\mu},U_{\nu})\rightarrow\kappa_{\mu\nu}+1$ for
$\forall(U_{\mu},U_{\nu})\in E(G)$, where $\kappa_{\mu\nu}$ is the number of
non-homotopic loops between $U_{\mu}$ and $U_{\nu}$ for $\mu,\nu\in\Lambda$.

Notice that such an edge labeled graph $G^{L}$ can be simply viewed as a graph
$H_{G}$ with multiple edges by defining.

$V(H_{G})=V(G^{L})$;

$E(H_{G})=\\{(U_{\mu},U_{\nu})_{i},\ 1\leq i\leq\kappa_{\mu\nu}+1\ |\ {\rm
for}\ \forall(U_{\mu},U_{\nu})\in E(G),\ \mu,\nu\in\Lambda\\}$.

Convention $2.9$ An edge labeled graph $G^{L}$ with an integer labeling
$L(e)\geq 1$ for $\forall e\in E(G)$ is viewed as a multiple graph $H_{G}$,
i.e., $G^{L}=H_{G}$ throughout this paper.

By this convention, such labeled graphs can be used essentially to
characterize non-homotopic loops of manifolds. For example, the torus is a
combinatorial Euclidean space $\mathscr{E}_{G}(2,\lambda=1,2)$ underlying a
dipole $D_{0,3,0}$, i.e., consists of two Euclidean space $U_{1},\
U_{2}\cong{\bf R}^{2}$ with $\kappa_{ij}=2$ non-homotopic loops and its
labeled graph on $P_{2}^{L}=D_{0,3,0}$, such as those shown in Fig.$2.2$.

Fig.$2.2$

Convention $2.9$ immediately implies the next result.

Theorem $2.10$ The number $\varpi(G^{L})$ of cycles basis of an edge labeled
graph $G^{L}$ with an integer labeling $L(e)\geq 1$ for $\forall e\in E(G)$ is

$\varpi(G^{L})=\varpi(G)+\sum\limits_{L(e)\geq 2,e\in E(G)}(L(e)-1).$

Whence, $\varpi(G^{L})=0$ if and only if $\varpi(G)=0$ and $L(e)=1$ for
$\forall e\in E(G)$.

Now denoted by $\mathscr{N}_{C}(\mathscr{E})$ and $\mathscr{N}_{C}(G^{L})$ the
sets of non-homotopic loops in the combinatorial Euclidean space
$\mathscr{E}_{G}(n,\lambda\in\Lambda)$ and cycles in its labeled dimensional
graph $\widetilde{M}^{n}[G^{L}]$, respectively. Then we know the following
interesting result.

Theorem $2.11$ There is a bijection $\vartheta:\
\mathscr{N}_{C}(\mathscr{E})\rightarrow\mathscr{N}_{C}(G^{L})$, i.e.,

$\pi(\mathscr{E}_{G}(n,\lambda\in\Lambda))\ \cong\
\pi(\widetilde{M}^{n}[G^{L}]).$

Proof We only need to prove that a loop $L\in\mathscr{N}_{C}(\mathscr{E})$ if
and only if there is a cycle $C_{L}\in\mathscr{N}_{C}(G^{L})$. The proof is
divided into two cases following.

Case 1. $L$ comes from the combinatorial structure $G$.

According to Theorems $2.1$ and $2.4$, we know that the underlying graphs of
$\mathscr{E}_{G}(n,\lambda\in\Lambda)$ and $\widetilde{M}^{n}[G^{L}]$ are
isomorphic. Whence there exists a cycle $C_{L}\in\mathscr{T}_{0}[G]$, i.e.,
$C_{L}\in\mathscr{N}_{C}(G^{L})$ correspondent to $L$ and verse via, for a
cycle $C\in\mathscr{T}_{0}[G]$, there also exists a loop
$L_{C}\in\mathscr{N}_{C}(\mathscr{E})$ for a cycle $C$. Whence, such a mapping
$\vartheta:\ L\rightarrow C_{L}$ is a bijection.

Case 2. $L$ comes from $U_{\mu}\cap U_{\nu}$ for two indexes
$\mu,\nu\in\Lambda$.

Assume there are $\kappa_{\mu\nu}$ non-homotopic loos in $U\cap U_{\nu}$. Not
loss of generality, let $L$ be the ith loop. By Definition $2.8$ and
convention $2.9$, we have a cycle $C_{L}$ consisted of multiple edges
$(U_{\mu},U_{\nu})_{i}$ with $(U_{\mu},U_{\nu})_{i+1}$ in the graph $G^{L}$.
Then $\vartheta:\ L\rightarrow C_{L}$ is a bijection by definition.

Combining the discussion of Cases $1$ and $2$, we get a bijection

$\vartheta:\ \mathscr{N}_{C}(\mathscr{E})\rightarrow\mathscr{N}_{C}(H_{G}).$

Notice that
$\pi(\mathscr{E}_{G}(n))=\left<\mathscr{N}_{C}(\mathscr{E})\right>$ and
$\pi(\widetilde{M}^{n}[G^{L}])=\pi(\widetilde{M}^{n}[H_{G}])=\left<\mathscr{N}_{C}(G^{L})\right>$.
The bijection $\vartheta:\
\mathscr{N}_{C}(\mathscr{E})\rightarrow\mathscr{N}_{C}(G^{L})$ naturally
induces an isomorphism $\vartheta*:\
\pi(\mathscr{E}_{G}(n))\rightarrow\pi(\widetilde{M}^{n}[G^{L}])$ by defining
$\vartheta*(L_{1}+L_{2})=\vartheta(L_{1})+\vartheta(L_{2})$ for $L_{1},\
L_{2}\in\mathscr{N}_{C}(\mathscr{E})$ in the field $\mathbb{Z}_{2}$. Hence, we
conclude that

$\hskip 142.26378pt\pi(\mathscr{E}_{G}(n))\ \cong\
\pi(\widetilde{M}^{n}[G^{L}]).\hskip 142.26378pt\Box$

Combining Theorems $2.1$, $2.4$ and $2.11$, we get an important result on
$n$-manifold with finite non-homotopic loops following.

Theorem $2.12$ For an $n$-manifold $M^{n}$ with finite non-homotopic loops,
there always exists a labeled $n$-dimensional graph $\widetilde{M}^{n}[G^{L}]$
such that

$\pi(M^{n})\cong\pi(\widetilde{M}^{n}[G^{L}]).$

Proof According to Theorem $2.1$, there is a combinatorial Euclidean space
$\mathscr{E}_{G}(n,\lambda\in\Lambda)$. Applying Theorems $2.4$ and $2.11$, we
get a labeled $n$-dimensional graph $\widetilde{M}^{n}[G^{L}]$ such that

$\pi(\mathscr{E}_{G}(n,\lambda\in\Lambda))\cong\pi(\widetilde{M}^{n}[G^{L}]),$

where $G^{L}$ is defined in Definition $2.8$. Whence, we know that

$\pi(M^{n})\cong\pi(\mathscr{E}_{G}(n,\lambda\in\Lambda))\cong\pi(\widetilde{M}^{n}[G^{L}]).$

This completes the proof. $\Box$

Applying Theorems $2.4$ and $2.12$, we get more useful conclusions following.

Corollary $2.13$ The number of non-homotopic loops in an $n$-manifold $M^{n}$
with finite non-homotopic loops is equal to the dimension of cycle space of
its $\mathscr{T}_{0}[H_{G[M^{n}]}]$.

Corollary $2.14$ For an integer $n\geq 2$, a compact $n$-manifold $M^{n}$ is
homotopy sphere if and only if $\widetilde{M}^{n}[G^{L}[M^{n}]]$ is a finite
$n$-dimensional tree with labeling $L:\ e\rightarrow 1$ for $\forall e\in
E(G^{L}[M^{n}])$.

Proof By definition, $M^{n}$ is compact if and only if $|G^{L}|$ is finite.
Applying Theorems $2.10$ and $2.12$, we know that $\pi(M^{n})$ is trivial if
and only if $\pi(\widetilde{M}^{n}[G^{L}])$ is trivial, i.e., each labeling of
edges is $1$. Now by Corollary $2.5$, there must be
$\pi(\mathscr{T}_{0}[H_{G}],v_{0})=\\{v_{0}\\}$ for
$v_{0}\in\mathscr{T}_{0}[H_{G}]\cap M^{n}$. But this can happens only if
$\mathscr{T}_{0}[H_{G}]$ is a finite tree by Theorem $2.6$. Whence,
$\widetilde{M}^{n}[G^{L}[M^{n}]]$ is an $n$-dimensional tree with a labeling
$L(e)=1$ for $\forall e\in E(G^{L}[M^{n}])$. $\Box$

Corollary $2.14$ enables us to obtain an unified proof for the generalized
Poincaré conjecture, i.e., any homotopy $n$-sphere is homeomorphic to $S^{n}$
following.

[Proof of Theorem $1.4$]

For any integer $n\geq 1$, by Corollary $2.14$ an $n$-manifold $M^{n}$ is
homotopy $n$-sphere if and only if $\widetilde{M}[G^{L}[M^{n}]]$ defined in
Definition $2.8$ is a finite labeled $n$-dimensional tree with $L(e)=1$ for
$\forall e\in E(G)$, i.e., a finite $n$-dimensional tree. Applying Theorem
$2.7$, such an $n$-dimensional tree $\widetilde{M}[G^{L}[M^{n}]]$ can
deformation retract to a point $v_{0}\in M^{n}$. Whence, $M^{n}$ is
homeomorphic to an $n$-sphere $S^{n}$. $\Box$

§$3.$ Listing $n$-Manifolds by Labeled Graphs

Theorem $3.1$ Let $\mathscr{A}[M^{n}]=\\{\ (U_{\lambda};\varphi_{\lambda})\ |\
\lambda\in\Lambda\\}$ be a atlas of a locally compact $n$-manifold $M^{n}$.
Then the dimensional graph $\widetilde{M}^{n}[G^{L}_{|\Lambda}|]$ of $M^{n}$
is a topological invariant on $|\Lambda|$, i.e., if
$\widetilde{M}^{n}[H^{L_{1}}_{|\Lambda|}]$ and
$\widetilde{M}^{n}[G^{L_{2}}_{|\Lambda|}]$ are two labeled $n$-dimensional
graphs of $M^{n}$, then there exists a self-homeomorphism $h:M^{n}\rightarrow
M^{n}$ such that $h:H^{L_{1}}_{|\Lambda|}\rightarrow G^{L_{2}}_{|\Lambda|}$
naturally induces an isomorphism of graph.

Proof Let

$\mathscr{A}^{1}_{|\Lambda|}[M^{n}]=\\{\ (U_{\lambda};\varphi_{\lambda})\ |\
\lambda\in\Lambda_{1}\\}$

and

$\mathscr{A}^{2}_{|\Lambda|}[M^{n}]=\\{\ (V_{\lambda};\phi_{\lambda})\ |\
\lambda\in\Lambda_{2}\\}$

be two minimum atlases of a locally compact $n$-manifold $M^{n}$ with labeled
graphs $H^{L_{1}}_{|\Lambda|}$ and $G^{L_{2}}_{|\Lambda|}$, respectively,
where $\Lambda_{1}=\Lambda_{2}=\\{1,2,3,\cdots,k,\cdots\\}$ are countable
index sets. Notice that $\varphi_{\lambda}:U_{\lambda}\rightarrow{\bf R}^{n}$
and $\phi_{\lambda}:V_{\iota}\rightarrow{\bf R}^{n}$ are homeomorphisms for
$\forall\lambda\in\Lambda_{1}$ and $\iota\in\Lambda_{2}$. So there is a
homeomorphism
$\tau_{\lambda}:\varphi_{\lambda}(U_{\lambda})\rightarrow\phi_{\lambda}(V_{\lambda})$
for $\lambda\in\Lambda_{1}=\Lambda_{2}$. Now define

$h_{\lambda}=\phi_{\lambda}^{-1}\tau_{\lambda}\varphi_{\lambda}:x\rightarrow\phi_{\lambda}^{-1}(\tau_{\lambda}(\varphi_{\lambda}(x)))\
\ {\rm for}\ \ x\in U_{\lambda}.$

Then $h_{\lambda}$ with its inverse
$h^{-1}=\varphi_{\lambda}^{-1}\tau_{\lambda}^{-1}\phi_{\lambda}$ is continuous
on $M^{n}$. By the compactness of $M^{n}$ there exists a partition of unity
$c_{\lambda}:U_{\lambda}\rightarrow{\bf R}^{n}$, $\lambda\in\Lambda$ on the
atlas $\mathscr{A}^{1}_{|\Lambda|}[M^{n}]$. Define a function
$h:M^{n}\rightarrow M^{n}$ by

$h=\sum\limits_{\lambda\in\Lambda}\phi_{\lambda}^{-1}\tau_{\lambda}\varphi_{\lambda}c_{\lambda},$

where $A_{\lambda}={\rm supp}(\varphi_{\lambda})$ and

$\phi_{\lambda}^{-1}\tau_{\lambda}\varphi_{\lambda}c_{\lambda}(x)=\left\\{\begin{array}[]{lr}c_{\lambda}(x)\phi_{\lambda}^{-1}\tau_{\lambda}\varphi_{\lambda}(x)&{\rm
if}\ x\in U_{\lambda},\\\ {\bf 0}=(0,\cdots,0)&{\rm if}\ x\in
U_{\lambda}-A_{\lambda}.\end{array}\right.$

Then $h:M^{n}\rightarrow M^{n}$ is a homeomorphism with
$h(U_{\lambda})=V_{\lambda}$ for $\lambda\in\Lambda_{1}=\Lambda_{2}$. Hence,
$h:\ V(H_{|\Lambda|}^{L})\rightarrow V(G^{L}_{|\Lambda|})$ is a bijection by
definition.

Now if there are $\kappa_{\mu\nu}$ non-homotopic loops between $U_{\mu}$ and
$U_{\nu}$, then there are must be $\kappa_{\mu\nu}$ non-homotopic loops
between $V_{\mu}$ and $V_{\nu}$ and vice via by the homeomorphic property.
Therefore, $L_{1}:(U_{\mu},U_{\nu})\rightarrow\kappa_{\mu\nu}+1$ in
$H^{L}_{|\Lambda|}$ if and only if
$L_{2}:(V_{\mu},V_{\nu})\rightarrow\kappa_{\mu\nu}+1$ in
$G^{L_{2}}_{|\Lambda|}$, i.e., $h(U_{\mu},U_{\nu})=(h(U_{\mu}),h(V_{\nu}))$
with $L_{1}(U_{\mu},U_{\nu})=L_{2}(h(U_{\mu}),h(U_{\nu}))$. By definition, two
labeled graphs $G^{L}_{1}$ and $G^{L}_{2}$ with labeling mappings $L_{1}$ and
$L_{2}$ are said to be isomorphic if there is an isomorphism $\varpi:\
G_{1}\rightarrow G_{2}$ with $\varpi L_{1}=L_{2}\varpi$. Whence, $h:\
H^{L}_{|\Lambda|}\rightarrow G^{L}_{|\Lambda|}$ naturally induces an
isomorphism between labeled graphs $H^{L_{1}}_{|\Lambda|}$ and
$G^{L_{2}}_{|\Lambda|}$. $\Box$

We get a conclusion by Theorem $3.1$ following.

Corollary $3.2$ The labeled graph $G^{L}_{|\Lambda|}$ of a locally compact
$n$-manifold $M^{n}$ is unique dependent on $|\Lambda|$.

For classifying $n$-manifolds by applying Theorem $3.1$, we introduce the
conception of minimum atlas following.

Definition $3.3$ An atlas

$\mathscr{A}[M^{n}]=\\{\ (U_{\lambda};\varphi_{\lambda})\ |\
\lambda\in\Lambda\\}$

of an $n$-manifold $M^{n}$ is minimal if there are no indexes $\mu,\
\nu\in\Lambda$ and a continuous mapping $\varphi_{\mu\nu}$ with
$\varphi_{\mu\nu}:\ U_{\mu}\cup U_{\nu}\rightarrow{\bf R}^{n}$ such that

$\mathscr{A}^{\prime}=\\{\ (U_{\lambda};\varphi_{\lambda}),(U_{\mu}\cup
U_{\nu},\varphi_{\mu\nu})\ |\ \lambda\in\Lambda\setminus\\{\mu,\nu\\}\\}$

is also an atlas of $M^{n}$.

An atlas $\mathscr{A}[M^{n}]$ of an $n$-manifold $M^{n}$ is minimum if it has
minimum cardinality among all of its minimal atlases. Denoted such a minimal
atlas by $\mathscr{A}_{min}[M^{n}]$ and its labeled $n$-dimensional graph by
$\widetilde{M}^{n}[G^{L}_{min}]$.

The next result characterizes minimal atlases of an $n$-manifold.

Theorem $3.4$ Let

$\mathscr{A}[M^{n}]=\\{\ (U_{\lambda};\varphi_{\lambda})\ |\
\lambda\in\Lambda\\}$

be an atlas of a locally compact $n$-manifold $M^{n}$. Then

($i$) $\mathscr{A}[M^{n}]$ is minimal if and only if there are no indexes
$\mu,\ \nu\in\Lambda$ such that $U_{\mu}\cap U_{\nu}$ is arcwise connected,
i.e., $L(e)\geq 2$ for $\forall e\in E(G^{L}_{min})$.

($ii$) $M^{n}$ is with finite non-homotopic loops if and only if
$\mathscr{A}_{min}[M^{n}]$ is finite.

Proof ($i$) If the result ($i$) is not true, then there exist indexes $\mu,\
\nu\in\Lambda$ such that $U_{\mu}\cap U_{\nu}$ is arcwise connected. Assume

$\varphi_{\mu}(U_{\mu}\cap U_{\nu})=S\subset{\bf R}^{n}\ \ {\rm and}\ \
\varphi_{\nu}(U_{\mu}\cap U_{\nu})=T\subset{\bf R}^{n}.$

Notice that $S$ and $T$ are homeomorphic to ${\bf R}^{n}$ by definition. We
can always choose a continuous mapping $\tau:\ S\rightarrow T$, i.e.,
$\tau(S)=T$. Define

$\varphi_{\mu}^{\tau}=\tau\varphi_{\mu}$.

Then we get that $\varphi_{\mu}^{\tau}|_{U_{\mu}\cap
U_{\nu}}=\varphi_{\nu}|_{U_{\mu}\cap U_{\nu}}$. Whence, there is a continuous
mapping $\varphi_{\mu\nu}:\ U_{\mu}\cup U_{\nu}\rightarrow{\bf R}^{n}$.
Therefore,

$\mathscr{A}^{\prime}=\\{\ (U_{\mu}\cup
U_{\nu};\varphi_{\mu\nu}),(U_{\lambda};\varphi_{\lambda})\ |\
\lambda\in\Lambda\setminus\\{\mu,\nu\\}\\}$

is also an atlas of $M^{n}$ but with $|\mathscr{A}^{\prime}|=|\Lambda|-1$.
This contradicts to the minimality of $\Lambda$. So ($i$) holds.

($ii$) If $\mathscr{A}_{min}[M^{n}]$ is finite, then $M^{n}$ is obvious only
with finite non-homotopic loops by the assumption of its locally compactness.
Now let

$\mathscr{A}_{min}[M^{n}]=\\{\ (U_{\lambda};\varphi_{\lambda})\ |\
\lambda\in\Lambda\\}$

be a minimum atlas of a locally compact manifold $M^{n}$ with an infinite
index set $\Lambda$. By ($i$), there are no indexes $\mu,\nu$ in $\Lambda$
such that $U_{\mu}\cap U_{\nu}$ is arcwise connected. In other words,
$U_{\mu}\cap U_{\nu}=\emptyset$ or with more than $2$ arcwise components for
$\forall\mu,\nu\in\Lambda$, i.e., $L(e)\geq 2$ for $\forall e\in
E(G^{L}_{min})[M^{n}]$. Applying Theorem $2.10$, we know that the number

$\varpi(G^{L}_{min}[M^{n}])=\varpi(G_{min}[M^{n}])+\sum\limits_{e\in
E(G_{min}[M^{n}])}(L(e)-1)$

of cycle basis of $G^{L}_{min}[M^{n}]$ is greater than any sufficient larger
number $N>0$, which contradicts the assumption that $M^{n}$ is only with
finite non-homotopic loops. $\Box$

Combining Corollary $3.2$ with that of Theorem $3.4$, we get a conclusion
following, which enables us to list locally compact $n$-manifolds with finite
non-homotopic loops by labeled graphs.

Corollary $3.5$ If the minimum labeled graph $G^{L}_{min}[M^{n}_{1}]$ of a
locally compact $n$-manifold $M^{n}_{1}$ is not isomorphic to the minimum
labeled graph $G^{L}_{min}[M^{n}_{2}]$ of $M_{2}^{n}$, then $M^{n}_{1}$ is not
homeomorphic to $M^{n}_{2}$.

Now by Theorem $3.4$, let

$\mathscr{A}_{min}[M^{n}]=\\{\ (U_{\lambda};\varphi_{\lambda})\ |\
\lambda\in\Lambda,|\Lambda|<+\infty\\}$

be a minimum atlas of locally compact $n$-manifolds $M^{n}$. Then we can list
$n$-manifolds following by Corollary $3.5$.

(1) $|\Lambda|=1$

In this case, $\mathscr{A}_{min}[M^{n}]=\\{(U;\varphi)\\}$, i.e., $M^{n}={\bf
R}^{n}$.

(2) $|\Lambda|=2$

In this case,

$\mathscr{A}_{min}[M^{n}]=\\{(U_{1};\varphi_{1}),\ (U_{2};\varphi_{2})\\}$

and $M^{n}$ is double covered, which can be classified by labeled graphs
$D^{L}_{0,1,0}$ shown in Fig.$3.1$,

Fig.$3.1$

For example, if $n=2$, i.e., compact $2$-manifolds ${\bf S}$, then
$\kappa_{12}$ is the genus of ${\bf S}$ with $\kappa_{12}\geq 1$.

(3) $|\Lambda|=3$

In this case,

$\mathscr{A}_{min}[M^{n}]=\\{(U_{1};\varphi_{1}),\ (U_{2};\varphi_{2}),\
(U_{3};\varphi_{3})\\}$

and we can list such $n$-manifolds $M^{n}$ by labeled graphs in Fig.$3.2$,
where integers $k,l,s\geq 2$.

Fig.$3.2$

(4) $|\Lambda|=4$

In this case,

$\mathscr{A}_{min}[M^{n}]=\\{(U_{1};\varphi_{1}),\ (U_{2};\varphi_{2}),\
(U_{3};\varphi_{3}),\ (U_{4};\varphi_{4})\\}$

and we can list such $n$-manifolds $M^{n}$ by labeled graphs in Fig.$3.3$,
where integers $k,l,r,q,s,t\geq 2$.

Fig.$3.3$

$\cdots\cdots\cdots\cdots$

(p) $|\Lambda|=p$

In this case,

$\mathscr{A}_{min}[M^{n}]=\\{(U_{i};\varphi_{i}),1\leq i\leq p\\}.$

and there are $G_{1},G_{2},\cdots,G_{k(p)}$ such non-isomorphic graphs known
in graph theory, where

$k(p)\sim\frac{2^{\displaystyle\frac{p(p-1)}{2}}}{n!}$

is the number of non-isomorphic graphs of order $p$ with

$p$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$
---|---|---|---|---|---|---|---|---|---
$k(p)$ | $1$ | $1$ | $2$ | $6$ | $21$ | $112$ | $853$ | $11117$ | $261080$

for $p\leq 9$. Then we can list such $n$-manifolds $M^{n}$ by labeled graphs
following:

$G^{L}_{1},G^{L}_{2},\cdots,G^{L}_{k(p)},\ \ \ L_{i}(e_{ij})\geq 2\ \ {\rm
for}\ \ \forall e_{i}\in E(G_{i}),1\leq i\leq k(p).$

References

[1] R.Abraham and J.E.Marsden, Foundation of Mechanics(2nd edition), Addison-
Wesley, Reading, Mass, 1978.

[2] M.H.Freedman, The topology of four-dimensional manifolds, J.Diff.Geom., 17
(1982), 357-453.

[3] S.Gadgil and H.Seshadri, Ricci flow and Perelman’s proof of the Poincaré
conjecture, Current Science, Vol.91, No.10, 25 Nov. 2006, 1326-1334.

[5] J.L.Gross and T.W.Tucker, Topological Graph Theory, John Wiley & Sons,
1987.

[6] L.F.Mao, Automorphism Groups of Maps, Surfaces and Smarandache Geometries,
American Research Press, 2005.

[7] L.F.Mao, Smarandache Multi-Space Theory, Hexis. Phoenix, USA 2006.

[8] L.F.Mao, Geometrical theory on combinatorial manifolds, JP J.Geometry and
Topology, Vol.7, No.1(2007),65-114.

[9] L.F.Mao, Combinatorial speculation and combinatorial conjecture for
mathematics, International J.Math. Combin., Vol.1(2007), 1-19.

[10] L.F.Mao, Combinatorial fields - an introduction, International J.Math.
Combin. Vol.3 (2009), 01-22.

[11] L.F.Mao, Combinatorial Geometry with Applications to Field Theory,
InfoQuest, USA, 2009.

[12] L.F.Mao, A combinatorial decomposition of Euclidean space ${\bf R}^{n}$
with contribution to visibility, International J.Math.Combin., Vol.1(2009).

[13] W.S.Massey, Algebraic Topology: An Introduction, Springer-Verlag, New
York, etc.(1977).

[14] J.Milnor, Towards the Poincaré conjecture and the classification of
$3$-manifolds, Notices AMS, 50(2003), 1226-1233.

[15] G.Perelman, The entropy formula for the Ricci flow and its geometric
applications, arXiv: math.DG/0211159v1, 11 Nov. 2002.

[16] G.Perelman, Ricci flow with surgery on 3-manifolds, arXiv:
math.DG/0303109, 10 Mar 2003.

[17] G.Perelman, Finite extinction time for the solutions to the Ricci flow on
certain three-manifolds, arXiv: math.DG/0307245, 17 Jul. 2003.

[18] S.Smale, Generalized Poincaré conjecture in dimensions greater than four,
Annals of Math., 74(1961), 391-406.

[19] W.P.Thurston, Three-Dimensional Geometry and Topology, Vol.1, ed. by
Silvio Levy, Princeton Mathematical Series 35, Princeton University Press,
1997.


# Do not think about pink elephant!

Kyomin Hwang Equally contributing authors. Suyoung Kim11footnotemark: 1 Junhoo
Lee11footnotemark: 1 Nojun Kwak Corresponding author. Seoul National
University
{kyomin98, ksyo96, mrjunoo<EMAIL_ADDRESS>
###### Abstract

Large Models (LMs) have heightened expectations for the potential of general
AI as they are akin to human intelligence. This paper shows that recent large
models such as Stable Diffusion and DALL-E3 also share the vulnerability of
human intelligence, namely the “white bear phenomenon”. We investigate the
causes of the white bear phenomenon by analyzing their representation space.
Based on this analysis, we propose a simple prompt-based attack method, which
generates figures prohibited by the LM provider’s policy. To counter these
attacks, we introduce prompt-based defense strategies inspired by cognitive
therapy techniques, successfully mitigating attacks by up to 48.22%.

##  1 Introduction

Large Models (LMs) have dramatically advanced artificial intelligence,
elevating it to, or beyond, the level of human cognitive abilities in some
areas. For instance, [1, 5, 14, 19] sparked hopes for the achievement of
Artificial General Intelligence (AGI). However, does this cognitive similarity
to humans guarantee success? Human cognition has weaknesses: ironic mental
processes [17], also known as the “white bear phenomenon” or “pink elephant
paradox” is notable. It occurs when trying not to think about something
results in thinking about it more. For example, attempting not to think about
a pink elephant inevitably brings a pink elephant to mind. This happens
because avoiding a concept requires recognizing it; in that recognition, we
inadvertently focus more cognitive effort on it.

Interestingly, as shown in Fig. 1, we found that the white bear phenomenon
also exists in LMs such as visual models like DALL-E3 [2] and Stable Diffusion
[12]. This paper analyzes this phenomenon, exploring how it emerges and how to
suppress it. We discovered that this effect arises in large models because
they struggle to learn negative concepts due to the architectural
characteristics and the ‘well-trained’ representation space of these models.

Figure 1: Example images that illustrate the “white bear phenomenon.” The
image generation model does not understand the negative word. Images are
generated by using DALL-E3.

We have identified that this effect could be exploited for effective prompt
attacks. Negative prompt for unwanted objects might seem acceptable to human
users, but in fact, this prompt becomes a strong prior in the model that
encourages the generation of those objects. For example, in Fig. 1, suppose we
are given a prompt that seems to suppress unwanted features like elephants,
however, suppressing the object itself can significantly heighten the
probability of its creation. Furthermore, we propose defense strategies
inspired by cognitive science to counteract this phenomenon.

More specifically, we point out that attention-based architectures [15, 12],
which recent LMs embody, induce this phenomenon due to its inherent inability
to correctly process the ‘negation’ operation; our research shows that this
effect is due to the linearity in a well-trained model’s representation space,
where concept of absence plays a minor role. This is demonstrated measuring
similarity between the presence and absence in representation space. We also
design a prompt attack exploiting the white bear phenomenon and introduce a
dataset that quantitatively measures the white bear phenomenon in large
models. The efficacy of this attack is tested in the Stable Diffusion [12].
Also, we report that this scenario is also effective in LLM-augmented models
such as DALL-E3 and ideogram [2, 10]. As defense strategies, we devise methods
intuited by cognitive science to counteract the aforementioned problem. These
include specifying existing concepts using a definition of given word or
employing alternative concepts instead of the attacked concept (white bear).

##  2 Preliminaries and Related works

Figure 2: Histogram of the cosine similarity between two distinct CLS tokens
obtained through the CLIP text encoder, with measurements presented in units
of 1e-3. ‘$w^{hyp}$’ is the hypernym of ‘$w$’ and ‘$w^{syn}$’ is a synonym to
‘$w$’.

#### Generative Models

The field of image generation [2, 12, 13, 18] has seen a substantial
improvement in model performance, highlighted by groundbreaking advancements
that have significantly expanded the capabilities and applications of
generative models. Large generative models, such as Stable Diffusion and
DALL-E3, are built on an attention-based structure. These large models
generally possess good representation, indicating that the representation
space they have learned exhibits linearity. Based on these characteristics, we
analyze the inability of large models to understand the concept of negativity.

#### Responsible AI

With the enhanced performance of image generation models, there has been a
corresponding increase in research focused on preventing their misuse and
ensuring ethical utilization. This line of work ranges from developing content
moderation tools to integrating ethical guidelines within AI systems.
Noteworthy contributions in this area include [3, 4, 7, 9], which foster a
responsible approach to AI development and application. In this paper, we warn
about the weak aspect of text-guided generation models [2, 12] by
demonstrating the novel attack mechanism that generates images against the LM
provider’s guideline (_e.g_. porn images or harmful images) through specially
crafted prompts. Furthermore, we propose a novel defense mechanism that
utilizes prompt engineering to mitigate such attacks, ensuring the models’
integrity and ethical use. This approach not only addresses the immediate
concerns of security and misuse but also contributes to the broader discourse
on the responsible development and deployment of generative AI technologies.

#### Notation

Before presenting our analysis, we clarify the notations to be used in the
subsequent section. Specifically, $w_{abs}$ and $w_{con}$ denote an abstract
word and a concrete word, respectively. Here, a ‘concrete word’ is defined as
the one that describes a specific object perceivable by the senses, such as
‘apple’ or ‘house.’ In contrast, an ‘abstract word’ refers to the one that
conveys a characteristic or concept, like ‘sadness’ or ‘joy’. $w^{hyp}$,
$w^{syn}$, and $w^{def}$ denote hypernym, synonym and definition,
respectively. $\sim$ denotes a word carrying the meaning of negation such as
‘not’.

##  3 Analysis on white bear phenomenon in LMs

Figure 3: Visualization by t-SNE of CLIP embeddings for various sentences.
Table 1: The cosine similarity between two CLS token vectors extracted through
the text encoder of CLIP. S(x, y) denotes the cosine similarity between
vectors $x$ and $y$. The mean and standard deviation are presented in units of
1e-3.

Distance(cosine similarity) | Mean | Std
---|---|---
S( f ($w_{abs}$ not $w_{con}$), f ($w_{abs}$) $-$ f ($w_{con}$) ) | 0.273 | 0.247
S( f ($w_{abs}$ not $w_{con}$), f ($w_{abs}$) $+$ f ($w_{con}$) ) | 0.792 | 0.176
S( f ($w_{abs}$ not $w_{con}$), f ($w_{abs}$) $+$ f (not) $+$ f ($w_{con}$) ) | 0.859 | 0.121

Figure 4: The images provided illustrate the application of our proposed
attack and defense strategies on the Stable Diffusion model. The image on the
left demonstrates the outcome of an attack using the prompt “draw $w_{abs}$
without $w_{con}$” while the those on the right show the results of employing
defense strategies with the prompts “draw $w_{abs}$, which is $w_{abs}^{def}$,
without $w_{con}$” and “draw $w_{abs}$, include $w_{con}^{1}$, instead of
$w_{con}^{2}$”, respectively.

Understanding the distinction between the ‘absence’ and ‘presence’ of concepts
is pivotal in grasping the “white bear phenomenon,” where trying not to think
of something results in it dominating our thoughts. This phenomenon raises an
intriguing question regarding large models: Do they also struggle to
differentiate between the absence and presence of objects or concepts? Fig. 2
suggests that they do. If they understand logical algebra, the hypernym
$w^{hyp}$ should be closer to $w$ and $\sim\sim w$ should be the same as $w$.
However, it is not. Furthermore, it shows $\sim w$ is much closer to $w$ than
its synonym ${w}^{syn}$. Additionally, Fig. 3 demonstrates that CLIP
embeddings are unable to comprehend negative concepts in language. In the CLIP
encoder, the representation of an object’s absence is similar to its presence.
Indicating that, like humans, LMs blur the lines between absence and presence.

#### Algebra to represent absence

To understand this phenomenon, it is essential to examine which operation is
needed to model the ‘absence’ of a concept. Intuitively, representing
‘absence’ involves removing ‘presence’. This can be achieved through two
primary operations: subtraction and projection. Subtraction simply means
subtracting an object’s information from its overall representation. For
instance, subtracting ‘apple’ from ‘there is an apple’ intuitively models the
absence of the apple. Projection, on the other hand, involves excluding
specific attributes from the representation space, akin to ignoring certain
data by projecting the original data into the orthogonal space to the
attribute we want to suppress.

#### Large model cannot represent absence

But why is this challenging for large models? The crux of the issue lies in
the models’ reliance on linearity for representation [16]. The manifold
hypothesis posits that data, despite appearing complex in high dimensions, can
be simplified in a low-dimensional manifold, benefiting from linear
classification and smooth style transformations via linear interpolation.
However, this linearity is insufficient for representing absence. When
attempting to represent “not A,” the aim is to eliminate any association with
A from the embedding. However, simply adding the token “not” does not
eliminate the concept from A in the embedding space, resulting in a new
embedding A alongside the novel information of the word “not”. Additionally,
the attention-based architecture which aggregates token information via
weighted averages lacks the capability for direct subtraction or projection
between tokens, further complicating the accurate representation of absence
(See Table 1).

#### Absence in LLM-Augmented-Models

This leads to the question: How effective is this logic in environments where
prompts are crafted by LLMs, such as Dall-E, compared to models like stable
diffusion, which use directly inserted prompts? Although LLMs like GPT-4
adjust prompts before submitting them to models like DALL-E3, the challenge of
accurately capturing the concept of absence persists, as demonstrated in Fig.
5, indicating that despite modification of prompts by LLMs, the fundamental
issue of representing absence remains. We provide a detailed analysis of LLM-
Augmented Models in the Appendix B.

##  4 Experiments

Table 2: Experiment results of the success rate of defending prompt-based
attacks.

input prompt template | success rate(%)
---|---
draw $w_{abs}$ without $w_{con}$ | 24.46
draw $w_{abs}$, which is $w_{abs}^{def}$, without $w_{con}$ | 34.93
draw $w_{abs}$, include $w_{con}^{1}$, instead of $w_{con}^{2}$ | 48.22

In the analysis presented in Section 3, we demonstrated that LMs inherently
cannot grasp the concept of negation due to their structural characteristics
and the linearity of representation space. Building on this insight, we
propose: 1) the potential for prompt-based attacks and 2) experimental defense
strategies to counteract these attacks. In our experiments, we utilized two
versions of LM: DALL-E3, which alters the user’s input prompt before
processing, and Stable Diffusion, which processes the user’s input as
provided.

#### Datasets

Due to the lack of a benchmark dataset for validating our claims, we
constructed a dataset comprising abstract words ($w_{abs}$), concrete words
($w_{con}$), and their definitions ($w^{def}$) using ChatGPT and the Oxford
Dictionary [11]. A detailed description of this dataset is provided in the
Appendix A.

#### Attack strategy

To explore the viability of leveraging LMs’ inability to comprehend negation
for prompt-based attacks, we crafted prompts in the format of ‘draw $w_{abs}$
without $w_{con}$’ and submitted them to Stable Diffusion. We manually
assessed whether the output images included the specified concrete word. The
presence of the concrete word in the generated image indicates a successful
attack. Table 2 shows the success rates of defending these attacks, determined
by the aforementioned criteria.

The findings reveal that 75.54% of attacks are successful for Stable
Diffusion. This phenomenon stems from the model’s structural inability to
negate the vector for the concrete word and its innate linear representation
space. More realistic attack examples can be found in Appendix Fig. 6.

Figure 5: The results of utilizing our proposed attack method on the DALL-E3,
which uses LLM augmented prompts. Despite rephrasing the user input prompt
through the LLM, it is observed that the model still generates a white bear.

#### Defence strategy

To address this phenomenon, we drew inspiration from cognitive behavioral
therapy and proposed two prompt-based defense strategies: 1) the first
strategy involves incorporating the definition of the abstract word (‘draw
$w_{abs}$, which is $w_{abs}^{def}$, without $w_{con}$’), and 2) the second
strategy requires the inclusion of alternative concrete words as a means of
defense (‘draw $w_{abs}$, include $w_{con}^{1}$, instead of $w_{con}^{2}$’).
The first strategy is inspired by the philosophy of mindfulness [6], which
aims to eliminate negative thoughts by concentrating on the sensory aspects of
the present. The second strategy draws inspiration from attention diversion
techniques [8], which aim to replace negative thoughts with positive or
neutral ones.
As shown in Table 2, the first method resulted in a 10.25%p improvement over
the baseline (‘draw $w_{abs}$ without $w_{con}$’), while the second method
yielded a 23.54%p enhancement. These findings confirm that our proposed
methods can effectively defend against prompt-based attacks through carefully
crafted input prompts, eliminating the need for additional training.

##  5 Conclusion

In this paper, we discuss for the first time the phenomenon of the “white bear
phenomenon” in large models, exploring its origin and proposing the hypothesis
that the issue stems from the nature of the way representations are made in
LMs. We demonstrate how this white bear phenomenon can act as a prompt attack,
generating images that contradict the policies of generation models. To
mitigate this, we drew inspiration from cognitive therapy methodologies aimed
at addressing the white bear phenomenon, presenting methods that can alleviate
its impact. However, we acknowledge the limitations of our approach, noting
that it does not fundamentally solve the problem. A deeper elucidation of why
the white bear phenomenon occurs along with the absence of an architecture
capable of learning about absence is required for a more foundational
resolution.

## Acknowledgement

This work was supported by NRF grant (2021R1A2C3006659) and IITP grants
(2022-0-00320, 2021-0-01343), all funded by MSIT of the Korean Government.

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Supplementary Material

Table 3: Examples of generated datasets from ChatGPT4 and Oxford Dictionary. $w_{abs}$ | $w_{abs}^{def}$ | $w_{con}$
---|---|---
despair | The complete loss or absence of hope. | cat
wisdom | The quality of having experience, knowledge, and good judgment; the quality of being wise. | shoe
loneliness | Sadness because one has no friends or company. | turtle
inspiration | The process of being mentally stimulated to do or feel something, especially to do something creative. | lion
thrill | An intense feeling of excitement or happiness. | bear

##  Appendix A Implementation Details

###  A.1 Models

In this paper, we utilized DALL-E3 and Stable Diffusion. DALL-E3 was accessed
through the official ChatGPT site provided by OpenAI, while Stable Diffusion
was employed using Stable Diffusion v2.1 available on Hugging Face.

###  A.2 Datasets

Due to the lack of an open-source benchmark dataset to test our hypothesis, we
generated suitable data using ChatGPT4 and the Oxford Dictionary. Abstract
words and concrete words were extracted through ChatGPT4, and definitions of
abstract words were obtained from the Oxford Dictionary. Table 3 showcases
examples from the dataset we created.

##  Appendix B Case of LLM-Augmented Prompts

Figure 7 illustrates the outcomes observed with DALL-E3, the model that
augments prompts with LLM capabilities. Here, we can use a sequence of prompts
continuously in the attached session. We guided DALL-E3 to draw ’without’ a
pink elephant in sequence. The concept of a pink elephant (an image that
should be excluded) becomes exaggerated when DALL-E3 is guided to exclude pink
elephants. We present additional examples in Figure 7. These examples
highlight the limitation of relying solely on prompt editing for effectively
excluding specific tasks.

Figure 6: Examples of realistic attacks generated through ideogram. Through
the prompt-based attack, images not desired by the user (e.g., copyrighted
characters, specific brands, or inappropriate images) can be generated. Images
are generated through Ideogram1.0 [10]. We did not use magicprompt(LLM-guided
prompt augmentation method that ideogram provides.) Figure 7: Example of a
prompt-based attack using a series of successive prompts within an attached
session.