# Does the system entanglement care about the readout efficiency of quantum
measurement?

Christian Carisch  Institute for Theoretical Physics, ETH Zürich, CH-8093
Zürich, Switzerland.  Oded Zilberberg  Department of Physics, University of
Konstanz, 78464 Konstanz, Germany.  Alessandro Romito  Department of Physics,
Lancaster University, Lancaster LA1 4YB, United Kingdom.

(September 1, 2024)

###### Abstract

Monitored quantum systems evolve along stochastic trajectories correlated with
the observer’s knowledge of the system’s state. Under such dynamics, certain
quantum resources like entanglement may depend on the observer’s state of
knowledge. Here, we quantify the entanglement for a particle on a 1d quantum
random walk under inefficient monitoring using a mixed state-entanglement
measure – the configuration coherence. We find that the system’s maximal mean
entanglement at the measurement-induced quantum-to-classical crossover is
suppressed in different ways by the measurement strength and inefficiency. In
principle, strong measurements can lower the amount of entanglement
indefinitely. However, at a given measurement strength, efficient readout can
crucially increase the system entanglement, making high-fidelity detectors
essential for successful quantum computing.

††preprint: APS/123-QED

In today’s NISQ era of quantum computing [1], quantum measurement has become
of crucial practical importance, as it is imperative to harness the potential
powers of quantum devices and it can be used to steer quantum systems into
states with desirable properties [2, 3, 4]. Quantum measurements involve a
detector performing the measurement and an observer reading out the
corresponding measurement outcome. Due to the coupling of the detector with
the system, this process is accompanied by an inherent back-action onto the
system [5, 6]. Subsequently, the observer infers the system’s state from the
measurement record. Hence, a quantum state reflects the observer’s knowledge
of the system. The amount of the observer’s information gained on the system’s
state is determined by the efficiency of the measurement process [2, 7]. For
example, efficient measurements do not degrade the state-of-knowledge of the
observer while incomplete information about the measurement outcome for
inefficient measurements will lead to mixed states of (classical)
probabilities associated with possible post-measurement quantum states. The
average over such realizations gives the average system state, which is
independent of the readout efficiency and evolves according to a continuous
master equation.

From the perspective of quantum processing, certain computational resources
associated with a system’s state, like entanglement, depend non-linearly on
the system state [8]. Therefore, the entanglement depends on the readout
efficiency and, consequently, on the observer’s available information. One
prominent manifestation of such effects is the measurement-induced
entanglement phase transition, where a quantum system under fully efficient
measurements undergoes a transition from a volume (critical) law to an area
law (Zeno) phase with increasing measurement strength [9, 10, 11, 12, 13].
Such measurement-induced transition has been observed in some pioneering
experiments [14, 15, 16]. Strikingly, this transition is strictly absent for
the average state, which develops into a complete statistical mixture for any
finite measurement strength. This discrepancy has been analyzed in
measurement-induced transitions at finite inefficiency in specific models [17,
18] and quantified using mixed-state entanglement, where it was found that, at
intermediate times, both the fully efficient and fully inefficient
descriptions feature a measurement-strength dependent coherence length [19].
Apart from these two extreme limits, the dependence of the state’s
entanglement on the efficiency of the quantum measurement remains to be
explored.

In this Letter, we quantify how the measurement and the observer’s knowledge
of its outcome impact the system entanglement. We consider inefficient local
density measurements described by a stochastic master equation [7]. The
measurement tends to localize the system, whereas inefficient readout leads to
mixed state quantum trajectories. We analyze these effects using mixed state
entanglement, which we measure by the recently developed configuration
coherence [20, 21]. First, we find that for a weakly measured particle
coherently oscillating between two sites, the stationary entanglement solely
depends on the readout efficiency. At high measurement strengths, localization
drastically suppresses entanglement. Second, we consider the measurement-
induced quantum-to-classical crossover of a particle performing a quantum
random walk on a chain. The system’s mean entanglement assumes a maximal value
at this crossover. We find that this maximal mean entanglement depends on both
the measurement strength as well as the readout efficiency, albeit in
quantitatively different ways. Our results showcase that the entanglement of a
monitored system depends non-trivially on the observer’s knowledge of the
measurement outcomes, an observation with far reaching consequences on how
entanglement can be understood and harnessed as a computational resource in
present and future quantum devices.

We consider a spinless particle hopping on a chain subject to local
inefficient density measurements, see Fig. 1(a). The particle’s free evolution
follows the Hamiltonian

$H\leavevmode\nobreak\
=\frac{J}{2}\sum_{i=1}^{L-1}a_{i}^{\dagger}a_{i+1}^{\phantom{\dagger}}+h.c.\,,$
(1)

with $J$ the hopping amplitude, $a_{i}^{\dagger}$ ($a_{i}$) the creation
(annihilation) operators, and $L$ the chain’s length. For convenience, we set
$J=1$. We now couple the sites to inefficient detectors that measure the local
densities $\langle n_{i}\rangle=\text{Tr}[\rho a_{i}^{\dagger}a_{i}],\
i=1,\dots,L$, with strength $k\geq 0$. The detector efficiency $0\leq\eta\leq
1$ describes the amount of measurement outcome information available to the
observer. With this, the particle’s dynamics is described by the stochastic
master equation (SME) [7]

$\displaystyle d\rho=-i[H,\rho]dt$
$\displaystyle+\sum_{i=1}^{L}k\left(n_{i}\rho
n_{i}-\frac{1}{2}\\{n_{i},\rho\\}\right)dt$ $\displaystyle+\sqrt{\eta
k}\left(n_{i}\rho+\rho n_{i}-2\langle n_{i}\rangle\rho\right)dW^{i}_{t}\,,$
(2)

where the $dW^{i}_{t}$ are independent Wiener increments with
$\overline{dW^{i}_{t}dW^{j}_{t^{\prime}}}=\delta_{i,j}\delta_{t,t^{\prime}}dt$.
The second term on the right-hand side in Eq. (Does the system entanglement
care about the readout efficiency of quantum measurement?) describes the
measurement’s backaction on the system, which only depends on the measurement
strength. On the other hand, the third term on the right hand side in the SME
characterizes the change of the state due the observer’s information gain from
the measurement readout.

Due to the randomness underlying the measurement, this term is stochastic and
depends on both the measurement strength and the readout efficiency $\eta$.
The evolution of the state $\rho_{m}$ under a single sequence of measurement
outcomes (a single realization of increments $dW^{i}_{t}$) is called a quantum
trajectory. Due to $\overline{dW_{t}^{i}}=0$, the ensemble state
$\overline{\rho}\equiv(\sum_{m=1}^{M}\rho_{m})/M$ averaged over many
trajectories, $M\gg 1$, is independent of the individual measurement outcomes,
and evolves according to the standard (continuous) Lindblad master equation
[22, 2]. Therefore, the average state is independent of the efficiency $\eta$.
An alternative interpretation of Eq. (Does the system entanglement care about
the readout efficiency of quantum measurement?) involves two density
measurements per site, one with a perfectly efficient detector ($\eta_{1}=1$)
and strength $k_{1}$, and a second with an inefficient detector ($\eta_{2}=0$)
and strength $k_{2}$. This two-measurement picture relates to the inefficient
single measurement by $k=k_{1}+k_{2}$, $\eta=k_{1}/(k_{1}+k_{2})$. Moreover,
the inefficient measurement with strength $k_{2}$ is equivalent to capacitive
coupling to a dephasing environment.

Figure 1:  (a) Setup: spinless particle hopping on a chain of length $L$ with
amplitude $J$. The particle is measured with strength $k$ and efficiency
$\eta$ at each site, cf. Eq. (Does the system entanglement care about the
readout efficiency of quantum measurement?). (b-e) Probability densities, $p$,
of the stationary state of trajectories for $L=2$ [see rectangle in (a)]. The
Bloch sphere poles correspond to the localized states $|1\rangle$ (north) and
$|2\rangle$ (south). (b) For high efficiency, $\eta=0.99$, and small
measurement strength, $k=0.01$, the trajectories remain almost pure and can
cover the full surface of the Bloch sphere. (c) For $\eta=0.99$ and high
measurement strength, $k=10$, the particle gets projected onto the poles of
the Bloch sphere (Zeno effect). (d) For $\eta=k=0.01$, the trajectories are
highly mixed but distributed symmetrically around the fully mixed state
$\mathds{1}/2$. (e) For $\eta=0.01$ and $k=10$, the trajectories get mixed and
at the same time projected on the $z$-axis due to the measurement.

To understand the simultaneous effects of measurement and readout efficiency,
we first consider the simple example of two sites, see rectangle selection in
Fig. 1(a). Without measurement, the particle performs Rabi oscillations with
frequency $J=1$ between the two sites. In general, the measurement tends to
localize individual trajectories to one of the two sites (quantum Zeno effect
[23]), whereas inefficient readout will lead to an increasingly mixed state
due to the observer’s lack of information on the particle’s exact location. To
illustrate, we employ a Bloch sphere picture, where the $z$-poles correspond
to the localized states $|1\rangle=a_{1}^{\dagger}|0\rangle$ and
$|2\rangle=a_{2}^{\dagger}|0\rangle$. For a localized initial state, the
$x$-component of the Bloch vector vanishes at all times, and the sphere can be
reduced to a disk [24].

At high efficiencies, the trajectories following (Does the system entanglement
care about the readout efficiency of quantum measurement?) remain almost pure,
i.e., on the surface of the Bloch sphere, see Figs. 1(b,c). This is a
consequence of the wavefunction collapse due to the efficient readout: the
observer knows which is the specific pure state that corresponds to the
measurement outcome with high fidelity. At small measurement strengths, the
Rabi oscillations persist, and at long times the trajectories reach any state
on the surface of the Bloch sphere with approximately equal probability [Fig.
1(b)]. Conversely, strong measurements localize the particle on one of the two
sites, and the probability distribution at long times is strongly focussed at
the poles [cf. Fig. 1(c)]. For small efficiencies, the restricted amount of
the observer’s knowledge on the measurement outcomes leads to a mixed state
trajectory, see Figs. 1(d,e). In combination with a small measurement
strength, the trajectories end up close to and symmetrically distributed
around the maximally mixed state $\mathds{1}/2=(\sum_{i=1}^{2}|i\rangle\langle
i|)/2$ located at the center of the Bloch sphere [cf. Fig. 1(d)]. At high
measurement strengths, the individual trajectories’ mixing is suppressed by
Zeno localization, and the probability distribution skews towards the poles
[cf. Fig. 1(e)]. To quantize these effects, we are interested in the
entanglement between the two sites. Interestingly, as we shall see, both the
localization and the mixing tend to decrease the amount of entanglement in the
system. Next, we move to discuss how entanglement scales with the measurement
strength and the readout efficiency, with consequences on the interpretation
of entanglement; we find that it is a resource depending on the amount of
information available to the observer rather than a system-inherent quantity.

Figure 2:  (a,b) Time evolution of the configuration coherence for different
measurement strengths $k$ and efficiencies $\eta=1,\,0.5\,,0$ (top to bottom).
(a) For small measurement strength $k=0.1$, the configuration coherence
strongly oscillates at first and then saturates at a finite value (for
$\eta>0$, marker \raisebox{-.9pt} {1}⃝) or vanishes (for $\eta=0$, marker
\raisebox{-.9pt} {2}⃝). (b) For high measurement strength $k=2$, the
entanglement approaches the stationary value in damped way. For efficiencies
$\eta=0,\,0.5$, there is a maximum at intermediate times (markers
\raisebox{-.9pt} {3}⃝, \raisebox{-.9pt} {4}⃝). (c) Stationary state
configuration coherence as a function of measurement strength and efficiency.
Small efficiencies and high measurement strengths suppress the entanglement.
Horizontal lines highlight the values of the efficiency used in (d). (d)
Stationary state configuration coherence from (c) for specific values of the
efficiency $\eta=1,\,0.75,\,0.5,\,0.25$ (top to bottom). At small measurement
strengths, the entanglement only depends on the efficiency, and is equal to
the average pure state entanglement $\overline{C}_{N}=1/4$ for $\eta=1$
(dashed line). At large measurement strengths, the measurement-induced
localization suppresses the entanglement.

For a general mixed state $\rho$ of the chain, bipartite entanglement
quantifies the quantum correlations across a cut at some bond $1\leq b\leq
L-1$. It can be quantified by the configuration coherence [20, 21],

$C_{N}(\rho,b)=2\sum_{\mathclap{\begin{subarray}{c}i=1,\dots,b\\\
j=b+1,\dots,L\end{subarray}}}|\langle i|\rho|j\rangle|^{2}\,,$ (3)

where $|j\rangle=a_{j}^{\dagger}|0\rangle$. For systems with a fixed number of
particles and hermitian jump operators such as the SME (Does the system
entanglement care about the readout efficiency of quantum measurement?), the
configuration coherence is a convex entanglement measure. For a single
particle, it relates to the negativity $\mathcal{N}(\rho)$ [25] as
$\mathcal{N}(\rho)=\sqrt{C_{N}(\rho)/2}$. For our example of two sites, the
configuration coherence reads $C_{N}\leavevmode\nobreak\ =\leavevmode\nobreak\
2|\langle 1|\rho|2\rangle|^{2}$. Without measurement, the Rabi oscillations
lead to alternating coherent (entangled) and incoherent (non-entangled)
configurations of the particle on the two sites. This persists for small
measurement strengths at short times [see Fig. 2(a)] before the entanglement
saturates at a stationary value depending on the measurement strength and the
efficiency. As expected, pure Lindblad evolution ($\eta=0$) drives the state
to the separable (non-entangled) infinite temperature state $\mathds{1}/2$. In
conjunction with the persisting Rabi oscillations, the stationary entanglement
for $k=0.1$ and $\eta=1$ is $C_{N}\approx 1/4$, which corresponds to the
average entanglement on the surface of the Bloch sphere [24].

Whereas the stationary entanglement vanishes for all $k>0$ when $\eta=0$, it
remains finite if the observer obtains any information about the measurement
outcome [cf. Fig. 2(b)]. As discussed above, both high measurement strengths
and low efficiencies decrease the stationary entanglement, the former due to
localization and the latter due to mixing, see Fig. 2(c). Interestingly, for
small measurement strengths $k\lesssim 0.3$, the entanglement solely depends
on the efficiency, see Fig. 2(d). Above a critical measurement strength, the
measurement succeeds in increasingly localizing the particle. This can be
understood in the following way: the measurement damps the Rabi oscillation
and gives it a lifetime. When the lifetime becomes shorter than half the Rabi
cycle necessary to bring the particle into a coherent superposition between
the two sites, the measurement starts to decrease the stationary entanglement
below the average value $\eta/4$. This observation of an efficiency-only
dependent stationary entanglement at small-to-intermediate measurement
strengths is a first main result of our work.

Figure 3:  (a-c) Quantum random walk in local density of quantum trajectories
of a single particle injected on site 11 on a chain of length $L=21$, measured
with strength $k=0.5$ and efficiencies $\eta=0,\,0.5,\,1$. (d-f) Configuration
coherence $C_{N}$ of the random walks of (a-c). (g-i) Average configuration
coherence $\overline{C}_{N}$ for many runs of trajectories like in (a-c). For
$\eta=0$ (first column), the particle undergoes a coherent-to-diffusive
crossover. Thereunder, the evolution changes from ballistic to diffusive, and
the entanglement is absent in the latter. There is no difference between a
single trajectory and the average state. For $\eta=0.5$ (second column) and
$\eta=1$ (third column), the trajectories’ evolution is interrupted by
stochastic collapses. In contrast to $\eta=0$, the average configuration
coherence in (h,i) shows long-time entanglement. (j) Maximal mean entanglement
$\overline{C}_{N,{\rm max}}$ for the chain of length $L=21$ for different
values of efficiency $\eta$ and measurement strength $k$.

After having analyzed the effect of measurement and its readout efficiency on
the illustrative example of two sites, we consider the single particle hopping
on a chain of length $L$. In that case, the particle performs a quantum random
walk [26] governed by the competition between hopping and measurement, see
Figs. 3(a-c). For vanishing efficiency, $\eta=0$, the measurement backaction
increasingly mixes the state. As a consequence, the particle undergoes a
quantum-to-classical crossover from a coherent ballistic evolution at short
times $t\lesssim 1/k$ to a classical diffusive evolution at long times
$t\gtrsim 1/k$ [27] [cf. Fig. 3(a)]. With knowledge of the measurement
outcomes, $\eta>0$, the quantum random walk becomes interrupted by stochastic
jumps whenever the particle is measured at a specific site with high fidelity
[cf. Figs. 3(b,c)]. As the crossover is a consequence of the mixing of the
state, the evolution between jumps retains diffusive characteristics for
$\eta<1$ [cf. Fig. 3(b)]. At full efficiency $\eta=1$, however, the
intermediate evolution is ballistic, see Fig. 3(c). If the measurement is
strong enough, the particle does not have time for coherent spreading between
jumps and localizes. To summarize, the three effects at play act as follows:
(i) the hopping tries to coherently spread the particle, (ii) the measurement
suppresses such spreading, and (iii) the readout efficiency determines whether
the measurement leads to diffusive spreading (low efficiency) or localization
(high efficiency). One natural candidate to analyze such effects is
entanglement, where it has been found that the particle has a measurement-
dependent coherence length for efficiencies $\eta=0,\,1$ [19].

Notably, the quantum-to-classical crossover for $\eta=0$ manifests in the
entanglement dynamics of the particle: the ballistic evolution is supported by
entanglement, whereas in the classical diffusive evolution, entanglement is
absent, see Figs. 3(d,g). Absence of stationary entanglement is expected as
the mixed state approaches the separable (non-entangled) infinite temperature
state, $(\sum_{i=1}^{L}|i\rangle\langle i|)/L=\mathds{1}/L$, for any finite
measurement strength $k>0$. For $\eta>0$, however, the intermediate ballistic
evolution between jumps leads to finite long-time entanglement [cf. Figs.
3(e,f,h,i)]. To characterize this effect, we consider the mean entanglement
over all bonds,
$\overline{C}_{N}(t)\equiv\sum_{b=1}^{L-1}\overline{C}_{N}(\rho(t),b)/(L-1)$.
Interestingly, at intermediate times, the maximum mean entanglement,
$\overline{C}_{N,{\rm max}}\equiv\max_{t}\overline{C}_{N}(t)$ can assume a
maximum before reaching a stationary value, similar to the single-bond
entanglement in Fig. 2(b) for $\eta=0,\,0.5$. For small measurement strengths
and high readout efficiency, this maximal value is assumed not at intermediate
times, but in the stationary limit. Fig. 3(j) shows the effect of the
measurement and the readout efficiency on the maximal mean entanglement. We
find that in general, increasing measurement strength suppresses the maximal
mean entanglement in the system. For a fixed value of the measurement
strength, the maximal mean entanglement scales exponentially with the readout
efficiency [24]. However, due to the readout efficiency being bounded,
$0\leq\eta\leq 1$, the total entanglement is limited even at full access to
the measurement record. On the other hand, for a given efficiency, the maximal
mean entanglement follows a power-law decline with increasing measurement
strength, $\overline{C}_{N,{\rm max}}\propto 1/k^{a}$, for some exponent $a$
[24]. As the measurement strength can be increased indefinitely, $0\leq
k<\infty$, the maximal mean entanglement can be lowered below any limit even
at perfect readout efficiency. The differentiated analysis of the effect of
quantum measurement and its readout efficiency on the entanglement of the one
dimensional quantum random walk is the second main result of our work.

In conclusion, we have analyzed inefficient quantum measurements of a particle
on a 1d chain. To quantify the effect of the measurement and the corresponding
readout efficiency, we employed the configuration coherence, a convex mixed
state entanglement measure. The maximal amount of entanglement in a system can
be limited at will using high measurement strengths. For a fixed measurement
strength, however, we find that the observer’s knowledge of the measurement
outcome exponentially increases the maximal available amount of entanglement.
Our results show that the effect of quantum measurement on the system
entanglement depends strongly on the observer’s access to the measurement
records, over the full range of zero knowledge to perfect readout efficiency.
This has strong implications on how quantum measurement influences the ability
of a system to perform tasks demanding entanglement as a resource. In future
work, we will address the effect of inefficient measurements for many-body and
higher dimensional systems.

_Acknowledgments._ The authors thank A. Blessing for help with stochastic
calculus and acknowledge financial support by ETH Research Grant ETH-51 201-1
and the Deutsche Forschungsgemeinschaft (DFG) - project number 449653034.

Supplemental Material for
Does the system entanglement care about the readout efficiency of quantum
measurement?

Christian Carisch1, Oded Zilberberg2, and Alessandro Romito3

1Institute for Theoretical Physics, ETH Zürich, 8093 Zürich, Switzerland
3Department of Physics, University of Konstanz, D-78457 Konstanz, Germany
2Department of Physics, Lancaster University, Lancaster LA1 4YB, United
Kingdom

##  S1 1\. Kraus operator evolution scheme for the stochastic master equation

In the main text, we introduce the stochastic master equation (SME) for local
inefficient density measurements [7]

$\displaystyle d\rho=-i[H,\rho]dt$
$\displaystyle+\sum_{i=1}^{L}k\left(n_{i}\rho
n_{i}-\frac{1}{2}\\{n_{i},\rho\\}\right)dt$ $\displaystyle+\sqrt{\eta
k}\left(n_{i}\rho+\rho n_{i}-2\langle n_{i}\rangle\rho\right)dW^{i}_{t}\,.$
(S1)

Here, we explain the numerical procedure used to time evolve a state according
to (S1). To this end, we make use of the more general stochastic master
equation with (not necessarily hermitian) jump operators $L_{i}$,
$i=1,\dots,m$ and local inefficiencies $\eta_{i}$, $i=1,\dots,m$,

$\displaystyle d\rho$
$\displaystyle=-i[H,\rho]dt+\sum_{i=1}^{m}\left(L_{i}\rho
L_{i}^{\dagger}-\frac{1}{2}\\{L_{i}^{\dagger}L_{i},\rho\\}\right)dt$
$\displaystyle+\sqrt{\eta_{i}}\left(L_{i}\rho+\rho
L_{i}^{\dagger}-\Tr(L_{i}\rho+\rho L_{i}^{\dagger})\right)dW^{i}_{t}\,.$ (S2)

The SME (S1) follows from (S1) by $L_{i}=\sqrt{k}n_{i}$ and $\eta_{i}=\eta$.
In general, Eq. (S1) can be integrated using any stochastic differential
equation solver, for example, Euler or Euler-Milstein-methods [28]. Such
methods, however, lead to unphysical states because they do not generally
conserve the unit trace and the positivity of the density matrix due to
numerical errors. To guarantee physical states in the evolution, we use the
tool of Kraus operators [8]. The Kraus operator update of Eq. (S1) is given by
[29, 30]

$\displaystyle\rho+d\rho=\frac{\tilde{M}\rho\tilde{M}^{\dagger}+\sum_{i=1}^{m}(1-\eta_{i})L_{i}\rho
L_{i}^{\dagger}dt}{\Tr\left[\tilde{M}\rho\tilde{M}^{\dagger}+\sum_{i=1}^{m}(1-\eta_{i})L_{i}\rho
L_{i}^{\dagger}dt\right]}\,,$ (S3)

with Kraus operator

$\displaystyle\tilde{M}$
$\displaystyle=\mathds{1}-\left(iH+\frac{1}{2}\sum_{i=1}^{m}L_{i}^{\dagger}L_{i}\right)dt$
$\displaystyle+\sum_{i=1}^{m}\sqrt{\eta_{i}}L_{i}\left(\sqrt{\eta_{i}}\Tr(L_{i}\rho+\rho
L_{i}^{\dagger})dt+dW_{t}^{i}\right)$
$\displaystyle+\sum_{i,j=1}^{m}\frac{\sqrt{\eta_{i}\eta_{j}}}{2}L_{i}L_{j}(dW_{t}^{i}dW_{t}^{j}-\delta_{ij}dt)\,.$
(S4)

This update automatically conserves the unit trace and positivity of the
density matrix. If the time evolution operator $U=\exp(-iH\Delta t/2)$ is
available, the update scheme can be improved to [31]

$\displaystyle\rho+d\rho=\frac{U\left(MU\rho
U^{\dagger}M^{\dagger}+\sum_{i=1}^{m}(1-\eta_{i})L_{i}U\rho
U^{\dagger}L_{i}^{\dagger}dt\right)U^{\dagger}}{\Tr\left[MU\rho
U^{\dagger}M^{\dagger}+\sum_{i=1}^{m}(1-\eta_{i})L_{i}U\rho
U^{\dagger}L_{i}^{\dagger}dt\right]}\,,$ (S5)

where $M=\tilde{M}+iHdt$. Inserting $L_{i}=\sqrt{k}n_{i}$, $i=1,\dots,L$ and
using $n_{i}n_{j}=\delta_{ij}n_{i}$, $n_{i}^{\dagger}=n_{i}$, as well as
$\sum_{i=1}^{L}n_{i}=\mathds{1}$, we finally find the update scheme

$\displaystyle\rho+d\rho=\frac{U\left(MU\rho
U^{\dagger}M^{\dagger}+(1-\eta)k\sum_{i=1}^{L}n_{i}U\rho
U^{\dagger}n_{i}dt\right)U^{\dagger}}{\Tr\left[MU\rho
U^{\dagger}M^{\dagger}+(1-\eta)k\sum_{i=1}^{L}n_{i}U\rho
U^{\dagger}n_{i}dt\right]}\,,$ (S6)

with

$\displaystyle M$
$\displaystyle=M^{\dagger}=\left(1-\frac{kdt}{2}\right)\mathds{1}$
$\displaystyle+\sqrt{\eta k}\sum_{i=1}^{L}n_{i}\left(2\sqrt{\eta k}\langle
n_{i}\rangle dt+dW_{t}^{i}\right)$ $\displaystyle+\frac{\eta
k}{2}\sum_{i=1}^{L}n_{i}((dW_{t}^{i})^{2}-dt)\,,$ (S7)

where we have introduced the expectation value $\langle n_{i}\rangle=\Tr[\rho
n_{i}]$.

Figure S1:  Bloch sphere trajectories following Eq. (S1) for different values
of the measurement strength $k$ and the efficiency $\eta$. (a) For small
measurement strength and high efficiency, the trajectory performs Bloch
oscillations around the surface of the sphere. (b) For high measurement
strength and high efficiency, the trajectory can still reach all states on the
surface of the sphere, but ends up close to one of the poles due to the Zeno
effect. (c) For small measurement strength and small efficiency, the Bloch
oscillations of (a) get damped by the mixing, and the trajectory ends up close
to the center of the sphere (corresponding to the maximally mixed state
$\mathds{1}/2$. (d) For high measurement strength and small efficiency, the
trajectory moves between close to the $|1\rangle$ pole and the maximally mixed
state due to the competition of the measurement’s Zeno effect and the mixing
due to the inefficiency.

##  S2 2\. Bloch sphere description of two-site trajectories

Figure S2: Configuration coherence as a function of the efficiency $\eta$ for
several values of the measurement strength $k$. For small $\eta$, the scaling
approaches the analytical result, $C_{N}=\eta/2$.

In the main text, we study the probability distribution of the stationary
state of trajectories of a single particle on two sites subject to the SME
(S1) [cf. Figs. 1(b-e) in the main text]. Here, we derive the Bloch sphere
picture and show the dynamics of individual trajectories and how the competing
measurement and efficiency effects explain their evolution, see Fig. S1.

First, we note that for a single particle on $L=2$ sites, the Hilbert space
reduces to $\text{span}(|1\rangle,\,|2\rangle)$ due to the particle number
conserving nature of (S1). We can then employ a Bloch sphere interpretation of
the general 2-site mixed state in the basis $\\{|1\rangle,\,|2\rangle\\}$,

$\displaystyle\rho=\frac{1}{2}\begin{bmatrix}1+z&x-iy\\\
x+iy&1-z\end{bmatrix}\,,$ (S8)

and the vector $\vec{v}=(x,y,z)$ with
$|\vec{v}|^{2}=x^{2}+y^{2}+z^{2}=\Tr\rho^{2}\leq 1$ describes a point on (or
inside) the Bloch sphere.

In terms of $x,y,z$, the SME (S1) can be written as

$\displaystyle dx$ $\displaystyle=-xkdt-xz\sqrt{2\eta k}dW_{t}$ $\displaystyle
dy$ $\displaystyle=-ykdt-zdt-yz\sqrt{2\eta k}dW_{t}$ (S9) $\displaystyle dz$
$\displaystyle=ydt+(1-z^{2})\sqrt{2\eta k}dW_{t}\,,$

where $dW_{t}=(dW_{t}^{1}-dW_{t}^{2})/\sqrt{2}$ is a Wiener increment with
$\langle
dW_{t}^{2}\rangle=(\langle(dW_{t}^{1})^{2}\rangle+\langle(dW_{t}^{2})^{2}\rangle)/2=dt$.
Eq. S2 describes coupled stochastic differential equations with a quadratic
diffusion term.

Next, we inject the particle in the left site, $\rho(t=0)=|1\rangle\langle
1|$. For this initial state, we have $x=0$, $\forall t$, and we can restrict
the Bloch sphere to the $x=0$ disk. For small measurement strength and high
efficiency, the particle performs Rabi oscillations between the two sites, and
remains almost in a pure state, see Fig. S1(a). If, on the other hand, the
particle gets measured strongly, the particle can initially still tunnel
between the two sites, but gets localized closed to one of the sites. In Fig.
S1(b), the localization happens at the left site. For small efficiencies, the
state becomes more mixed due to lack of information from the measurement
readout. For small measurements, this results in Rabi oscillations towards the
center of the Bloch sphere, i.e., the maximally mixed state
$\rho=(|1\rangle\langle 1|+|2\rangle\langle 2|)/2$, see Fig. S1(c). For high
measurements and small efficiencies, the measurement projects the state onto
the z-axis of the Bloch sphere, and the small efficiency leads to highly mixed
states, as seen in Fig. S1(d).

For small efficiencies, we can simplify the stochastic differential equations
(S2) in the steady-state regime. It is known that for $\eta=0$, the steady-
state is $\rho=\mathds{1}/2$, i.e., $x=y=z=0$. We can therefore assume that
for $\eta\ll 1$, the steady-state is close to the identity and we can neglect
the second order terms $\propto z^{2}$ and $\propto yz$. As $x=0$, we find the
following set of equations,

$\displaystyle dy$ $\displaystyle=-ykdt-zdt$ (S10) $\displaystyle dz$
$\displaystyle=ydt+\sqrt{2\eta k}dW_{t}\,,$

and the stochastic equations for the quadratic terms are (we use the rules of
Itô calculus, $dW_{t}^{2}=dt$, and neglect all terms of higher order than
$dt$),

$\displaystyle dy^{2}$ $\displaystyle=2ydy+(dy)^{2}=-2y^{2}kdt-2yzdt$
$\displaystyle dz^{2}$ $\displaystyle=2zdz+(dz)^{2}$
$\displaystyle=2zydt+2z\sqrt{2\eta k}dW_{t}+2\eta kdt$ (S11) $\displaystyle
d(yz)$ $\displaystyle=zdy+ydz+dydz=-yzkdt-z^{2}dt+y^{2}dt\,.$

Next, we calculate the expectation values of the quadratic terms
$\operatorname{\mathbb{E}}[y^{2}],\,\operatorname{\mathbb{E}}[z^{2}]$, and
$\operatorname{\mathbb{E}}[yz]$ in the stationary limit, where the left hand
sides of the Eqs. (S2) vanish. Using $\operatorname{\mathbb{E}}[dW_{t}]=0$, we
find

$\displaystyle 0$
$\displaystyle=-2\operatorname{\mathbb{E}}[y^{2}]k-2\operatorname{\mathbb{E}}[yz]$
$\displaystyle 0$ $\displaystyle=2\operatorname{\mathbb{E}}[yz]+2\eta k$ (S12)
$\displaystyle 0$
$\displaystyle=-\operatorname{\mathbb{E}}[yz]k-\operatorname{\mathbb{E}}[z^{2}]+\operatorname{\mathbb{E}}[y^{2}]\,,$

with solution $\operatorname{\mathbb{E}}[yz]=-\eta k$,
$\operatorname{\mathbb{E}}[y^{2}]=\eta$,
$\operatorname{\mathbb{E}}[z^{2}]=\eta(1\leavevmode\nobreak\
-\leavevmode\nobreak\ k)$. The solution justifies the negligence of the
quadratic terms from the stochastic differential equations for $\eta\ll 1$. In
terms of the trajectory-averaged entanglement, this gives
$\overline{C}_{N}=2\overline{|\langle
1|\rho|2\rangle|^{2}}=\operatorname{\mathbb{E}}[y^{2}]/2=\eta/2$. We find that
for small $\eta$, the numerics approaches this result independent of the
measurement strength $k$, see Fig. S2.

##  S3 3\. Average pure state entanglement

In the main text, we find that for small values of the measurement strength
$k$, the long-time entanglement of the single particle on two sites at
$\eta=1$ is $C_{N}\approx 1/4$. Here, we show that this is the average
entanglement of a pure state, in agreement with the fact that for small
measurement strengths, the trajectories can explore the whole surface of the
Bloch sphere. Again, we can restrict the derivation to the $x=0$-disk if we
assume an initial state $|\psi(t=0)\rangle=|1\rangle$, i.e., we can
parametrize any pure state on the circle using a single angle $\theta$,
$|\psi(\theta)\rangle=\cos(\theta)|1\rangle+i\sin(\theta)|2\rangle$. This
state’s configuration coherence is given as
$C_{N}(\theta)=2\cos(\theta)^{2}\sin(\theta)^{2}$. For the average state, we
find

$\displaystyle\overline{C}_{N}$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta
2\cos(\theta)^{2}\sin(\theta)^{2}$
$\displaystyle=\frac{1}{8\pi}\int_{0}^{2\pi}d\theta(1-\cos(4\theta))=\frac{1}{4}\,,$
(S13)

in conjunction with our numerical findings for small measurement strengths and
high efficiency in the main text.

Figure S3:  (a) Maximal mean entanglement $\overline{C}_{N,{\rm max}}$ as a
function of measurement strength $k$ for different values of readout
efficiency $\eta$. (b) $\overline{C}_{N,{\rm max}}$ as a function of readout
efficiency $\eta$ for different values of measurement strength $k$.

##  S4 4\. Maximal mean entanglement of the chain

In the main text, we consider the maximal mean entanglement
$\overline{C}_{N,{\rm max}}$ for the chain of length $L$. Here, we discuss the
dependency of the maximal mean entanglement on the measurement strength $k$
and the readout efficiency $\eta$ in more detail.

As expected, the maximal mean entanglement decreases with increasing
measurement strength $k$. A loglog-plot [cf. Fig. S3(a)] reveals almost
straight lines, suggesting a power-law dependence, $\overline{C}_{N,{\rm
max}}\propto 1/k^{a}$ for some weakly $\eta$-dependent exponent $a$. On the
other hand, the maximal mean entanglement increases with increasing
efficiency, and a semilogy-plot [cf. Fig. S3(b)] hints at an exponential
dependence, $\overline{C}_{N,{\rm max}}\propto\exp(-b\eta)$. Note that at
small measurement strengths, the particle coherently experiences the walls of
the chain, leading to finite-size effects.

##  S5 5\. Details of the numerical implementation

Table S1 provides numerical details for the simulations leading to the results
in the Figures of the main text as well as the appendix. The numerical
simulations follow the Kraus operator update (S6) with varying values of the
timestep $dt$, stopping times $t_{f}$, and number of trajectories to average
over. The timestep has to be small compared to the measurement strength, i.e.,
$kdt\ll 1$, and the number of trajectories to average over for a faithful
average statement has to be higher for higher efficiency $\eta$, as the
stochastic terms in the increment $d\rho$ scale with $\sqrt{\eta k}$.

In order to make statements about the stationary state of average quantities,
one has to run the simulation until the quantities converge. For the two-site
example of Figs. S1 and Figs. 1(b-e), 2(c,d) of the main text, we make use of
the analytical solution of (S1) at efficiency $\eta=0$. From this solution,
the configuration coherence can be calculated as [19]

$C_{N}(t)=\begin{cases}\frac{e^{-kt}}{2\,(1-(k/2)^{2})}\sin^{2}\left(\sqrt{1-(k/2)^{2}}t\right),\
k<2\\\ \frac{1}{2}t^{2}e^{-2t},\ k=2\\\
\frac{e^{-kt}}{2\,((k/2)^{2}-1)}\sinh^{2}\left(\sqrt{(k/2)^{2}-1}t\right),\
k>2\,.\end{cases}$ (S14)

We use this result to estimate a stopping time $t_{f}$ for our trajectory
simulations for $k<2$. As the average state of the trajectories follows the
SME (S1) independent of the efficiency $\eta$, the trajectories’ stationary
state is reached when the average state has reached it’s steady state, i.e.,
when the configuration coherence (S14) is very close to zero. We set as a
condition for $t_{f}$ that $C_{N}(t_{f})\leq 10^{-3}$ and find

$\displaystyle\frac{e^{-kt_{f}}}{2\,(1-(k/2)^{2})}\leq C_{N}(t_{f})\leq
10^{-3}$ $\displaystyle\implies t_{f}=-\log(2\times
10^{-3}(1-(k/2)^{2}))/k\,.$ (S15)

For measurement strenghts $k\geq 2$, we find good convergence for $t_{f}=5k$.

For the chain of length $L>2$, an analytical solution does not exist. However,
if we are interested in the intermediate time dynamics rather than the
stationary limit, we can infer a reasonable stopping time $t_{f}$ from the
dynamics. In the main text, we are interested in the maximal mean
entanglement, $\overline{C}_{N,{\rm max}}$. At each bond, the maximal value
will be realized shortly after the particle reaches that bond (the
entanglement damping due to the measurement will thereafter monotonously
decrease the entanglement). Until the quantum-to-classical crossover at
$t\approx 1/k$, the particle evolves ballistically with velocity $J=1$. For
small measurement strengths, the particle will thus hit the wall (and thereby
pass the outermost bonds) at $t_{f}=L/2$. If the quantum-to-classical
crossover happens before the particle reaches the wall, the subsequent
evolution is diffusive, with diffusion constant $D=4/k$ [32]. Then, the
particle reaches the wall at time $t_{f}=1/D(L/2)^{2}=k(L/4)^{2}$. The
numerics for the main text example of the chain with length $L=21$ shows good
results for a stopping time $t_{f}=\text{max}((L/4)^{2}k,L)$ taking into
account both types of dynamics.

Fig. | $dt$ | $t_{f}$ | # of trajectories
---|---|---|---
S1 | $\text{min}(1,1/k)\times 10^{-3}$ | $\begin{cases}-\log(2\times 10^{-3}(1-(k/2)^{2}))/k,\,k<2\\\ 5k,\,\text{else}\end{cases}$ | 1
1(b-e) | $\text{min}(1,1/k)\times 10^{-3}$ | $\begin{cases}-\log(2\times 10^{-3}(1-(k/2)^{2}))/k,\,k<2\\\ 5k,\,\text{else}\end{cases}$ | 10000
2(a) | $10^{-3}$ | $10/k$ | $\begin{cases}1,\,\eta=0\\\ 1800,\,\eta=0.5\\\ 2200,\,\eta=1\end{cases}$
2(b) | $5\times 10^{-4}$ | $10/k$ | $\begin{cases}1,\,\eta=0\\\ 1900,\,\eta=0.5\\\ 1700,\,\eta=1\end{cases}$
2(c,d), S2 | $\text{min}(1,1/k)\times 10^{-3}$ | $\begin{cases}-\log(2\times 10^{-3}(1-(k/2)^{2}))/k,\,k<2\\\ 5k,\,\text{else}\end{cases}$ | $\text{max}(1000\eta,100)$
3(a-f) | $10^{-3}$ | $15$ | 1
3(g-i) | $10^{-3}$ | $15$ | 100
3(j) | $\text{min}(1,1/k)\times 10^{-2}$ | $\text{max}((L/4)^{2}k,L)$ | $\begin{cases}1,\,\eta=0\\\ 200\lceil 10\eta k\rceil,\,\text{else}\end{cases}$
Table S1: Details of the numerical implementation for the Figs. in the main
text and the Supplemental Material.

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A Generalization of

Seifert-Van Kampen Theorem for Fundamental Groups

Linfan MAO

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

E-mail<EMAIL_ADDRESS>

Abstract: As we known, the Seifert-Van Kampen theorem handles fundamental
groups of those topological spaces $X=U\cup V$ for open subsets $U,\ V\subset
X$ such that $U\cap V$ is arcwise connected. In this paper, this theorem is
generalized to such a case of maybe not arcwise-connected, i.e., there are
$C_{1}$, $C_{2}$,$\cdots,\ C_{m}$ arcwise-connected components in $U\cap V$
for an integer $m\geq 1$, which enables one to find fundamental groups of
combinatorial spaces by that of spaces with theirs underlying topological
graphs, particularly, that of compact manifolds by their underlying graphs of
charts.

Key Words: Fundamental group, Seifert-Van Kampen theorem, topological space,
combinatorial manifold, topological graph.

AMS(2010): 51H20.

§$1.$ Introduction

All spaces $X$ considered in this paper are arcwise-connected, graphs are
connected topological graph, maybe with loops or multiple edges and interior
of an arc $a:(0,1)\rightarrow X$ is opened. For terminologies and notations
not defined here, we follow the reference [1]-[3] for topology and [4]-[5] for
topological graphs.

Let $X$ be a topological space. A fundamental group $\pi_{1}(X,x_{0})$ of $X$
based at a point $x_{0}\in X$ is formed by homotopy arc classes in $X$ based
at $x_{0}\in X$. For an arcwise-connected space $X$, it is known that
$\pi_{1}(X,x_{0})$ is independent on the base point $x_{0}$, that is, for
$\forall x_{0},y_{0}\in X$,

$\pi_{1}(X,x_{0})\cong\pi_{1}(X,y_{0}).$

Find the fundamental group of a space $X$ is a difficult task in general.
Until today, the basic tool is still the Seifert-Van Kampen theorem following.

Theorem $1.1$(Seifert and Van-Kampen) Let $X=U\cup V$ with $U,\ V$ open
subsets and let $X,\ U,\ V$, $U\cap V$ be non-empty arcwise-connected with
$x_{0}\in U\cap V$ and $H$ a group. If there are homomorphisms

$\phi_{1}:\pi_{1}(U,x_{0})\rightarrow H\ \ {and}\ \
\phi_{2}:\pi_{1}(V,x_{0})\rightarrow H$

and

with $\phi_{1}\cdot i_{1}=\phi_{2}\cdot i_{2}$, where $i_{1}:\pi_{1}(U\cap
V,x_{0})\rightarrow\pi_{1}(U,x_{0})$, $i_{2}:\pi_{1}(U\cap
V,x_{0})\rightarrow\pi_{1}(V,x_{0})$,
$j_{1}:\pi_{1}(U,x_{0})\rightarrow\pi_{1}(X,x_{0})$ and
$j_{2}:\pi_{1}(V,x_{0})\rightarrow\pi_{1}(X,x_{0})$ are homomorphisms induced
by inclusion mappings, then there exists a unique homomorphism $\Phi:\
\pi_{1}(X,x_{0})\rightarrow H$ such that $\Phi\cdot j_{1}=\phi_{1}$ and
$\Phi\cdot j_{2}=\phi_{2}$.

Applying Theorem $1.1$, it is easily to determine the fundamental group of
such spaces $X=U\cup V$ with $U\cap V$ an arcwise-connected following.

Theorem $1.2$(Seifert and Van-Kampen theorem, classical version) Let spaces
$X,U,V$ and $x_{0}$ be in Theorem $1.1$. If

$j:\pi_{1}(U,x_{0})*\pi_{1}(V,x_{0})\rightarrow\pi_{1}(X,x_{0})$

is an extension homomorphism of $j_{1}$ and $j_{2}$, then $j$ is an
epimorphism with kernel Ker$j$ generated by $i_{1}^{-1}(g)i_{2}(g),\
g\in\pi_{1}(U\cap V,x_{0})$, i.e.,

$\pi_{1}(X,x_{0})\cong\frac{\pi_{1}(U,x_{0})*\pi_{1}(V,x_{0})}{\left[i_{1}^{-1}(g)\cdot
i_{2}(g)|\ g\in\pi_{1}(U\cap V,x_{0})\right]},$

where $\left[A\right]$ denotes the minimal normal subgroup of a group
$\mathscr{G}$ included $A\subset\mathscr{G}$.

Now we use the following convention.

Convention $1.3$ Assume that

($1$) $X$ is an arcwise-connected spaces, $x_{0}\in X$;

($2$) $\\{U_{\lambda}:\lambda\in\Lambda\\}$ is a covering of $X$ by arcwise-
connected open sets such that $x_{0}\in U_{\lambda}$ for
$\forall\lambda\in\Lambda$;

($3$) For any two indices $\lambda_{1},\lambda_{2}\in\Lambda$ there exists an
index $\lambda\in\Lambda$ such that $U_{\lambda_{1}}\cap
U_{\lambda_{2}}=U_{\lambda}$

If $U_{\lambda}\subset U_{\mu}\subset X$, then the notation

$\phi_{\lambda\mu}:\pi_{1}(U_{\lambda},x_{0})\rightarrow\pi_{1}(U_{\mu},x_{0})\
\ {\rm and}\ \
\phi_{\lambda}:\pi_{1}(U_{\lambda},x_{0})\rightarrow\pi_{1}(X,x_{0})$

denote homomorphisms induced by the inclusion mapping $U_{\lambda}\rightarrow
U_{\mu}$ and $U_{\lambda}\rightarrow X$, respectively. It should be noted that
the Seifert-Van Kampen theorem has been generalized in [2] under Convention
$1.3$ by any number of open subsets instead of just by two open subsets
following.

Theorem $1.4$([2]) Let $X,U_{\lambda},\ \lambda\in\Lambda$ be arcwise-
connected space with Convention $1.3$ satisfies the following universal
mappping condition: Let $H$ be a group and let
$\rho_{\lambda}:\pi_{1}(U_{\lambda},x_{0})\rightarrow H$ be any collection of
homomorphisms defined for all $\lambda\in\Lambda$ such that the following
diagram is commutative for $U_{\lambda}\subset U_{\mu}$:

Then there exists a unique homomorphism $\Phi:\pi_{1}(X,x_{0})\rightarrow H$
such that for any $\lambda\in\Lambda$ the following diagram is commutative:

Moreover, this universal mapping condition characterizes $\pi_{1}(X,x_{0})$ up
to a unique isomorphism.

Theorem $1.4$ is useful for determining the fundamental groups of CW-
complexes, particularly, the adjunction of $n$-dimensional cells to a space
for $n\geq 2$. Notice that the essence in Theorems $1.2$ and $1.4$ is that
$\cap_{\lambda\in\Lambda}U_{\lambda}$ is arcwise-connected, which limits the
application of such kind of results. The main object of this paper is to
generalize the Seifert-Van Kampen theorem to such an intersection maybe non-
arcwise connected, i.e., there are $C_{1}$, $C_{2}$,$\cdots,\ C_{m}$ arcwise-
connected components in $U\cap V$ for an integer $m\geq 1$. This enables one
to determine the fundamental group of topological spaces, particularly,
combinatorial manifolds introduced in [6]-[8] following which is a special
case of Smarandache multi-space ([9]-[10]).

Definition $1.4$ 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

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

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

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 $\mu,\nu\in\Lambda,\ \mu\not=\nu$,

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

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

Definition $1.5$ 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 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 a combinatorial Euclidean space
$\mathscr{E}_{G}(n_{\nu};\nu\in\Lambda)$ with $\Lambda=\\{1,2,\cdots,m\\}$.
Furthermore, If $m=1$ and $n_{1}=n$, or $n_{\nu}=n$ for $\nu\in\Lambda$, then
$\widetilde{M}(n_{1},n_{2},\cdots,n_{m})$ or
$\mathscr{E}_{G}(n_{\nu};\nu\in\Lambda)$ is exactly a manifold $M^{n}$ by
definition.

§$2.$ Topological Space Attached Graphs

A topological graph $G$ is itself a topological space formally defined as
follows.

Definition $2.1$ A topological graph $G$ is a pair $(S,S^{0})$ of a Hausdorff
space $S$ with its a subset $S^{0}$ such that

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

($2$) $S-S^{0}$ 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 $A\subset G$ is open if and only if $A\cap\overline{e}_{i}$ is
open for $1\leq i\leq m$.

Fig.$2.1$

Notice that a topological graph maybe with semi-edges, i.e., those edges
$e^{+}\in E(G)$ with $e^{+}:[0,1)\ {\rm or}\ (0,1]\rightarrow S$. A
topological space $X$ attached with a graph $G$ is such a space $X\odot G$
such that

$X\bigcap G\not=\emptyset,\ \ G\not\subset X$

and there are semi-edges $e^{+}\in(X\bigcap G)\setminus G$, $e^{+}\in
G\setminus X$. An example for $X\odot G$ can be found in Fig.$2.1$. In this
section, we characterize the fundamental groups of such topological spaces
attached with graphs.

Theorem $2.2$ Let $X$ be arc-connected space, $G$ a graph and $H$ the subgraph
$X\cap G$ in $X\odot G$. Then for $x_{0}\in X\cap G$,

$\pi_{1}(X\odot
G,x_{0})\cong\frac{\pi_{1}(X,x_{0})*\pi_{1}(G,x_{0})}{\left[i_{1}^{-1}(\alpha_{e_{\lambda}})i_{2}(\alpha_{e_{\lambda}})|\
e_{\lambda}\in E(H)\setminus T_{span})\right]},$

where $i_{1}:\pi_{1}(H,x_{0})\rightarrow X$,
$i_{2}:\pi_{1}(H,x_{0})\rightarrow G$ are homomorphisms induced by inclusion
mappings, $T_{span}$ is a spanning tree in $H$,
$\alpha_{\lambda}=A_{\lambda}e_{\lambda}B_{\lambda}$ is a loop associated with
an edge $e_{\lambda}=a_{\lambda}b_{\lambda}\in H\setminus T_{span}$, $x_{0}\in
G$ and $A_{\lambda}$, $B_{\lambda}$ are unique paths from $x_{0}$ to
$a_{\lambda}$ or from $b_{\lambda}$ to $x_{0}$ in $T_{span}$.

Proof This result is an immediately conclusion of Seifert-Van Kampen theorem.
Let $U=X$ and $V=G$. Then $X\odot G=X\cup G$ and $X\cap G=H$. By definition,
there are both semi-edges in $G$ and $H$. Whence, they are opened. Applying
the Seifert-Van Kampen theorem, we get that

$\pi_{1}(X\odot
G,x_{0})\cong\frac{\pi_{1}(X,x_{0})*\pi_{1}(G,x_{0})}{\left[i_{1}^{-1}(g)i_{2}(g)|\
g\in\pi_{1}(X\cap G,x_{0})\right]},$

Notice that the fundamental group of a graph $H$ is completely determined by
those of its cycles ([2]), i.e.,

$\pi_{1}(H,x_{0})=\left<\alpha_{\lambda}|e_{\lambda}\in E(H)\setminus
T_{span}\right>,$

where $T_{span}$ is a spanning tree in $H$,
$\alpha_{\lambda}=A_{\lambda}e_{\lambda}B_{\lambda}$ is a loop associated with
an edge $e_{\lambda}=a_{\lambda}b_{\lambda}\in H\setminus T_{span}$, $x_{0}\in
G$ and $A_{\lambda}$, $B_{\lambda}$ are unique paths from $x_{0}$ to
$a_{\lambda}$ or from $b_{\lambda}$ to $x_{0}$ in $T_{span}$. We finally get
the following conclusion,

$\hskip 56.9055pt\pi_{1}(X\odot
G,x_{0})\cong\frac{\pi_{1}(X,x_{0})*\pi_{1}(G,x_{0})}{\left[i_{1}^{-1}(\alpha_{e_{\lambda}})i_{2}(\alpha_{e_{\lambda}})|\
e_{\lambda}\in E(H)\setminus T_{span})\right]}\hskip 56.9055pt\Box$

Corollary $2.3$ Let $X$ be arc-connected space, $G$ a graph. If $X\cap G$ in
$X\odot G$ is a tree, then

$\pi_{1}(X\odot G,x_{0})\cong\pi_{1}(X,x_{0})*\pi_{1}(G,x_{0}).$

Particularly, if $G$ is graphs shown in Fig.$2.2$ following

Fig.$2.2$

and $X\cap G=K_{1,m}$, Then

$\pi_{1}(X\odot B_{m}^{T},x_{0})\cong\pi_{1}(X,x_{0})*\left<L_{i}|1\leq i\leq
m\right>,$

where $L_{i}$ is the loop of parallel edges $(x_{0},x_{i})$ in $B_{m}^{T}$ for
$1\leq i\leq m-1$ and

$\pi_{1}(X\odot S_{m}^{T},x_{0})\cong\pi_{1}(X,x_{0}).$

Theorem $2.4$ Let $\mathscr{X}_{m}\odot G$ be a topological space consisting
of $m$ arcwise-connected spaces $X_{1},X_{2},\cdots,X_{m}$, $X_{i}\cap
X_{j}=\emptyset$ for $1\leq i,j\leq m$ attached with a graph $G$,
$V(G)=\\{x_{0},x_{1},\cdots,x_{l-1}\\}$, $m\leq l$ such that $X_{i}\cap
G=\\{x_{i}\\}$ for $0\leq i\leq l-1$. Then

$\displaystyle\pi_{1}(\mathscr{X}_{m}\odot G,x_{0})$ $\displaystyle\cong$
$\displaystyle\left(\prod\limits_{i=1}^{m}\pi_{1}(X_{i}^{*},x_{0})\right)*\pi_{1}(G,x_{0})$
$\displaystyle\cong$
$\displaystyle\left(\prod\limits_{i=1}^{m}\pi_{1}(X_{i},x_{i})\right)*\pi_{1}(G,x_{0}),$

where $X_{i}^{*}=X_{i}\bigcup(x_{0},x_{i})$ with
$X_{i}\cap(x_{0},x_{i})=\\{x_{i}\\}$ for $(x_{0},x_{i})\in E(G)$, integers
$1\leq i\leq m$.

Proof The proof is by induction on $m$. If $m=1$, the result is hold by
Corollary $2.3$.

Now assume the result on $\mathscr{X}_{m}\odot G$ is hold for $m\leq k<l-1$.
Consider $m=k+1\leq l$. Let $U=\mathscr{X}_{k}\odot G$ and $V=X_{k+1}$. Then
we know that $\mathscr{X}_{k+1}\odot G=U\cup V$ and $U\cap V=\\{x_{k+1}\\}$.

Applying the Seifert-Van Kampen theorem, we find that

$\displaystyle\pi_{1}(\mathscr{X}_{k+1}\odot G,x_{k+1})$ $\displaystyle\cong$
$\displaystyle\frac{\pi_{1}(U,x_{k+1})*\pi_{1}(V,x_{k+1})}{\left[i_{1}^{-1}(g)i_{2}(g)|\
g\in\pi_{1}(U\cap V,x_{k+1})\right]}$ $\displaystyle\cong$
$\displaystyle\frac{\pi_{1}(\mathscr{X}_{k}\odot
G,x_{0})*\pi_{1}(X_{k+1},x_{k+1})}{\left[i_{1}^{-1}(g)i_{2}(g)|\ g\in\\{{\bf
e}_{x_{k+1}}\\}\right]}$ $\displaystyle\cong$
$\displaystyle\left(\left(\prod\limits_{i=1}^{k}\pi_{1}(X_{i}^{*},x_{0})\right)*\pi_{1}(G,x_{0})\right)*\pi_{1}(X_{k+1},x_{k+1})$
$\displaystyle\cong$
$\displaystyle\left(\prod\limits_{i=1}^{k+1}\pi_{1}(X_{i}^{*},x_{0})\right)*\pi_{1}(G,x_{0})$
$\displaystyle\cong$
$\displaystyle\left(\prod\limits_{i=1}^{m}\pi_{1}(X_{i},x_{i})\right)*\pi_{1}(G,x_{0}),$

by the induction assumption. $\Box$

Particularly, for the graph $B_{m}^{T}$ or star $S_{m}^{T}$ in Fig.$2.2$, we
get the following conclusion.

Corollary $2.5$ Let $G$ be the graph $B_{m}^{T}$ or star $S_{m}^{T}$. Then

$\displaystyle\pi_{1}(\mathscr{X}_{m}\odot B_{m}^{T},x_{0})$
$\displaystyle\cong$
$\displaystyle\left(\prod\limits_{i=1}^{m}\pi_{1}(X_{i}^{*},x_{0})\right)*\pi_{1}(B_{m}^{T},x_{0})$
$\displaystyle\cong$
$\displaystyle\left(\prod\limits_{i=1}^{m}\pi_{1}(X_{i},x_{i-1})\right)*\left<L_{i}|1\leq
i\leq m\right>,$

where $L_{i}$ is the loop of parallel edges $(x_{0},x_{i})$ in $B_{m}^{T}$ for
integers $1\leq i\leq m-1$ and

$\pi_{1}(\mathscr{X}_{m}\odot
S_{m}^{T},x_{0})\cong\prod\limits_{i=1}^{m}\pi_{1}(X_{i}^{*},x_{0})\cong\prod\limits_{i=1}^{m}\pi_{1}(X_{i},x_{i-1}).$

Corollary $2.6$ Let $X=\mathscr{X}_{m}\odot G$ be a topological space with
simply-connected spaces $X_{i}$ for integers $1\leq i\leq m$ and $x_{0}\in
X\cap G$. Then we know that

$\pi_{1}(X,x_{0})\cong\pi_{1}(G,x_{0}).$

§$3.$ A Generalization of Seifert-Van Kampen Theorem

These results and graph $B_{m}^{T}$ shown in Section $2$ enables one to
generalize the Seifert-Van Kampen theorem to the case of $U\cap V$ maybe not
arcwise-connected.

Theorem $3.1$ Let $X=U\cup V$, $U,V\subset X$ be open subsets, $X,\ U,\ V$
arcwise-connected and let $C_{1},C_{2},\cdots,C_{m}$ be arcwise-connected
components in $U\cap V$ for an integer $m\geq 1$, $x_{i-1}\in C_{i}$,
$b(x_{0},x_{i-1})\subset V$ an arc $:I\rightarrow X$ with
$b(0)=x_{0},b(1)=x_{i-1}$ and $b(x_{0},x_{i-1})\cap U=\\{x_{0},x_{i-1}\\}$,
$C_{i}^{E}=C_{i}\bigcup b(x_{0},x_{i-1})$ for any integer $i,\ 1\leq i\leq m$,
$H$ a group and there are homomorphisms

$\phi_{1}^{i}:\pi_{1}(U\bigcup b(x_{0},x_{i-1}),x_{0})\rightarrow H,\ \
\phi_{2}^{i}:\pi_{1}(V,x_{0})\rightarrow H$

such that

with $\phi_{1}^{i}\cdot i_{i1}=\phi_{2}^{i}\cdot i_{i2}$, where
$i_{i1}:\pi_{1}(C_{i}^{E},x_{0})\rightarrow\pi_{1}(U\cup
b(x_{0},x_{i-1}),x_{0})$,
$i_{i2}:\pi_{1}(C_{i}^{E},x_{0})\rightarrow\pi_{1}(V,x_{0})$ and
$j_{i1}:\pi_{1}(U\cup b(x_{0},x_{i-1},x_{0}))\rightarrow\pi_{1}(X,x_{0})$,
$j_{i2}:\pi_{1}(V,x_{0}))\rightarrow\pi_{1}(X,x_{0})$ are homomorphisms
induced by inclusion mappings, then there exists a unique homomorphism $\Phi:\
\pi_{1}(X,x_{0})\rightarrow H$ such that $\Phi\cdot j_{i1}=\phi_{1}^{i}$ and
$\Phi\cdot j_{i2}=\phi_{2}^{i}$ for integers $1\leq i\leq m$.

Proof Define $U^{E}=U\bigcup\\{\ b(x_{0},x_{i})\ |\ 1\leq i\leq m-1\\}$. Then
we get that $X=U^{E}\cup V$, $U^{E},V\subset X$ are still opened with an
arcwise-connected intersection $U^{E}\cap V=\mathscr{X}_{m}\odot S_{m}^{T}$,
where $S_{m}^{T}$ is a graph formed by arcs $b(x_{0},x_{i-1})$, $1\leq i\leq
m$.

Notice that $\mathscr{X}_{m}\odot Sm^{T}=\bigcup\limits_{i=1}^{m}C_{i}^{E}$
and $C_{i}^{E}\bigcap C_{j}^{E}=\\{x_{0}\\}$ for $1\leq i,j\leq m,\ i\not=j$.
Therefore, we get that

$\pi_{1}(\mathscr{X}_{m}\odot
S_{m}^{T},x_{0})=\bigotimes\limits_{i=1}^{m}\pi_{1}(C_{i}^{E},x_{0}).$

This fact enables us knowing that there is a unique $m$-tuple
$(h_{1},h_{2},\cdots,h_{m})$, $h_{i}\in\pi_{1}(C_{i}^{E},x_{i-1}),\ 1\leq
i\leq m$ such that

$\mathscr{I}=\prod\limits_{i=1}^{m}h_{i}$

for $\forall\mathscr{I}\in\pi_{1}(\mathscr{X}_{m}\odot S_{m}^{T},x_{0})$.

By definition,

$i_{i1}:\pi_{1}(C_{i}^{E},x_{0})\rightarrow\pi_{1}(U\cap
b(x_{0},x_{i-1}),x_{0}),$
$i_{i2}:\pi_{1}(C_{i}^{E},x_{0})\rightarrow\pi_{1}(V,x_{0})$

are homomorphisms induced by inclusion mappings. We know that there are
homomorphisms

$i_{1}^{E}:\pi_{1}(\mathscr{X}_{m}\odot
S_{m}^{T},x_{0})\rightarrow\pi_{1}(U^{E},x_{0}),$
$i_{2}^{E}:\pi_{1}(\mathscr{X}_{m}\odot
S_{m}^{T},x_{0})\rightarrow\pi_{1}(V,x_{0})$

with $i_{1}^{E}|_{\pi_{1}(C_{i}^{E},x_{0})}=i_{i1}$,
$i_{2}^{E}|_{\pi_{1}(C_{i}^{E},x_{0})}=i_{i2}$ for integers $1\leq i\leq m$.

Similarly, because of

$\pi_{1}(U^{E},x_{0})=\bigcup\limits_{i=1}^{m}\pi_{1}(U\cup
b(x_{0},x_{i-1},x_{0}))$

and

$j_{i1}:\pi_{1}(U\cup b(x_{0},x_{i-1},x_{0}))\rightarrow\pi_{1}(X,x_{0}),$
$j_{i2}:\pi_{1}(V\rightarrow\pi_{1}(X,x_{0})$

being homomorphisms induced by inclusion mappings, there are homomorphisms

$j_{1}^{E}:\pi_{1}(U^{E},x_{0})\rightarrow\pi_{1}(X,x_{0}),\ \
j_{2}^{E}:\pi_{1}(V,x_{0})\rightarrow\pi_{1}(X,x_{0})$

induced by inclusion mappings with $j_{1}^{E}|_{\pi_{1}(U\cup
b(x_{0},x_{i-1},x_{0}))}=j_{i1}$, $j_{2}^{E}|_{\pi_{1}(V,x_{0})}=j_{i2}$ for
integers $1\leq i\leq m$ also.

Define $\phi_{1}^{E}$ and $\phi_{2}^{E}$ by

$\phi_{1}^{E}(\mathscr{I})=\prod\limits_{i=1}^{m}\phi_{1}^{i}(i_{i1}(h_{i})),\
\
\phi_{2}^{E}(\mathscr{I})=\prod\limits_{i=1}^{m}\phi_{2}^{i}(i_{i2}(h_{i})).$

Then they are naturally homomorphic extensions of homomorphisms
$\phi_{1}^{i},\ \phi_{2}^{i}$ for integers $1\leq i\leq m$. Notice that
$\phi_{1}^{i}\cdot i_{i1}=\phi_{2}^{i}\cdot i_{i2}$ for integers $1\leq i\leq
m$, we get that

$\displaystyle\phi_{1}^{E}\cdot i_{1}^{E}(\mathscr{I})$ $\displaystyle=$
$\displaystyle\phi_{1}^{E}\cdot
i_{1}^{E}\left(\prod\limits_{i=1}^{m}h_{i}\right)$ $\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{m}\left(\phi_{1}^{i}\cdot
i_{i1}(h_{i})\right)=\prod\limits_{i=1}^{m}\left(\phi_{2}^{i}\cdot
i_{i2}(h_{i})\right)$ $\displaystyle=$ $\displaystyle\phi_{2}^{E}\cdot
i_{2}^{E}\left(\prod\limits_{i=1}^{m}h_{i}\right)=\phi_{2}^{E}\cdot
i_{2}^{E}(\mathscr{I}),$

i.e., the following diagram

is commutative with $\phi_{1}^{E}\cdot i_{1}^{E}=\phi_{2}^{E}\cdot i_{2}^{E}$.
Applying Theorem $1.1$, we know that there exists a unique homomorphism
$\Phi:\ \pi_{1}(X,x_{0})\rightarrow H$ such that $\Phi\cdot
j_{1}^{E}=\phi_{1}^{E}$ and $\Phi\cdot j_{2}^{E}=\phi_{2}^{E}$. Whence,
$\Phi\cdot j_{i1}=\phi_{1}^{i}$ and $\Phi\cdot j_{i2}=\phi_{2}^{i}$ for
integers $1\leq i\leq m$. $\Box$

The following result is a generalization of the classical Seifert-Van Kampen
theorem to the case of maybe non-arcwise connected.

Theorem $3.2$ Let $X$, $U$, $V$, $C_{i}^{E}$, $b(x_{0},x_{i-1})$ be arcwise-
connected spaces for any integer $i,\ 1\leq i\leq m$ as in Theorem $3.1$,
$U^{E}=U\bigcup\\{\ b(x_{0},x_{i})\ |\ 1\leq i\leq m-1\\}$ and $B_{m}^{T}$ a
graph formed by arcs $a(x_{0},x_{i-1})$, $b(x_{0},x_{i-1})$, $1\leq i\leq m$,
where $a(x_{0},x_{i-1})\subset U$ is an arc $:I\rightarrow X$ with
$a(0)=x_{0},a(1)=x_{i-1}$ and $a(x_{0},x_{i-1})\cap V=\\{x_{0},x_{i-1}\\}$.
Then

$\pi_{1}(X,x_{0})\cong\frac{\pi_{1}(U,x_{0})*\pi_{1}(V,x_{0})*\pi_{1}(B_{m}^{T},x_{0})}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}(g)|\ g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i}^{E},x_{0})\ \right]},$

where $i_{1}^{E}:\pi_{1}(U^{E}\cap V,x_{0})\rightarrow\pi_{1}(U^{E},x_{0})$
and $i_{2}^{E}:\pi_{1}(U^{E}\cap V,x_{0})\rightarrow\pi_{1}(V,x_{0})$ are
homomorphisms induced by inclusion mappings.

Proof Similarly, $X=U^{E}\cup V$, $U^{E},V\subset X$ are opened with
$U^{E}\cap V=\mathscr{X}_{m}\odot S_{m}^{T}$. By the proof of Theorem $3.1$ we
have known that there are homomorphisms $\phi_{1}^{E}$ and $\phi_{2}^{E}$ such
that $\phi_{1}^{E}\cdot i_{1}^{E}=\phi_{2}^{E}\cdot i_{2}^{E}$. Applying
Theorem $1.2$, we get that

$\pi_{1}(X,x_{0})\cong\frac{\pi_{1}(U^{E},x_{0})*\pi_{1}(V,x_{0})}{\left[(i_{1}^{E})^{-1}(\mathscr{I})\cdot
i_{2}^{E}(\mathscr{I})|\mathscr{I}\in\pi_{1}(U^{E}\cap V,x_{0})\right]}.$

Notice that $U^{E}\cap V^{E}=\mathscr{X}_{m}\odot S_{m}^{T}$. We have known
that

$\pi_{1}(U^{E},x_{0})\cong\pi_{1}(U,x_{0})*\pi_{1}(B_{m}^{T},x_{0})$

by Corollary $2.3$. As we have shown in the proof of Theorem $3.1$, an element
$\mathscr{I}$ in $\pi_{1}(\mathscr{X}_{m}\odot S_{m}^{T},x_{0})$ can be
uniquely represented by

$\mathscr{I}=\prod\limits_{i=1}^{m}h_{i},$

where $h_{i}\in\pi_{1}(C_{i}^{E},x_{0}),\ 1\leq i\leq m$. We finally get that

$\hskip
56.9055pt\pi_{1}(X,x_{0})\cong\frac{\pi_{1}(U,x_{0})*\pi_{1}(V,x_{0})*\pi_{1}(B_{m}^{T},x_{0})}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i}^{E},x_{0})\
\right]}.\hskip 56.9055pt\Box$

The form of elements in $\pi_{1}(\mathscr{X}_{m}\odot S_{m}^{T},x_{0})$
appeared in Corollary $2.5$ enables one to obtain another generalization of
classical Seifert-Van Kampen theorem following.

Theorem $3.3$ Let $X$, $U$, $V$, $C_{1},C_{2},\cdots,C_{m}$ be arcwise-
connected spaces, $b(x_{0},x_{i-1})$ arcs for any integer $i,\ 1\leq i\leq m$
as in Theorem $3.1$, $U^{E}=U\bigcup\\{\ b(x_{0},x_{i-1})\ |\ 1\leq i\leq
m\\}$ and $B_{m}^{T}$ a graph formed by arcs $a(x_{0},x_{i-1})$,
$b(x_{0},x_{i-1})$, $1\leq i\leq m$. Then

$\pi_{1}(X,x_{0})\cong\frac{\pi_{1}(U,x_{0})*\pi_{1}(V,x_{0})*\pi_{1}(B_{m}^{T},x_{0})}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i},x_{i-1})\right]},$

where $i_{1}^{E}:\pi_{1}(U^{E}\cap V,x_{0})\rightarrow\pi_{1}(U^{E},x_{0})$
and $i_{2}^{E}:\pi_{1}(U^{E}\cap V,x_{0})\rightarrow\pi_{1}(V,x_{0})$ are
homomorphisms induced by inclusion mappings.

Proof Notice that $U^{E}\cap V=\mathscr{X}_{m}\odot S^{T}_{m}$. Applying
Corollary $2.5$, replacing

$\pi_{1}(\mathscr{X}_{m}\odot S_{m}^{T},x_{0})=\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i}^{E},x_{0})\right]$

by

$\pi_{1}(\mathscr{X}_{m}\odot S_{m}^{T},x_{0})=\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i},x_{i-1})\right]$

in the proof of Theorem $3.2$. We get this conclusion. $\Box$

Particularly, we get corollaries following by Theorems $3.1$, $3.2$ and $3.3$.

Corollary $3.4$ Let $X=U\cup V$, $U,V\subset X$ be open subsets and $X,\ U,\
V$ and $U\cap V$ arcwise-connected. Then

$\pi_{1}(X,x_{0})\cong\frac{\pi_{1}(U,x_{0})*\pi_{1}(V,x_{0})}{\left[i_{1}^{-1}(g)\cdot
i_{2}(g)|\ g\in\pi_{1}(U\cap V,x_{0})\right]},$

where $i_{1}:\pi_{1}(U\cap V,x_{0})\rightarrow\pi_{1}(U,x_{0})$ and
$i_{2}:\pi_{1}(U\cap V,x_{0})\rightarrow\pi_{1}(V,x_{0})$ are homomorphisms
induced by inclusion mappings.

Corollary $3.5$ Let $X$, $U$, $V$, $C_{i}$, $a(x_{0},x_{i})$, $b(x_{0},x_{i})$
for integers $i,\ 1\leq i\leq m$ be as in Theorem $3.1$. If each $C_{i}$ is
simply-connected, then

$\pi_{1}(X,x_{0})\cong\pi_{1}(U,x_{0})*\pi_{1}(V,x_{0})*\pi_{1}(B_{m}^{T},x_{0}).$

Proof Notice that $C_{1}^{E},C_{2}^{E},\cdots,C_{m}^{E}$ are all simply-
connected by assumption. Applying Theorem $3.3$, we easily get this
conclusion. $\Box$

Corollary $3.6$ Let $X$, $U$, $V$, $C_{i}$, $a(x_{0},x_{i})$, $b(x_{0},x_{i})$
for integers $i,\ 1\leq i\leq m$ be as in Theorem $3.1$. If $V$ is simply-
connected, then

$\pi_{1}(X,x_{0})\cong\frac{\pi_{1}(U,x_{0})*\pi_{1}(B_{m}^{T},x_{0})}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i}^{E},x_{0})\ \right]},$

where $i_{1}^{E}:\pi_{1}(U^{E}\cap V,x_{0})\rightarrow\pi_{1}(U^{E},x_{0})$
and $i_{2}^{E}:\pi_{1}(U^{E}\cap V,x_{0})\rightarrow\pi_{1}(V,x_{0})$ are
homomorphisms induced by inclusion mappings.

§$4.$ Fundamental Groups of Combinatorial Spaces

4.1 Fundamental groups of combinatorial manifolds

By definition, a combinatorial manifold $\widetilde{M}$ is arcwise-connected.
So we can apply Theorems $3.2$ and $3.3$ to find its fundamental group
$\pi_{1}(\widetilde{M})$ up to isomorphism in this section.

Definition $4.1$ Let $\widetilde{M}$ be a combinatorial manifold underlying a
graph $G[\widetilde{M}]$. An edge-induced graph $G^{\theta}[\widetilde{M}]$ is
defined by

$V(G^{\theta}[\widetilde{M}])=\\{x_{M},x_{M^{\prime}},x_{1},x_{2},\cdots,x_{\mu(M,M^{\prime})}|\
for\ \forall(M,M^{\prime})\in E(G[\widetilde{M}])\\},$

$E(G^{\theta}[\widetilde{M}])=\\{(x_{M},x_{M^{\prime}}),(x_{M},x_{i}),(x_{M^{\prime}},x_{i})|\
1\leq i\leq\mu(M,M^{\prime})\\},$

where $\mu(M,M^{\prime})$ is called the edge-index of $(M,M^{\prime})$ with
$\mu(M,M^{\prime})+1$ equal to the number of arcwise-connected components in
$M\cap M^{\prime}$.

By the definition of edge-induced graph, we finally get
$G^{\theta}[\widetilde{M}]$ of a combinatorial manifold $\widetilde{M}$ if we
replace each edge $(M,M^{\prime})$ in $G[\widetilde{M}]$ by a subgraph
$TB_{\mu(M,M^{\prime})}^{T}$ shown in Fig.$4.1$ with $x_{M}=M$ and
$x_{M^{\prime}}=M^{\prime}$.

Fig.$4.1$

Then we have the following result.

Theorem $4.2$ Let $\widetilde{M}$ be a finitely combinatorial manifold. Then

$\pi_{1}(\widetilde{M})\cong\frac{\left(\prod\limits_{M\in
V(G[\widetilde{M}])}\pi_{1}(M)\right)*\pi_{1}(G^{\theta}[\widetilde{M}])}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{(M_{1},M_{2})\in
E(G[\widetilde{M}])}\pi_{1}(M_{1}\bigcap M_{2})\right]},$

where $i_{1}^{E}$ and $i_{2}^{E}$ are homomorphisms induced by inclusion
mappings $i_{M}:\pi_{1}(M\cap M^{\prime})\rightarrow\pi_{1}(M)$,
$i_{M^{\prime}}:\pi_{1}(M\cap M^{\prime})\rightarrow\pi_{1}(M^{\prime})$ such
as those shown in the following diagram:

for $\forall(M,M^{\prime})\in E(G[\widetilde{M}])$.

Proof This result is obvious for $|G[\widetilde{M}]|=1$. Notice that
$G^{\theta}[\widetilde{M}]=B_{\mu(M,M^{\prime})+1}^{T}$ if
$V(G[\widetilde{M}])=\\{M,\ M^{\prime}\\}$. Whence, it is an immediately
conclusion of Theorem $3.2$ for $|G[\widetilde{M}]|=2$.

Now let $k\geq 3$ be an integer. If this result is true for
$|G[\widetilde{M}]|\leq k$, we prove it hold for $|G[\widetilde{M}]|=k$. It
should be noted that for an arcwise-connected graph $H$ we can always find a
vertex $v\in V(H)$ such that $H-v$ is also arcwise-connected. Otherwise, each
vertex $v$ of $H$ is a cut vertex. There must be $|H|=1$, a contradiction.
Applying this fact to $G[\widetilde{M}]$, we choose a manifold $M\in
V(G[\widetilde{M}])$ such that $\widetilde{M}-M$ is arcwise-connected, which
is also a finitely combinatorial manifold.

Let $U=\widetilde{M}\setminus(M\setminus\widetilde{M})$ and $V=M$. By
definition, they are both opened. Applying Theorem $3.2$, we get that

$\pi_{1}(\widetilde{M})\cong\frac{\pi_{1}(\widetilde{M}-M)*\pi_{1}(M)*\pi_{1}(B_{m}^{T})}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i})\ \right]},$

where $C_{i}$ is an arcwise-connected component in $M\cap(\widetilde{M}-M)$
and

$m=\sum\limits_{(M,M^{\prime})\in E(G[\widetilde{M}])}\mu(M,M^{\prime}).$

Notice that

$\pi_{1}(B_{m}^{T})\cong\prod\limits_{(M,M^{\prime})\in
E(G[\widetilde{M}]}\pi_{1}(TB_{\mu(M,M^{\prime})}).$

By the induction assumption, we know that

$\pi_{1}(\widetilde{M}-M)\cong\frac{\left(\displaystyle\prod\limits_{M\in
V(G[\widetilde{M}-M])}\pi_{1}(M)\right)*\pi_{1}(G^{\theta}[\widetilde{M}-M])}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ \displaystyle g\in\prod\limits_{(M_{1},M_{2})\in
E(G[\widetilde{M}-M])}\pi_{1}(M_{1}\cap M_{2})\right]},$

where $i_{1}^{E}$ and $i_{2}^{E}$ are homomorphisms induced by inclusion
mappings $i_{M_{1}}:\pi_{1}(M_{1}\cap M_{2})\rightarrow\pi_{1}(M_{1})$,
$i_{M_{2}}:\pi_{1}(M_{1}\cap M_{2})\rightarrow\pi_{1}(M_{2})$ for
$\forall(M_{1},M_{2})\in E(G[\widetilde{M}-M])$. Therefore, we finally get
that

$\displaystyle\pi_{1}(\widetilde{M})$ $\displaystyle\cong$
$\displaystyle\frac{\pi_{1}(\widetilde{M}-M)*\pi_{1}(M)*\pi_{1}(B_{m}^{T})}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ \displaystyle g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i})\
\right]}$ $\displaystyle\cong$
$\displaystyle\frac{\frac{\left(\displaystyle\prod\limits_{M\in
V(G[\widetilde{M}-M])}\pi_{1}(M)\right)\displaystyle*\pi_{1}(G^{\theta}[\widetilde{M}-M])}{\left[\displaystyle(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{(M_{1},M_{2})\in
E(G[\widetilde{M}-M])}\pi_{1}(M_{1}\cap
M_{2})\right]}}{\left[\displaystyle(i_{1}^{E})^{-1}(g)\cdot i_{2}(g)|\
g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i})\ \right]}$ $\displaystyle*$
$\displaystyle\frac{\pi_{1}(M)*\displaystyle\prod\limits_{(M,M^{\prime})\in
E(G[\widetilde{M}]}\pi_{1}(TB_{\mu(M,M^{\prime})})}{\left[\displaystyle(i_{1}^{E})^{-1}(g)\cdot
i_{2}(g)|\ g\in\prod\limits_{i=1}^{m}\pi_{1}(C_{i})\ \right]}$
$\displaystyle\cong$ $\displaystyle\frac{\left(\displaystyle\prod\limits_{M\in
V(G[\widetilde{M}])}\pi_{1}(M)\right)*\pi_{1}(G^{\theta}[\widetilde{M}])}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ \displaystyle g\in\prod\limits_{(M_{1},M_{2})\in
E(G[\widetilde{M}])}\pi_{1}(M_{1}\bigcap M_{2})\right]}$

by facts

$\left(\mathscr{G}/\mathscr{H}\right)*H\cong\mathscr{G}*H/\mathscr{H}$

for groups $\mathscr{G,\ H}$, $G$ and

$G^{\theta}[\widetilde{M}]=G^{\theta}[\widetilde{M}-M]\bigcup\limits_{(M,M^{\prime})\in
E(G[\widetilde{M}]}TB_{\mu(M,M^{\prime})},$
$\pi_{1}(G^{\theta}[\widetilde{M}])=\pi_{1}(G^{\theta}[\widetilde{M}-M])*\prod\limits_{(M,M^{\prime})\in
E(G[\widetilde{M}]}\pi_{1}(TB_{\mu(M,M^{\prime})}),$ $\prod\limits_{M\in
V(G[\widetilde{M}])}\pi_{1}(M)=\left(\prod\limits_{M\in
V(G[\widetilde{M}-M])}\pi_{1}(M)\right)*\pi_{1}(M),$

where $i_{1}^{E}$ and $i_{2}^{E}$ are homomorphisms induced by inclusion
mappings $i_{M}:\pi_{1}(M\cap M^{\prime})\rightarrow\pi_{1}(M)$,
$i_{M^{\prime}}:\pi_{1}(M\cap M^{\prime})\rightarrow\pi_{1}(M^{\prime})$ for
$\forall(M,M^{\prime})\in E(G[\widetilde{M}])$. This completes the proof.
$\Box$

Applying Corollary $3.5$, we get a result known in [8] by noted that
$G^{\theta}[\widetilde{M}]=G[\widetilde{M}]$ if $\forall(M_{1},M_{2})\in
E(G^{L}[\widetilde{M}])$, $M_{1}\cap M_{2}$ is simply connected.

Corollary $4.3$([8]) Let $\widetilde{M}$ be a finitely combinatorial manifold.
If for $\forall(M_{1},M_{2})\in E(G^{L}[\widetilde{M}])$, $M_{1}\cap M_{2}$ is
simply connected, then

$\pi_{1}(\widetilde{M})\cong\left(\bigoplus\limits_{M\in
V(G[\widetilde{M}])}\pi_{1}(M)\right)\bigoplus\pi_{1}(G[\widetilde{M}]).$

4.2 Fundamental groups of manifolds

Notice that $\pi_{1}({\bf R}^{n})=identity$ for any integer $n\geq 1$. If we
choose $M\in V(G[\widetilde{M}])$ to be a chart
$(U_{\lambda},\varphi_{\lambda})$ with
$\varphi_{\lambda}:U_{\lambda}\rightarrow{\bf R}^{n}$ for $\lambda\in\Lambda$
in Theorem $4.2$, i.e., an $n$-manifold, we get the fundamental group of
$n$-manifold following.

Theorem $4.4$ Let $M$ be a compact $n$-manifold with charts
$\\{(U_{\lambda},\varphi_{\lambda})|\
\varphi_{\lambda}:U_{\lambda}\rightarrow{\bf R}^{n},\lambda\in\Lambda)\\}$.
Then

$\pi_{1}(M)\cong\frac{\pi_{1}(G^{\theta}[M])}{\left[(i_{1}^{E})^{-1}(g)\cdot
i_{2}^{E}(g)|\ g\in\prod\limits_{(U_{\mu},U_{\nu})\in
E(G[M])}\pi_{1}(U_{\mu}\cap U_{\nu})\right]},$

where $i_{1}^{E}$ and $i_{2}^{E}$ are homomorphisms induced by inclusion
mappings $i_{U_{\mu}}:\pi_{1}(U_{\mu}\cap
U_{\nu})\rightarrow\pi_{1}(U_{\mu})$, $i_{U_{\nu}}:\pi_{1}(U_{\mu}\cap
U_{\nu})\rightarrow\pi_{1}(U_{\nu})$, $\mu,\nu\in\Lambda$.

Corollary $4.5$ Let $M$ be a simply connected manifold with charts
$\\{(U_{\lambda},\varphi_{\lambda})|\
\varphi_{\lambda}:U_{\lambda}\rightarrow{\bf R}^{n},\lambda\in\Lambda)\\}$,
where $|\Lambda|<+\infty$. Then $G^{\theta}[M]=G[M]$ is a tree.

Particularly, if $U_{\mu}\cap U_{\nu}$ is simply connected for
$\forall\mu,\nu\in\Lambda$, then we obtain an interesting result following.

Corollary $4.6$ Let $M$ be a compact $n$-manifold with charts
$\\{(U_{\lambda},\varphi_{\lambda})|\
\varphi_{\lambda}:U_{\lambda}\rightarrow{\bf R}^{n},\lambda\in\Lambda)\\}$. If
$U_{\mu}\cap U_{\nu}$ is simply connected for $\forall\mu,\nu\in\Lambda$, then

$\pi_{1}(M)\cong\pi_{1}(G[M]).$

Therefore, by Theorem $4.4$ we know that the fundamental group of a manifold
$M$ is a subgroup of that of its edge-induced graph $G^{\theta}[M]$.
Particularly, if $G^{\theta}[M]=G[\widetilde{M}]$, i.e., $U_{\mu}\cap U_{\nu}$
is simply connected for $\forall\mu,\nu\in\Lambda$, then it is nothing but the
fundamental group of $G[\widetilde{M}]$.

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