text:
\section*{Acknowledgments}

We would like to thank Costas Bachas, Tresa Bautista, Antoine Bourget and Atish Dabholkar for useful discussions.
We are particularly grateful to Adel Bilal for carefully reading the draft of the manuscript.

\section{Classical two-dimensional gravity}
\label{sec:classical}

In this section we review some general aspects of (classical) $2d$ gravity coupled to some matter $\psi$.

Let $\mc M$ be a $2$-dimensional space with metric $g_{\mu\nu}$ with signature $(-+)$ and whose coordinates are denoted by~$\sigma^\mu$.
The total action of the matter fields $\psi$ coupled to $2d$ gravity is
\begin{equation}
	\label{2dgrav:action:gravity+matter}
	S[g, \psi] = S_{\text{grav}}[g] + S_m[g, \psi]
\end{equation}
where $S_{\text{grav}}$ is the action for pure gravity.
Only the case where the matter action $S_m[g, \psi]$ is obtained by making covariant the action $S_m[\eta, \psi]$ by minimal coupling ($\eta_{\mu\nu}$ being just the flat metric) is considered.
The action is required to satisfy the following criteria: renormalizability, invariance under diffeomorphisms and no more than second order derivatives.

Weyl transformations, which correspond to rescaling of the metric
\begin{equation}
	\label{2dgrav:sym:weyl}
	g_{\mu\nu} = \e^{2\omega(\sigma^\mu)} g'_{\mu\nu},
\end{equation}
will be a central element of the discussion.
In general this transformation is not a symmetry of the action.
A necessary condition for an action to be Weyl invariant is to be invariant under conformal transformations in flat space~\cite{Iorio:1997:WeylGaugingConformalInvariance}
This condition is sufficient when the action is at most quadratic in the first derivative, which is the case for the models considered in this paper: hence we assume that the action for conformal matter on curved space is Weyl invariant.

The gravitational action is given by the sum of two terms
\begin{equation}
	\label{2dgrav:action:grav}
	S_{\text{grav}} = S_{\text{EH}} + S_\mu
\end{equation}
as it is well-known that only two invariants fulfil the above conditions.
The first piece $S_{\text{EH}}$ is the Einstein–Hilbert action
\begin{equation}
	\label{2dgrav:action:einstein-hilbert}
	S_{\text{EH}}[g] = \int \dd^2 \sigma\, \sqrt{\abs{g}}\, R
		= 4\pi \chi, \qquad
	\chi = 2 - 2 h,
\end{equation}
which is a topological invariant in two dimensions and equal to the Euler number $\chi$, $h$ being the genus of the surface.
As a consequence it is not dynamical (equivalently the Einstein tensor is identically zero) and it can be ignored as long as one is not interested in topological properties (which we are not).
This action is also invariant under Weyl transformation \eqref{2dgrav:sym:weyl}.
The second allowed term is the cosmological constant\footnotemark{}
\footnotetext{The cosmological constant can be either positive or negative.
In the current convention $\mu > 0$ and $\mu < 0$ correspond respectively to anti-de Sitter and de Sitter.}
\begin{equation}
	\label{2dgrav:action:cosmological}
	S_\mu[g] = \mu \int \dd^2 \sigma\, \sqrt{\abs{g}} = \mu\, A[g]
\end{equation}
where $A$ is the area of $\mc M$ associated to the metric $g$.
It is not invariant under Weyl transformations \eqref{2dgrav:sym:weyl}.
The presence of this term has dramatic consequences in two dimensions, as will be exemplified in \cref{sec:dofs,sec:dynamics}.
In particular in the case of pure gravity the equation of motion reduces to $\mu = 0$ and it has no solution since $\mu$ is a fixed parameter of the model.

The classical equations of motion are given by varying the full action \eqref{2dgrav:action:gravity+matter} with respect to $g_{\mu\nu}$ and $\psi$
\begin{equation}
	\frac{\var S}{\var g^{\mu\nu}} = 0, \qquad
	\frac{\var S}{\var \psi} = 0.
\end{equation}
Without specifying the action for the matter it is not possible to go further with the second equation.
Nonetheless the first equation is dictating a lot of properties of the system and is responsible for its subtleties.

The energy--momentum tensors associated to $S$ and $S_m$ are defined by
\begin{equation}
	\label{2dgrav:eq:energy-tensor}
	T_{\mu\nu} = - \frac{4\pi}{\sqrt{\abs{g}}} \frac{\var S}{\var g^{\mu\nu}}, \qquad
	T^{(m)}_{\mu\nu} = - \frac{4\pi}{\sqrt{\abs{g}}} \frac{\var S_m}{\var g^{\mu\nu}}.
\end{equation}
Then the metric equation of motion implies that the total energy--momentum tensor \eqref{2dgrav:eq:energy-tensor} vanishes
\begin{equation}
	\label{2dgrav:eom:gravity+matter:g}
	T_{\mu\nu} = T^{(m)}_{\mu\nu} + 2\pi \mu\, g_{\mu\nu}
		= 0.
\end{equation}
These three independent equations provide constraints on the metric and matter fields since they do not contain derivatives of the metric (see section~\ref{sec:dofs} for details on the counting of degrees of freedom).
It is convenient to decompose the energy--momentum tensor into its trace $T$ and its traceless components $\bar T_{\mu\nu}$
\begin{equation}
	\bar T_{\mu\nu} \equiv T_{\mu\nu} - \frac{1}{2}\; T\, g_{\mu\nu}, \qquad
	T \equiv g^{\mu\nu} T_{\mu\nu}.
\end{equation}
In terms of these variables the equation of motion \eqref{2dgrav:eom:gravity+matter:g} reads
\begin{equation}
	\label{2dgrav:eom:gravity+matter:g:Tbar+trace}
	T = T^{(m)} + 4\pi \mu
		= 0, \qquad
	\bar T_{\mu\nu} = \bar T^{(m)}_{\mu\nu}
		= 0.
\end{equation}
An obvious advantage is that the cosmological constant has been decoupled from the traceless tensor, and more generally any matter potential that does not depend on the metric does not appear in $\bar T_{\mu\nu}$.

If $S_m[\eta, \psi]$ is conformally invariant, then the action $S_m[g, \psi]$ is Weyl invariant.
In this case its energy--momentum tensor \eqref{2dgrav:eq:energy-tensor} is traceless $T^{(m)} = 0$.
Then the trace of $T_{\mu\nu}$ reduces to the cosmological constant and gives the equation of motion $\mu = 0$.
This equation has no solution and solutions for conformally invariant matter coupled to gravity can exist only in the absence of a cosmological constant.
Then the equation of motion without cosmological constant is simply $T^{(m)}_{\mu\nu} = 0$ which provides two independent equations because it is symmetric and traceless.

\section{Degrees of freedom}
\label{sec:dofs}

In this section we provide a counting of the degrees of freedom for a general model linear in the inverse metric.
As we will show below, in this case the equations of motion for the metric are Weyl invariant even if the action is not, and as a consequence there are more constraints \emph{without} the Weyl symmetry than with it.
At the end we comment the general case.

In two dimensions the naive counting of on-shell degrees of freedom gives $-1$ because the system is over-constrained.
In general a gauge invariance leads to two constraints, one from the gauge fixing condition and one from the equation of motion ensuring that the first one is preserved in time.
In the present case the equations of motion are trivial and thus the second kind of constraint does not exist and then diffeomorphisms removes two of the three components of the metric.
Moreover we have seen that gravity with a cosmological constant is inconsistent.
Once this term is removed the action is invariant under Weyl transformations which allows to fix the last component and there is no on-shell degree of freedom.

When matter is present gravity reduces the number of degrees of freedom in $2$ dimensions, as already illustrated above and as we will now discuss for general matter fields $\psi$ with a total of $N$ on-shell degrees of freedom (before coupling to gravity).
The counting may seem useless in view of the absence of dynamics discussed in \cref{sec:dynamics} but it applies to more general situations (such as non-unitary matter).

Since the precise counting is sensitive to the form of the action we will restrict our attention to the simpler case when the matter Lagrangian is linear in the inverse metric (in particular it excludes couplings of the form $R \psi$).
\begin{equation}
	S_m = \frac{1}{2\pi} \int \dd^2 \sigma\, \sqrt{\abs{g}}\, \mc L, \qquad
	\mc L = - \frac{1}{2} \big( g^{\mu\nu} \mc L_{\mu\nu}(\psi) + V(\psi) \big).
\end{equation}
The first term is Weyl invariant because the transformation of $g^{\mu\nu}$ cancels the one of $\sqrt{\abs{g}}$.
The traceless and trace components of the energy--momentum tensor are
\begin{equation}
	\label{2dgrav:eom:linear:g}
	\bar T_{\mu\nu} = \mc L_{\mu\nu} - \frac{1}{2}\, g_{\mu\nu} \big( g^{\alpha\beta} \mc L_{\alpha\beta} \big), \qquad
	T = - V + 4\pi\mu.
\end{equation}
It can be seen that the traceless tensor is the energy--momentum tensor corresponding to the action $S_m$ where all the parameters breaking Weyl invariance have been set to zero.
As a consequence no dynamical component of the metric appears inside and the equation $\bar T_{\mu\nu} = 0$ gives two constraints on the matter.
The trace is also Weyl invariant since it does not contain the metric and it provides another constraint for the matter.
In total the number of on-shell degrees of freedom is reduced by three, giving $N - 3$.
After using the diffeomorphisms, the metric has still one off-shell degree of freedom which should be fixed by the equations of motion of the scalar fields since they are not Weyl invariant.

If the matter action is Weyl invariant, then the trace equation is removed and there is one constraint less, giving $N - 2$ on-shell degrees of freedom.
As in the case of pure gravity the last off-shell metric component is fixed by a Weyl transformation.

The conclusion is that even if Weyl invariance is not a symmetry of the action it is a symmetry of the equations of motion for $g_{\mu\nu}$.
This is similar to the electric-magnetic duality except that the matter equations of motion are breaking this duality.
It might be surprising to have a system where adding a gauge symmetry \emph{increases} the number of degrees of freedom, whereas the usual lore is that a gauge symmetry describes a redundancy.
This behaviour is very peculiar to two-dimensional gravity because gravity provides constraints together with degrees of freedom.
The metric components act as Lagrange multipliers and the Weyl symmetry removes one of those, which implies that one less constraint will be imposed.

\section{Dynamics of unitary matter}
\label{sec:dynamics}

The goal of this section is to show that a large class of models of unitary matter coupled to gravity has no dynamics.
It is well-known that the energy--momentum tensor can be rewritten as a sum of squares for free scalar fields, which implies that each term vanishes independently.
There is a difference between conformal and non-conformal matter since the cosmological constant and the additional potential may provide a negative term in $T_{\mu\nu}$.
In fact, using the decomposition \eqref{2dgrav:eom:gravity+matter:g:Tbar+trace}, the equations in both cases can be brought into a similar form and in some cases the behaviour of systems with a cosmological constant is even worse.

We will illustrate this using $N$ scalar fields $X_i$ with a potential $V(X_i)$
\begin{equation}
	S_m = - \frac{1}{4\pi} \int \dd^2 \sigma\, \sqrt{\abs{g}}\, \left( g^{\mu\nu} \pd_\mu X_i \pd_\nu X_i + V(X_i) \right), \qquad
	i = 1, \ldots, N,
\end{equation}
where the sum over the index $i$ is implicit and some of the $X_i$ may not appear in the potential.
This potential does not contain any constant term which would just correspond to a shift of the cosmological constant.
The equations of motion for the metric and the matter are
\begin{equation}
	\label{2dgrav:eom:unitary-model}
	T_{\mu\nu} = 0, \qquad
	- \lap X_i + \frac{1}{2} \frac{\pd V}{\pd X_i} = 0
\end{equation}
where $\lap$ is the curved space Laplacian for the metric $g$
\begin{equation}
	\label{conv:eq:laplacian}
	\lap = g^{\mu\nu} \grad_\mu \grad_\nu
		= \frac{1}{\sqrt{\abs{g}}}\, \big( \pd_\mu \sqrt{\abs{g}} g^{\mu\nu} \pd_\nu \big).
\end{equation}

Then the trace and the traceless part of the energy--momentum tensor read
\begin{equation}
	\label{2dgrav:eom:unitary-trace-Tbar}
	T = - V + 4\pi \mu, \qquad
	\bar T_{\mu\nu} = \pd_\mu X_i \pd_\nu X_i - \frac{1}{2} g_{\mu\nu} (g^{\alpha\beta} \pd_\alpha X_i \pd_\beta X_i).
\end{equation}
One can see that $\bar T_{\mu\nu}$ looks like the energy--momentum tensor of $N$ free scalar fields and one can use the usual strategy to solve the associated equation.
First one uses the diffeomorphisms to write the metric in the flat conformal gauge
\begin{equation}
	g_{\mu\nu} = \e^{2\phi} \eta_{\mu\nu}.
\end{equation}
Writing explicitly the components of $\bar T_{\mu\nu}$ one arrives at the equation
\begin{equation}
	2 (\bar T_{00} \pm \bar T_{01}) = (\pd_0 X_i \pm \pd_1 X_i)^2
		= 0.
\end{equation}
Since this is a sum of squares the only solution is the one where all terms vanish
\begin{equation}
	(\pd_0 \pm \pd_1) X_i = 0
	\quad \Longrightarrow \quad
	\pd_\mu X_i = 0
	\quad \Longrightarrow \quad
	X_i = X_i^0 = \cst.
\end{equation}
In the conformal gauge the matter equation \eqref{2dgrav:eom:unitary-model} and the trace equation give the following constraints on the values of $X^i_0$
\begin{equation}
	\label{2dgrav:eq:unitary:constraint-X0}
	\frac{\pd V}{\pd X_i}(X_i^0) = 0, \qquad
	V(X_i^0) = 4 \pi \mu.
\end{equation}
It can be noted that no equation allows to fix the value of $\phi$.

The previous computation shows that unitary matter coupled to $2d$ gravity does not have any dynamics since the solutions of the equations of motion is given by the trivial solution $X_i = \cst$ (with some constraints), even in the presence of interactions (if the potential does not contain the metric).
The constraints provided by the vanishing of the trace are too strong and kill all the dynamics (this is not the case for matter without gravity, whether the backgroud is curved or not).
Note that this argument does not apply to the $X^\mu$ fields of bosonic string theory since $X^0$ is a timelike boson.

Finally in order to give a specific example and to stress the difference between conformal and non-conformal matter, we consider free massive scalar fields
\begin{equation}
	V(X_i) = \sum_i m_i^2 X_i^2
\end{equation}
where some of the masses can vanish.
Then the first equation of \eqref{2dgrav:eq:unitary:constraint-X0} implies $X^0_i = 0$ if $m_i \neq 0$.
Introducing this into the second equation of \eqref{2dgrav:eq:unitary:constraint-X0} gives $\mu = 0$ and there is no solution at all.
Hence the behaviour is worse if one includes a cosmological constant and non-conformal matter fields since the system of equations is not even consistent, while for conformal matter there is at least the trivial solution $X_i = \cst$ (with the obvious condition that one did not include the cosmological constant).

\section{Introduction}

Two-dimensional gravity has been a toy model for four-dimensional gravity since several decades: it presents many simplifications while -- hopefully -- retaining some characteristic features.
Moreover two-dimensional gravity appears in the worldsheet formulation of string theory which provides a second incentive to study it.

Classical gravity is generically absent from the references about the previous two topics since it is deemed to be trivial and one is more interested in the quantum effects -- in particular when the matter is conformal~\cite{Polyakov:1981:QuantumGeometryBosonic}, which is the natural stage for string theory.
Nonetheless the classical theory presents some peculiarities that have not been discussed elsewhere (to our knowledge) and we believe that it is useful to be aware of those.
In particular one often blames theories for which the semi-classical limit of the quantized theory does not reproduce the classical theory, but one should recognize that two-dimensional gravity is such a theory.\footnotemark{}
\footnotetext{%
	The simplest trait of this being that renormalization introduces a cosmological constant even if the latter is forbidden at the classical level when the matter is conformal.
}

The goal of this paper is to discuss the degrees of freedom and the dynamics of two-dimensional gravity coupled to conformal and non-conformal matter.
In the latter case gravity displays some particularities that one does not find in higher dimensions -- and this is an important point to keep in mind when comparing two and four dimensions.
More specifically we show that for a wide class of Lagrangians there are \emph{less} degrees of freedom when the matter is massive (no Weyl symmetry).
This is a consequence of the invariance of the equations of motion for the metric under the Weyl symmetry, even if the action itself is not invariant.
The expected modifications due to the quantum dynamics are briefly commented.
Then we discuss the fact that unitary matter coupled to gravity does not generically admit dynamics, using a set of scalar fields with an arbitrary (metric-independent) potential as an example.
At best only a trivial solution is possible while in some cases there is no solution at all (in similarity with the well-known example of pure gravity with cosmological constant).

The paper is organized as follows.
In \cref{sec:classical} we describe the coupling of conformal and non-conformal matter to classical two-dimensional gravity.
Then in \cref{sec:dofs} we count the degrees of freedom.
Finally in \cref{sec:dynamics} we study the dynamics for a general model with scalar fields.
In \cref{app:quantum-gravity} we make the connection with the quantum theory.
\Cref{app:chiral-anomaly} compares this discussion with the more familiar setting of a $\group{U}(1)$ chiral gauge theory.

\section{Analogy: four-dimensional gauge anomaly}
\label{app:chiral-anomaly}

There exists a full analogy between the description of $2d$ gravity and $4d$ chiral gauge theory, for example one finds the emergence of a gauge symmetry (respectively Weyl and $\group{U}(1)$) through the choice of a convenient parametrization and the appearance of a Wess--Zumino action.

\subsection{Massive vector field}

Let's consider a massive vector field $\mc A_\mu$ (playing the role of $g_{\mu\nu}$) with Proca action (in $d$ dimensions)
\begin{equation}
	\label{chiral:action:proca}
	S_A = - \int \dd^d x \left(\frac{1}{4}\, \mc F_{\mu\nu} \mc F^{\mu\nu} + \frac{m^2}{2} \mc A_\mu \mc A^\mu \right).
\end{equation}
If $m^2 = 0$ then it enjoys a $\group{U}(1)$ gauge symmetry
\begin{equation}
	\label{chiral:sym:U1}
	\mc A_\mu = \mc A'_\mu + \pd_\mu \alpha,
\end{equation}
which reduces the $d$ components to $(d - 1)$ off-shell dofs, and furthermore to $(d - 2)$ on-shell dofs.
If $m^2 \neq 0$ then there is one (on-shell) constraint and thus there are only $(d-1)$ on-shell degrees of freedom.
In full similarity with $2d$ gravity matter terms can be invariant under this $\group{U}(1)$ or not.

It is convenient to adopt another parametrization (called Stückelberg) for the vector field where the spin $0$ component is separated from the spin $1$ component
\begin{equation}
	\label{chiral:eq:param-A}
	\mc A_\mu = A_\mu + \pd_\mu a
\end{equation}
where $A_\mu$ and $a$ (called the axion) play respectively the roles of $g_{0\mu\nu}$ and $\phi$ (from the point of view of Lorentz representations, the trace of $g_{\mu\nu}$ is similar to the divergence of $\mc A_\mu$), the only difference being that $A_\mu$ is dynamical.
In these variables the action reads
\begin{equation}
	\label{chiral:action:stuckelberg}
	S_A = - \int \dd^d x \left(\frac{1}{4}\, F_{\mu\nu} F^{\mu\nu} + \frac{m^2}{2} (A_\mu + \pd_\mu a) (A^\mu + \pd^\mu a) \right).
\end{equation}
In order to avoid introducing an additional degree of freedom ($A_\mu$ has $d$ components and $a$ has $1$) this system should be invariant under an emergent $\group{U}(1)_e$
\begin{equation}
	\label{chiral:sym:U1-emergent}
	A_\mu = A'_\mu + \pd_\mu \alpha, \qquad
	a = a' - \alpha.
\end{equation}
Note that this symmetry is not fundamental and is just a consequence of the parametrization \eqref{chiral:eq:param-A} that has been chosen.
The interpretation is slightly different from $2d$ gravity since another gauge symmetry (diffeomorphisms) is used to gauge fix the metric, whereas in the current setup \eqref{chiral:eq:param-A} is a parametrization.

\subsection{Effective action}

For the rest of this section we focus on $d = 4$.
The action for the system is given by
\begin{equation}
	S[\mc A, \psi] = S_A[\mc A] + S_f[\mc A, \psi]
\end{equation}
where $S_A$ is Proca action \eqref{chiral:action:proca} and $S_f$ is the action for a chiral fermion $\psi$ (the dependence on the conjugate $\bar\psi$ is implicit everywhere)
\begin{equation}
	S_f = \int \dd^4 x\, i \bar \psi \slashed \dc \psi, \qquad
	\dc_\mu = \pd_\mu - i g A_\mu.
\end{equation}
This model is discussed for example in~\cite{Preskill:1991:GaugeAnomaliesEffective}.
Since the fermion is chiral it obeys the relations $\psi = \gamma_5 \psi = P_L \psi$ where $P_L$ is the projector on left-handed chirality.
The partition function of the system is
\begin{equation}
	\label{chiral:partition:matter}
	Z = \int \dd \mc A_\mu\, \e^{- S_A[\mc A]}\, Z_f[\mc A], \qquad
	Z_f[\mc A] = \int \dd\psi \, \e^{- S_f[\mc A, \psi]}
\end{equation}
where $Z_f$ is the fermion partition function.
The terms $S_A$ and $Z_f$ plays respectively the roles of $S_\mu$ and $Z_m$ in \eqref{quant:partition:gravity}.

In terms of the parametrization \eqref{chiral:eq:param-A} the partition function becomes (we ignore the ghost action since it decouples)
\begin{equation}
	Z = \int \dd A_\mu \dd a\, \e^{- S_A[A, a] - S_{\text{WZ}}[A, a]}\, Z_f[A]
\end{equation}
where the Wess--Zumino (WZ) action is
\begin{equation}
	S_{\text{WZ}}[A, a] = - \ln \frac{Z_f[A + \pd a]}{Z_f[A]}.
\end{equation}
The main difference with $2d$ gravity is that $\mc A_\mu$ is dynamical and it does not drop from the path integral, but this does not modify the general argument.

While the full action is not invariant $\group{U}(1)$ if $m^2 \neq 0$, the fermion action is invariant under the transformation $\psi = \e^{i g \alpha(x)} \psi'$ together with \eqref{chiral:sym:U1}, and the associated current $J^\mu = \bar\psi \gamma^\mu \psi$ is conserved classically.
On the other hand there is a gauge anomaly due to the chirality of the theory and the current is not conserved quantum mechanically
\begin{equation}
	\label{chiral:eq:quantum-anomaly}

	\pd_\mu \mean{J^\mu} =  \frac{g^3}{48 \pi^2}\, \mc F_{\mu\nu} \widetilde{\mc F}^{\mu\nu}, \qquad
	\widetilde{\mc F}^{\mu\nu} = \frac{1}{2}\, \levi{^{\mu\nu\rho\sigma}} \mc F_{\rho\sigma}.
\end{equation}
This can be used to determine the $S_{\text{WZ}}$ in full similarity with the derivation of the Liouville action \eqref{quant:action:liouville}.
The quantum current is the variation of the effective action \eqref{chiral:partition:matter} with respect to the gauge field
\begin{equation}
	\label{chiral:eq:quantum-current}
	\mean{J^\mu} = - \frac{\var}{\var \mc A_\mu} \ln Z_f[\mc A].
\end{equation}
Then one can vary from $\mc A_\mu$ to $A_\mu$ continuously by parametrizing $\mc A_\mu = A_\mu + \pd_\mu \alpha$.
From the variation $\var \mc A_\mu = \pd_\mu \alpha$, the relation \eqref{chiral:eq:quantum-current} can be integrated from $\alpha = 0$ to $\alpha = a$ which results in (after an integration by part)
\begin{equation}
	S_{\text{WZ}}[A, a] = - \int \dd^4 x\, a\, \pd_\mu \mean{J^\mu}
		= - \frac{g^3}{48 \pi^2}\ \int \dd^4 x\, a \, F_{\mu\nu} \widetilde{F}^{\mu\nu},
\end{equation}
where the expression \eqref{chiral:eq:quantum-anomaly} and the identity $\mc F_{\mu\nu} = F_{\mu\nu}$ have been used.
This computation is possible because the mass term is outside the matter path integral which is effectively gauge invariant (classically), and in particular the anomalous contribution comes from the fermion measure (which does not depend on the mass).
This mirrors $2d$ gravity where the cosmological constant lies outside the matter path integral, the latter being Weyl invariant.

The total action in the parametrization \eqref{chiral:eq:param-A} reads
\begin{equation}
	S[A, a, \psi] = S_A[A, a] + S_{\text{WZ}}[A, a] + S_f[A, \psi].
\end{equation}
This action should be invariant under the emergent $\group{U}(1)_e$; in particular this implies that the variation of the Wess--Zumino (WZ) term $S_{\text{WZ}}$ is exactly the one necessary for cancelling the gauge anomaly of the fermion action \emph{in terms of the field} $A_\mu$ (the anomaly related to $\group{U}(1)$ is still present).
Note also that the fermions and the axion are not coupled to each other, in the same way that the matter and the Liouville mode do not couple in $2d$ gravity.

Another point where the analysis differs from $2d$ gravity is in the effect of the anomaly-generated term: the latter does not provide dynamics for the axion and the kinetic term for the latter comes from the mass term.
On the other hand one cannot start from $m^2 = 0$ since such a mass term would be generated at $1$-loop from the cubic WZ vertex and the tree mass term is necessary to remove the divergence.
A similar story holds for the cosmological constant in $2d$ gravity.

\section{Quantum two-dimensional gravity}
\label{app:quantum-gravity}

In this appendix we make contact between the classical and quantum regimes of gravity for the counting of the degrees of freedom.

The metric becomes dynamical due to quantum effects.
It is convenient to write the metric in the conformal gauge using the diffeomorphisms
\begin{equation}
	\label{2dgrav:eq:conformal-gauge}
	g = \e^{2\phi} g_0
\end{equation}
where $\phi$ is the Liouville mode and $g_0$ is a fixed background metric.
Indeed in this gauge the dynamics of the metric and of the matter decouples and both sectors are mixed only through constraints.
Since the physical metric is left invariant under the transformation
\begin{equation}
	\label{2dgrav:sym:emerging-weyl}
	g_0 = \e^{2\omega} g'_0, \qquad
	\phi = \phi' - \omega,
\end{equation}
it means that the system presents an "emerging" Weyl symmetry which is necessary to ensure that there is only one off-shell degree of freedom (three from $g_0$ and one from $\phi$, minus two from diffeomorphisms and one from this emerging symmetry).
Note that this emerging Weyl symmetry is not fundamental since it is very specific to the conformal gauge, at the opposite of Weyl symmetry \eqref{2dgrav:sym:weyl}: this last symmetry (when it exists) can be used in any gauge and truly reduce the total number of off-shell degrees of freedom, whereas the emerging Weyl is here only not to spoil the counting due to the redundant notation.
In particular any action should have this invariance in order to be background independent.

The total partition function is
\begin{equation}
	\label{quant:partition:gravity}
	Z = \int \dd_g g_{\mu\nu}\, \e^{- S_\mu[g]} Z_m|g], \qquad
	Z_m[g] = \int \dd_g \psi\, \e^{- S_m[g, \psi]}
\end{equation}
where $Z_m$ is the partition function of the matter and the index $g$ on the measure indicates that it depends on the metric.
Upon gauge fixing to \eqref{2dgrav:eq:conformal-gauge} one writes the partition function as (we ignore the ghosts since they can be dropped inside the matter and ultimately we are only interested in classical solutions for which they vanish)
\begin{equation}
	Z = \int \dd_g \phi\, \e^{- S_\mu[g_0, \phi] - S_{\text{grav}}[g_0, \phi]} Z_m|g_0]
\end{equation}
where
\begin{equation}
	\label{quant:action:grav}
	S_{\text{grav}}[g_0, \phi] = - \ln \frac{Z_m[\e^{2\phi} g_0]}{Z_m[g_0]}.
\end{equation}
where $S_{\text{grav}}[g_0, \phi]$ is the Wess--Zumino effective action coming from changing the metric in the matter partition function from $g$ to $g_0$.
Typically the leading terms are the Liouville action and the Mabuchi action~\cite{Ferrari:2012:GravitationalActionsTwo, Bilal:2016:2DQuantumGravity}.
The total action is then
\begin{equation}
	\label{quant:eq:quant-action}
	S_*[g_0, \phi, \psi] = S_\mu[g_0, \phi] + S_{\text{grav}}[g_0, \phi] + S_m[g_0, \psi].
\end{equation}

When the matter is conformal the gravitational action can be computed by parametrizing the metric as $g = \e^{2\omega} g_0$ and by integrating the trace of the quantum energy--momentum tensor over $\omega$ from $0$ to $\phi$.
The latter is given by the Weyl anomaly (quantities with a subscript $0$ are given in terms of the metric $g_0$)
\begin{equation}
	\mean{T^m} = \frac{2\pi}{\sqrt{g}} \frac{\var}{\var \omega} \ln Z_m[g]
		= - \frac{c_m}{12}\, R.
\end{equation}
Using the relation $R = R_0 - 2 \lap_0 \omega$ and integrating, the action \eqref{quant:action:grav} becomes the Liouville action
\begin{equation}
	\label{quant:action:liouville}
	S_{\text{grav}} = - \frac{c_m}{24\pi} \int \dd^2 \sigma\, \sqrt{g_0} \big(g_0^{\mu\nu} \pd_\mu \phi \pd_\nu \phi + R_0 \phi \big)
\end{equation}
(remember that the ghosts are included into the matter: they have $c_g = - 26$ and thus the coefficient of the action is positive as long as the central charge of the rest of the matter is less than $26$).

Studying the action \eqref{quant:eq:quant-action} as a classical action with background metric $g_0$ will give informations about the semi-classical properties of the theory.
In section~\ref{sec:dynamics} we have argued that classical $2d$ gravity with unitary matter is trivial, but this does not imply that the quantum behaviour is also trivial and that forbidden classical systems have no quantum dynamics.
As a specific example of this fact, consider Liouville theory: classically the cosmological constant is forbidden, but it is necessary to include it in order to compute the path integral (in any case it is a consistent counter-term and it will appear anyway).
In particular the semi-classical limit does not resemble the classical gravity one started from: the quantum effects are not negligible in this limit and make the dynamics non-trivial.

Invariance under the emergent Weyl symmetry \eqref{2dgrav:sym:emerging-weyl} links the trace of the energy--momentum tensor $T^*_{0\mu\nu}$ to the variation of the full action with respect to the Liouville mode
\begin{equation}
	\label{quant:eq:trace-energy-tensor-eom-phi}
	T_0^* = - \frac{1}{2}\, \frac{\var S_*}{\var \phi}.
\end{equation}

Another requirement is the one of background independence: the full action \eqref{quant:eq:quant-action} should be independent of the gauge choice \eqref{2dgrav:eq:conformal-gauge}, i.e.\ it should not depend on the background metric $g_0$ or on the specific decomposition.
This is true in particular if the variation of the total action with respect to $g_0$ vanishes, which implies that the full stress–energy tensor is zero
\begin{equation}
	\label{quant:eom:vanishing-quantum-T}
	T^*_{0\mu\nu}
		= 0.
\end{equation}
From \eqref{quant:eq:trace-energy-tensor-eom-phi} the trace equation is automatically satisfied if $\phi$ satisfies its equation of motion: as a consequence \eqref{quant:eom:vanishing-quantum-T} provides two constraints on the matter and this number does not depend on whether the matter is conformal or massive.

Finally the variations of \eqref{quant:eq:quant-action} give decoupled equations of motion for $\phi$ and $\psi$.
As a conclusion one sees that quantum effects have given dynamics to the Liouville field and there is a total of $N - 1$ degrees of freedom (matter plus gravity).


text:
\section{\large Supplemental Material}

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\subsection{A. The method of projector quantum Monte Carlo}
We employ the method of projector QMC to investigate the ground-state properties of the SSH phonon model at finite frequency described in \Eq{EqOriginalModel} and the effective Hamiltonian described in \Eq{EqHopSquare} in the anti-adiabatic limit. The algorithm is based on the principle that the ground-state expectation value of operator $\hat O$ in the exact ground state $\ket{\psi_G}$ can be computed exactly via projecting a trial wave function $\ket{\psi_T}$ along the imaginary time axis
\begin{equation}\label{EqPQMCexpectation}
\langle\hat{O}\rangle=\frac{\langle\psi_G|\hat{O}|\psi_G\rangle}{\langle\psi_G|\psi_G\rangle} =\lim_{\Theta\rightarrow\infty}\frac{\langle\psi_T|\E{-\Theta H}\hat{O}\E{-\Theta H}|\psi_T\rangle}{\expectation{\psi_T}{\E{-2\Theta H}}{\psi_T}}
\end{equation}
as long as the trial wave function $\ket{\psi_T}$ has a finite overlap with the true ground state $\ket{\psi_G}$.
The algorithm is intrinsically unbiased against the choice of $\ket{\psi_T}$ assuming that the trial wave function is not orthogonal to the true ground state, namely $\braket{\psi_T}{\psi_G}\ne 0$, which is generically satisfied for a quantum many-body system with finite size. In this paper, we choose $\ket{\psi_T}$ to be the ground-state wave function of the non-interacting part of the model under consideration. In practical QMC simulations, we set the projection parameter $\Theta$ to a finite but sufficiently large value so that the expectation value $\langle \hat O\rangle$ converges when larger $\Theta$ is considered. We set $\Theta=34/t, 36/t, 38/t,46/t,50/t$ for $L=6,8,10,12,14$ accordingly, each of which has been checked to be large enough for convergence. Similar to finite temperature algorithm, Trotter decomposition is implemented by discretizing $\Theta$ in spacing $\Delta\tau=\Theta/L_\tau$. The Trotter error scales as $\Delta\tau^2$. In this work we choose $\Delta\tau=0.1/t$. The convergence of the discretization has also been checked by comparing results with smaller $\Delta\tau$.

The SSH model in \Eq{EqOriginalModel} becomes quadratic in electron operators for a specific space-time phonon configuration. We compute the electron's Green's function for each phonon configuration, and sample the phonon configuration by Monte Carlo. As for the effective model in the anti-adiabatic limit in \Eq{EqHopSquare}, auxiliary fields are introduced into the Hamiltonian via a SU(2) symmetric Hubbard-Stratonovich decomposition \cite{Assaadnote}. The AF and VBS order parameters for a finite system size $N=L^2$ are given by
\begin{align}
\hat{O}_{\mathrm{AF}}\inc{\v{q}}&=\frac{1}{L^2}\sum_j \E{\hspace{1pt}\mathrm{i}\hspace{1pt} \v{q}\cdot \v{R}_j} \hat{S}_j^z \label{EqAFMorder}\\
\hat{O}_{\mathrm{VBS}}\inc{\v{q}}&=\frac{1}{L^2}\sum_j \E{\hspace{1pt}\mathrm{i}\hspace{1pt} \v{q}\cdot \v{R}_j} \inc{\hat{B}_{j,\hat{x}}+\hspace{1pt}\mathrm{i}\hspace{1pt} \hat{B}_{j,\hat{y}}} \label{EqVBSorder}
\end{align}
where $\hat{B}_{j,\delta}= c^\dagger_{j,{\sigma}}c_{j+\delta,{\sigma}}+h.c.$ is the kinetic operator on $\delta=\hat{x}, \hat{y}$ bonds. VBS breaks the lattice $\mathbb{Z}_4$ symmetry while AF breaks spin SU(2) rotational symmetry. The structure factors for each order parameter with momentum $\v{Q}$ is defined as $S(\v{Q})=\langle |\hat{O}(\v{Q})|^2\rangle$. For both AF and staggered VBS, the peaked momentum is $\v{Q}=\inc{\pi,\pi}$. The AF susceptibility can be computed as an integration over the imaginary-time interval $\tau_M$
\begin{equation}\label{EqSusceptibilityDef}
\chi_{\mathrm{AFM}}\inc{\v{q}}= \int_{\Theta-\tau_M/2}^{\Theta+\tau_M/2} \mathrm{d}\tau\,\langle\hat{O}_{\mathrm{AFM}} \inc{\v{q},\tau}\hat{O}_{\mathrm{AFM}}\inc{\v{q},0}\rangle.
\end{equation}
To compute $\chi_{\mathrm{AF}}\inc{\v{q}}$, the imaginary-time interval $\tau_M$ should be large enough such that $\chi$ is converged, while the correlators $\langle \hat{O}\inc{\v{q},\tau}\hat{O}\inc{\v{q},0}\rangle$ should be computed after sufficiently long projection. Thus, in practice we choose $\tau_M\approx\Theta/4$ and integrate over the correlators from $\Theta-\tau_M/2$ to $\Theta+\tau_M/2$.

\subsection{B. Notes on finite-size analysis}
In the main text, we performed the finite-size scaling to obtain the quantities in the thermodynamic limit. The scaling function we usually use is $C(L)=C(L\rightarrow\infty)+a/L^b$, where $C(L)$ stands for the quantity such as AF order parameter and spin gap computed for system size $L$. We estimate the order parameter in finite lattice as the square root of structure factor $M_\text{AF}=\sqrt{3\,S_\text{AF}(\pi,\pi)}$, where factor 3 comes from the equal contribution of $x,y,z$ components of the spin structure factor. In \Fig{figJAFMorder}, we presented the result of AF order parameters as a function of $\lambda$ in the anti-adiabatic limit. For $\lambda\approx 0.25$ and finite phonon frequency ($0<\omega<\infty$), the extrapolated AF order as a function of $\omega$ is shown in \Fig{FigOrderParameterg14}. The value at $t/\omega=0$ comes from the computation at anti-adiabatic limit. The critical frequency $\omega_c$ discussed in the main text is the phonon frequency where $M_\text{AF}$ vanishes.

\begin{figure}[t]
\centering
\includegraphics[width=3.265in] {AFM_order_parameter_vs_frequency_g=14.png}
\caption{The QMC-computed AF order parameter as a function of inverse phonon frequency at $\lambda\approx 0.25$ ($g=1.4$). The red dashed line represents value of corresponding critical frequency discussed in the main text.}
\label{FigOrderParameterg14}
\end{figure}

We compute spin gap $\Delta_s$ according to the asymptotic scaling behavior of time-dependent spin-spin correlation: $\avg{S^z(\v{Q},\tau)S^z(\v{Q},0)}\sim\E{-\Delta_s\tau}$, where $\v{Q}=\inc{\pi,\pi}$. The spin gap $\Delta_s(L)$ for different lattice size $L$ can be extracted by fitting the correlation when $\tau$ is sufficiently large. Then, $\Delta(L)$ is extrapolated to $L\to \infty$ via a power-law scaling function $\Delta(L)=\Delta+a/L^b$ to obtain the spin gap in the thermodynamic limit.

\subsection{C. Exact solution in the adiabatic limit}
In the adiabatic limit ($\omega=0$), phonon displacements $X_{ij}$ become classical quantities without quantum dynamics.  At zero-temperature, the phonon displacements are static and the fermions are described by a quadratic Hamiltonian depending on the phonon displacement $X=\{X_{ij}\}$. Consequently, an exact solution of the ground state of the SSH model in the adiabatic limit can be obtained in the variational sense. In the adiabatic limit, the SSH model \Eq{EqOriginalModel} is reduced to the following quadratic form depending on the phonon configuration $X$:
\begin{equation}\label{EqMeanfieldHamiltonian}
H\Inc{X}=\sum_{i,\delta}\left[ \frac{K}{2}X_{i,\delta}^2 +\inc{gX_{i,\delta}-t}(c^\dagger_{i,{\sigma}}c_{i+\delta,{\sigma}}+h.c.)\right],
\end{equation}
where $\delta=x,y$ refers to unit vectors on $x$ or $y$ directions and $X_{i,\delta}=X_{i,i+\delta}$ is the phonon displacement on the NN bond $\avg{i,i+\delta}$. The variational method is applied here by minimizing the ground-state energy, which yields the following self-consistent equation: $g\avg{(c^\dagger_{i,{\sigma}}c_{i+\delta,{\sigma}}+h.c.)}= -K X_{i,\delta}$.

Here we compute the energy assuming various different ansatz of the phonon displacement configuration $X$, including staggered, staircase, columnar, and plaquette VBS patterns. The ordering vector for staggered and staircase patterns is $(\pi,\pi)$ while the ordering vector of columnar and plaquette patterns is $(\pi,0)$ or $(0,\pi)$. Since the Fermi surface of the non-interacting electrons at half filling is perfectly nested by the wave vector $(\pi,\pi)$, it is expected that the VBS patterns with ordering vector $(\pi,\pi)$ is more favored than the ones with $(0,\pi)$ or $(\pi,0)$. Both the staggered and columnar VBS patterns break the lattice $\mathbb{C}_4$ symmetry while the staircase and plaquette ones develop dimer ordering on both $x$ and $y$ directions. We assume $X_{i,\delta}=m_\delta+\inc{-1}^{i_x+i_y} \Delta_\delta^\mathrm{stag}+\inc{-1}^{i_\delta} \Delta_\delta^\mathrm{col}$,
where $m_\delta$ is the uniform component of the phonon displacements such that the uniform hopping amplitude along $\delta=x,y$ direction is given by $t_\delta=t-gm_\delta$.
Our calculations clearly show that the SSH phonons in the adiabatic limit favors the staggered VBS state with momentum $\inc{\pi,\pi}$ and $\mathbb{C}_4$ symmetry breaking for any finite EPC constant $\lambda$. Besides, when the staggered VBS order parameter develops spontaneously in $x$-direction, namely $\Delta_x^\mathrm{stag}\neq 0$ while $\Delta_y^\mathrm{stag}=0$, the uniform hopping amplitude $t_x$ and $t_y$ exhibits the anisotropy $t_x>t_y$. Such anisotropy develops so that the shifted Fermi surface $-2t_x\cos{k_x}-2t_y\cos{k_y}=0$ does not cross the nodal lines of gap function $2\Delta_x^\mathrm{stag}\sin{k_x}=0$; namely the Fermi surface can be fully gapped to gain energy.

\subsection{D. The effect of repulsive Hubbard interaction}
As mentioned in the main text, the SSH model in \Eq{EqOriginalModel} respects the O(4) symmetry, which implies the degeneracy between AF and pseudospin-AF (namely CDW/SC) correlations. However, this degeneracy can be lifted by any finite (even infinitesimal) Hubbard interaction.
By turning on a weak repulsive Hubbard interaction $H_U$, AF is more favored than pseudospin-AF ordering so that the degeneracy between them is lifted. For instance, in \Fig{FigHubbardU} we present the results of turning on a weak Hubbard $U$ at $\omega=3t$ and $\lambda\approx 0.25$ ($g=1.4t$) where the ground state without $U$ is degenerate between AF and pseudospin-AF (namely CDW/SC) ordering. Once a weak Hubbard $U=0.5$ is turned on, pseudospin-AF (CDW/SC) order is suppressed to zero and AF is the only order in the ground state, as shown in \Fig{FigHubbardU}.
Thus, in the presence of any weak repulsive Hubbard interaction, AF is the only order when the phonon frequency is larger than a critical value or the EPC constant is less than a critical value.

\begin{figure}[t]
\includegraphics[width= 3.165in]{AFMCDW_correlation_ratio_FSS_g=14_w=30_U=05.png}
\caption{The finite size scaling of correlation ratio of AF and pseudospin-AF ordering at $\omega=3t$ and $\lambda\approx 0.25$ ($g=1.4t$) for $U=0.5$. }
\label{FigHubbardU}
\end{figure}

\end{document}

\subsection{E. The nature of the VBS-AF transition}
Here we provide the results of energy derivative with respect to $\omega$ and $g$ mentioned in the main text, which can be written as:
\begin{equation}\label{EqEnergyDeriv}
\begin{aligned}
\frac{\partial E}{\partial g}&=\sum_{\avg{ij}}\hat{X}_{\avg{ij}} \sum_{{\sigma}}\inc{c^\dagger_{i{\sigma}}c_{j{\sigma}}+\mathrm{h.c.}} \\
\frac{\partial E}{\partial\omega}&=-\sum_{\avg{ij}} \frac{K}{\omega^3}\avg{\inc{\partial_\tau\,\hat{X}_{\avg{ij}}}^2}
\end{aligned}
\end{equation}
We plot $\partial E/\partial\omega$ at $g=1.4$ and $\partial E/\partial g$ at $\omega=1.0$ in \Fig{FigEnergyDeriv}, corresponding to \Fig{FigRatiog14} and \Fig{FigRatiow10} in main text. The energy derivative displays a change of slope near the phase transition point, while discontinuity has not been observed for systems up to $L=14$. Thus, the VBS-AFM transition is possibly a second-order or weakly first-order transition.

\begin{figure}[ht]
\subfigure[]{\label{figEnergyDerivOmega}\includegraphics[width= 2.919in]{Energy_derivative_vs_omega_g=14.png}}
\subfigure[]{\label{figEnergyDerivCoupling}\includegraphics[width= 2.995in]{Energy_derivative_vs_coupling_w=10.png}}
\caption{The derivative of ground state energy. (a) $\partial E/\partial\omega$ as the function of $\omega$ at $g=1.4$. The corresponding phase transition point obtained from \Fig{FigRatiog14} is $\omega_c\approx1.6$. (b) $\partial E/\partial g$ as the function of $g$ at $\omega=1.0$. The corresponding phase transition point obtained from \Fig{FigRatiow10} is $g_c\approx1.2$. The largest system size available in our simulation is $L=14$. }
\label{FigEnergyDeriv}
\end{figure}

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

In a previous paper \cite{L}, we established a result which we termed
{\it strong Jordan separation}. This was a version of Jordan separation which
applied to maps $f:S^n\rightarrow S^{n+1}$ which aren't assumed to be injective.
Under some mild hypothesis, one could nevertheless ensure that the image
separated $S^{n+1}$, and that any continuous extension
$F: \mathbb D^{n+1}\rightarrow S^{n+1}$ surjects onto one of the connected components
of $S^{n+1}-f(S^n)$. Recently, Iwaniec and Onninen \cite{IO} found applications of this result
in the field of quasi-conformal hyperelasticity. In this short note, we extend our
result from \cite{L} to the broadest possible setting, by establishing the following two
results:

\begin{Thm} Let $X$ be a compact topological space, $f:X\rightarrow S^{n+1}$ a continuous map,
and $U\subset X$ an open subset homeomorphic
to an open $n$-disk $\mathbb D^n_\circ$.
Assume that
\begin{itemize}
\item the map $f: X\rightarrow S^{n+1}$ contains $U$ in its set
of injectivity (i.e. $U\subset Inj(f):= \{x\in X \hskip 2pt | \hskip 2pt f^{-1}(f(x))=x\}$), and
\item the map $\check{H} ^n(X; \mathbb Z_2)\rightarrow \check{H} ^n(X-U; \mathbb Z_2)$ on
 \v{C}ech cohomology groups  induced by the inclusion $X-U\hookrightarrow X$
 has a non-trivial kernel.
\end{itemize}
Then $f(X)$ separates $S^{n+1}$ into at least two connected components. Furthermore,
there are precisely two connected components $V_1,V_2$ of $S^{n+1}-f(X)$ having
the property that their closure $\bar V_i$ intersects $f(U)$. In fact, for these two
connected components, we have containments $f(U) \subset \bar V_i$.
\end{Thm}

In the previous theorem, one should think of the sets
$V_1,V_2$ as corresponding locally to the two ``sides'' of $f(U)\cong
\mathbb D^n_\circ$ in the ambient $S^{n+1}$.

\begin{Thm}
Under the hypotheses of the previous theorem, let us further assume that $X$ is a closed
subspace of an ambient topological space $\hat X$.
Define two subgroups of $H_n(X; \mathbb Z_2)$ by:
\begin{itemize}
\item $K=\ker \big(H_n(X; \mathbb Z_2)\rightarrow
H_n(\hat X ; \mathbb Z_2)\big)$, and
\item $J= \im \big(H_n(X-U; \mathbb Z_2)\rightarrow H_n(X ; \mathbb Z_2)\big)$.
\end{itemize}
where both maps are induced by the corresponding inclusions of spaces.
If $K\not \subseteq J$, then we have that for
any continuous extension $F:\hat X\rightarrow S^{n+1}$, $F$ surjects onto one of the two
open components $V_i$.
\end{Thm}

We will prove both these theorems in Section 2 of our paper.  Observe that,
other than the condition requiring $U$ to lie in $Inj(f)$, the hypotheses in both theorems
are {\it internal},
in the sense that they are statements purely about the (homology of the)
spaces $U, X, \hat X$, and do not involve the map $f$. In Section 3,
we will provide examples showing that the hypotheses of the theorems are necessary. We will
also exhibit examples of triples $(\hat X, X, U)$ satisfying the homological hypotheses of both theorems.

\vskip 10pt

\centerline{\bf Acknowledgments}

\vskip 5pt

Tadeusz Iwaniec asked whether the author's arguments for the strong Jordan separation
theorem in \cite{L} could be extended to the case where the source space is a manifold other than the
sphere. The present paper was motivated by, and answers, Iwaniec's question.

The author was partially supported by the NSF, under grant DMS-0606002, and by a
Sloan Research Fellowship.

\section{Proofs}

Throughout this section, all homology and cohomology groups are understood to have
coefficients in $\mathbb Z_2$. The proofs of both Theorems follow closely the proof of the strong
Jordan separation in \cite[Section 2]{L}.

\begin{Prf}[Theorem 1]
We start by recalling that Alexander duality provides us with an isomorphism:
$$\tilde H_0(S^{n+1} - f(X)) \cong \check H ^n(f(X))$$
hence to show that $f(X)$ separates $S^{n+1}$ it is sufficient to show that
$ \check H^n(f(X)) \neq 0$. Now consider the
decomposition of $X$ into two open sets, one obtained by shrinking $U$
slightly, the other obtained by enlarging $X-U$ slightly. The intersection of these
two open sets is an open subset homeomorphic to $S^{n-1}\times \mathbb R$. Corresponding
to this decomposition, we have an image decomposition of $f(X)$ into two open sets
(recall that $U\subset Inj(f)$). In particular, we can apply the Mayer-Vietoris sequence
in \v Cech cohomology to compute the cohomology of $f(X)$:
$$ \ldots \rightarrow \check H^{n-1}(S^{n-1})\rightarrow \check H^n(f(X)) \rightarrow
\check H^n(f(U))\oplus \check H^n(f(X-U)) \rightarrow \check H^n(S^{n-1}) \rightarrow \ldots$$
Putting in the known terms into the sequence above, we obtain:
\begin{equation}
\ldots \rightarrow \mathbb Z_2 \rightarrow \check H^n(f(X)) \rightarrow \check H^n(f(X-U)) \rightarrow 0
\end{equation}
so to show that $ \check H^n(f(X)) \neq 0$, it is sufficient to show that the $\mathbb Z_2 \cong
\check H^{n-1}(S^{n-1})$ injects into $ \check H^n(f(X))$. In order to show this, we compare
the Mayer-Vietoris sequence above with the corresponding Mayer-Vietoris sequence for
the decomposition of $X$. This gives us the following commutative diagram:
$$\xymatrix{
\ldots \ar[r]  &  \mathbb Z_2 \ar[r]^-{\phi} \ar[d]^{f^*}_{\cong} & \check H^n(f(X)) \ar[r] \ar[d]^{f^*} &
\check H^n(f(X-U)) \ar[r] \ar[d]^{f^*} & 0\\
\ldots \ar[r] &  \mathbb Z_2 \ar[r] ^-\psi & \check H^n(X) \ar[r] & \check H^n(X-U)
\ar[r] & 0 \\
}$$
Since we assumed that the map $\check H^n(X) \rightarrow \check H^n(X-U)$ has
non-trivial kernel, we see that $\psi$ is injective. This forces the composite
$\psi \circ f^* = f^*\circ \phi$ to be injective, and hence the map $\phi$ to be injective,
as desired. This allows us to conclude that
$0 \neq \check H ^n(f(X)) \cong \tilde H_0(S^{n+1} - f(X))$, and we see that
$f(X)$ separates $S^{n+1}$.

Now, the remainder of the proof is virtually identical to that given in \cite{L}.
We will briefly sketch out the arguments, referring the reader to \cite[Section 2.1]{L}
for more details.

To see that there are precisely two connected components $V_1,V_2$ whose closure
intersects $f(U)$, one considers
the inclusion $S^{n+1}-f(X) \hookrightarrow S^{n+1}-f(X-U)$. The latter space is obtained
from the former by ``adding in $f(U)$'', i.e. we have identifications
$S^{n+1}-f(X-U) = \big( S^{n+1} - f(X)\big) \cup f(U)$. By applying Alexander duality to the
exact sequence in (1), we obtain:
$$0\rightarrow \mathbb Z_2 \rightarrow \tilde H_0(S^{n+1}-f(X))\rightarrow
\tilde H_0\big(\big( S^{n+1} - f(X)\big) \cup f(U)\big)\rightarrow 0$$
This is a homological version of the statement ``there are precisely two components
of $S^{n+1}-f(X)$ which are incident to $f(U)$''.

In order to obtain the statement we desire,
we make use of the following elementary result from point set topology:
if $\{V_i\}$ is a collection of pairwise disjoint open sets in $\mathbb R^{n+1}$, and $Z$ is a connected
set which intersects the closure of each $V_i$, then $Z \cup \big( \bigcup V_i\big)$ is connected.
Now apply this to the situation where the $\{V_i\}$ are the connected components of $S^{n+1}-
f(X)$ whose closure intersects $f(U)$, and $Z=f(U)$. For any distinct pair of connected components
of $S^{n+1}-f(X)$ whose closure intersects $f(U)$, we will get a corresponding element in
$\tilde H_0(S^{n+1}-f(X))$ which maps to zero in $H_0(S^{n+1}-f(X-U))$. In other words, the
rank of the kernel is one less than the number of connected components of $S^{n+1}-f(X)$
whose closure intersects $f(U)$. Since we know that the kernel has rank one, we conclude that
there are precisely two connected components $V_1,V_2$ of $S^{n+1}-f(X)$ whose closure
intersects $f(U)$.

Finally, to see that $f(U) \subset \bar V_i$, take $p\in f(U)$ arbitrary, consider a shrinking sequence
of open metric balls $\{U_i\}$ centered at $f^{-1}(p)$. We can apply the same argument as
in the previous paragraph, but replacing $f(U)$ by $f(U_i)$. Observe that $f(X-U)\subset f(X-U_i)$
induces an isomorphism on all the \v Cech cohomology groups,
and so by Alexander duality, the homology groups of the complements are also unchanged.
This forces {\it the same
two components} $V_1,V_2$ to intersect {\it every $f(U_i)$}. Since the sets $f(U_i)$ are shrinking
down to $\{p\}$, this immediately gives us that
$p$ lies in the closure of both $V_i$, completing the proof of Theorem 1.
\end{Prf}

\begin{Prf}[Theorem 2] Before starting with the proof of the theorem, let us briefly discuss some
general background material. For $p\notin f(X)$, we will consider the homomorphism
$f_*:H_n(X)\rightarrow H_n(\mathbb R^{n+1} - p)\cong \mathbb Z_2$. Note that, since all the groups
$H_n(\mathbb R^{n+1} - p)$ are isomorphic to $\mathbb Z_2$, we see that for any $p,q \notin f(X)$,
there are canonical identifications between the groups $H_n(\mathbb R^{n+1} - p)$ and
$H_n(\mathbb R^{n+1} - q)$. In particular, it makes sense to talk about elements being ``the same''
or ``different'' in the groups $H_n(\mathbb R^{n+1} - p)$ and $H_n(\mathbb R^{n+1} - q)$. Finally,
let us fix some notation. Recall that $U\subset X$ is an open set
homeomorphic to an open disc $\mathbb D ^n_\circ$, which we identify with the unit disk in $\mathbb R^n$.
Fixing this identification, we now denote by $U(r)$ ($r<1$) the subset of $U$ which
corresponds to the open disk of radius $r$. We will use $O$ to denote the point in $U$
which corresponds to the origin.

Let us now argue by way of contradiction: assume that there exists a continuous
extension $F: \hat X \rightarrow S^{n+1}$ and points
$z_i\in V_i$ with $z_i\notin F(\hat X)$. By hypothesis, $K\not \subseteq J$, so
there exists a homology class $\alpha \in H_n(X)$ having the following two properties:
\begin{enumerate}
\item $\alpha \in \ker( H_n(X) \rightarrow H_n(\hat X))$, and
\item $\alpha \notin \im (H_n(X-U) \rightarrow H_n(X)).$
\end{enumerate}
Let us consider the image of the class $\alpha$ in each of the homology groups
$H_n(\mathbb R^{n+1} - z_i) \cong \mathbb Z_2$. To compute this, observe that since
$z_i\notin F(\hat X)$, we can
factor the map $f_*$ as the composition $F_*\circ i_*$, where $i: X\rightarrow \hat X$
is the inclusion. This forces the containment $\ker (i_*)\subseteq \ker (f_*)$, which
combined with property (1) in the choice of $\alpha$, ensures that
$\alpha$ maps to zero in each of the homology groups $H_n(\mathbb R^{n+1} - z_i) \cong \mathbb Z_2$.

So in order to obtain a contradiction, it is sufficient to show that the class $\alpha$ maps to {\it distinct}
elements in each of the two homology groups $H_n(\mathbb R^{n+1} - z_i) \cong \mathbb Z_2$. This is
considerably harder; we will proceed in several steps. We will first
replace the map $f$ by
another map $g$, obtained by locally perturbing $f$ on the interior of $U$. This new map $g$
will have the following properties:
\begin{itemize}
\item $g$ is tame on the interior of $U(1/2)$,
\item $g\equiv f$ on the complement of $U(3/4)$,
\item $U \subset Inj(g)$,
\item $f$ is homotopic to $g$ in the complement of the points $z_1,z_2$.
\end{itemize}
The construction of $g$ can be done by appealing to the important {\it codimension one
taming theorem} of
Ancel-Cannon \cite{AC} (when $n\geq 4$), Ancel \cite{A} (when $n=3$), Bing \cite{B} (when
$n=2$) and Schoenflies (when $n=1$). We refer the reader to \cite[pg. 689]{L} for details on
how to accomplish this perturbation.

Since $f\simeq g$ in the complement of the $z_i$, it is sufficient to show that $g_*$
maps $\alpha$ to distinct elements in $H_n(\mathbb R^{n+1} - z_i)$. Now for the map $g$, we
can again apply Theorem 1, and see that there are exactly two connected components
$V_1^\prime, V_2^\prime$ of $S^{n+1}-g(X)$ whose closure intersects $g(U)$.
Recall that $O\in U$ is the point corresponding to the origin
under the homeomorphism identifying $\mathbb D^n_\circ$ with $U$. We know that there
exists a sequence of points $x_i \in V_1^\prime$, $y_i \in V_2^\prime$, with the property
that $\lim \{x_i\}=\lim\{y_i\}=g(O)$. We claim that, for each $i$, the
$g_*(\alpha)$ form distinct elements in the two groups
$H_n(\mathbb R^{n+1}-x_i)$ and $H_n(\mathbb R^{n+1}-y_i)$.

To see this, we first note that, if the points $p,q$
are chosen in the same connected component of $\mathbb R^{n+1}-g(X)$, then the two homomorphisms
$g_*: H_n(X)\rightarrow H_n(\mathbb R^{n+1} - p)$ and $g_*: H_n(X)\rightarrow H_n(\mathbb R^{n+1} - q)$
are identical. In particular, it is enough to show that the images of $g_*(\alpha)$ are distinct for
a specific index $i$. But recall that the map $g$ is {\it tame} on $U(1/2)$, and hence there
exists a global homeomorphism $\phi: \mathbb R^{n+1}\rightarrow \mathbb R^{n+1}$ with the property that
$\phi\circ g$ maps $U(1/2)$ into the standard $\mathbb R^n \times \{0\} \subset \mathbb R^{n+1}$. Furthermore,
for $i$ sufficiently large, $\phi(x_i)$ (respectively $\phi(y_i)$) are points which are locally
immediately above (respectively below) the hyperplane $\mathbb R^n\times \{0\}$. It is of course
sufficient to show that $\phi_*(g_*(\alpha))$ represents distinct elements in the two homology
groups $H_n(\mathbb R^{n+1}-\phi(x_i))$ and $H_n(\mathbb R^{n+1}-\phi(y_i))$.

\vskip 10pt

Now consider a cycle representing the homology class $\alpha \in H_n(X)$; this is a formal linear
sum $\sum \sigma_k$ of finitely many maps $\sigma_k: \Delta ^n \rightarrow X$. Now the cycle
we are interested in is the image cycle $\phi_*(g_*(\alpha))$, which is represented by the formal
linear sum $\sum \tau_k$, where each $\tau_k= \phi\circ g\circ \sigma_k$. In particular, there
exists a point $p \in \phi(g(U(1/2)))$ with the property that $p$ lies solely in the image of the
{\it interior} of the finitely many simplices, and $p$ is a non-singular value of each of the
maps  $\tau_k$. Now note that
we can join $\phi(x_i)$ to $\phi(y_i)$ by a PL-curve $\eta$ which intersects $(\phi\circ g)(X)$ in a
single transverse intersection at the point $p$.
There are now two possibilities: either (A) the homology class $\phi_*(g_*(\alpha))$ represents
distinct elements in the two homology groups $H_n(\mathbb R^{n+1}-\phi(x_i))$ and
$H_n(\mathbb R^{n+1}-\phi(y_i))$, or (B) from intersection theory, we have that the number of
intersection points of $\eta$ with the maps $\tau_k$ is even.

Let us now argue that possibility (B) does not occur. By way of contradiction, if this was the
case, one could subdivide the finitely many simplices which intersect $\eta$, obtaining a
new cycle $\sum \tau_k^\prime$ having the additional property that all the maps $\tau_k^\prime$
whose image passes through $p$ coincide with a fixed map $\tau: \Delta ^n\rightarrow
\mathbb R^{n+1}$. At the cost of further subdividing, we may moreover assume that $\tau$ lies
entirely within the image of $g(U)$. Now, since there are an {\it even} number of copies
of $\tau$ in the cycle
$\sum \tau_k^\prime$, and since we are working with $\mathbb Z_2$-coefficients, we can remove
all occurences of this singular simplex from the cycle, resulting in a new cycle $\sum \tau_k ^{\prime
\prime}$, which still represents $\phi_*(g_*(\alpha))$ and with the property that {\it the image
of all the singular simplices avoid the point $p$}.

Now observe that all the subdivisions of the singular simplices $\tau_k$ gives rise to
subdivisions of the singular simplices $\sigma _k$ (recall that we have
$\tau_k= \phi\circ g\circ \sigma_k$), since a subdivision of a singular simplex
is actually performed at the level of the
source space. So corresponding to the cycle $\sum \tau_k ^{\prime}$ representing
$\phi_*(g_*(\alpha))$, we have a corresponding cycle $\sum
\sigma_k ^{\prime}$ representing the original $\alpha$. Let $\sigma$ be the singular
simplex corresponding to $\tau$, and recall that the subdivision was chosen fine enough so
that $\tau$ was contained inside the image of $g(U)$. In particular, this forces $\sigma (\Delta^n)
\subset U$, which we recall lies in the set of injectivity of the map $g$. This implies that
there are no ``accidental cancellations'' due to distinct singular simplices in the chain
$\sum \sigma_k ^{\prime}$ both getting mapped to $\tau$. Since $\tau$ occurred an
even number of times in the cycle $\sum \tau_k ^{\prime}$, we have that $\sigma$ likewise
occurs an even number of times in the cycle $\sum \sigma_k ^{\prime}$. Finally, working with
coefficients in $\mathbb Z_2$ means that we can drop all copies of $\sigma$, obtaining a new cycle
$\sum \sigma_k ^{\prime \prime}$ having the following two properties:
\begin{itemize}
\item the cycle $\sum \sigma_k ^{\prime \prime}$ represents the homology class $\alpha$, and
\item the cycle $\sum \sigma_k ^{\prime \prime}$ has image in $X$ which is disjoint from the point
$O\in U$.
\end{itemize}
But now observe that, since $U$ is homeomorphic to $\mathbb D^n_\circ$, we have that $X-U$ is a
deformation retract of $X-\{O\}$. Applying the deformation retraction to the cycle
$\sum \sigma_k ^{\prime \prime}$, we can now obtain a cycle representing $\alpha$, but whose
image is contained inside $X-U$. This forces $\alpha \in \im \big( H_n(X-U)\rightarrow
H_n(X)\big)$, contradicting property (2) in our choice of the class $\alpha$, and completing
our proof of Theorem 2.
\end{Prf}

\section{Optimality and examples}

Before discussing some specific examples covered by our two theorems, let us start by giving some simple non-examples:
\begin{itemize}
\item Take $X=S^1 \subset \mathbb R^2$, and let $f$ be the projection onto the interval
$[-1,1] \subset \mathbb R ^2 \subset S^2$. The set of injectivity does not contain any open set $U$.
\item Take $X\subset \mathbb R^2$ to be the union of the standard unit circle, along with the interval
$[1,2]$ on the $x$-axis. If $f$ is the projection onto the interval $[-1,2]\subset \mathbb R^2 \subset S^2$,
then we see that $f$ is injective on $U = (1,2)$, but the inclusion $X-U \hookrightarrow X$ induces an
isomorphism on $\check H^1$, so has trivial kernel.
\end{itemize}
In both cases, we see that the conclusion to Theorem 1 fails, i.e. $f(X) \subset S^2$ fails to separate.
Similarly, for Theorem 2, we can consider the following simple example: let $X=S^1$, and
$f:S^1\hookrightarrow S^2$ be the embedding into the equator. Let $\hat X = S^1 \times [0,1]$ be
an annulus,
with $X \subset \hat X$ corresponding to $S^1\times \{0\}$. Note that since $X\hookrightarrow \hat X$
is a homotopy equivalence, the group $K$ in Theorem 2 is automatically trivial, and hence
$K\subset J$ holds. It is also clear that there exist extensions $F: S^1\times [0,1] \rightarrow
S^2$ with the property that $F$ fails to surject onto either hemisphere. These simple examples
show that the hypotheses in Theorems 1 and 2 are indeed necessary.

\vskip 10pt

We now proceed to give some examples of triples $(\hat X, X, U)$ which satisfy the homological
conditions of both theorems. In particular, for each of the following examples of triples, we have
that if $f: X\rightarrow S^{n+1}$ is injective on $U$, then: (1) $f(X)$ automatically separates $S^{n+1}$,
(2) there are exactly two connected components $V_1,V_2 \subset S^{n+1}-f(X)$ whose closure
contain $f(U)$, and (3) any extension of $f$ to a map $F:\hat X \rightarrow S^{n+1}$ surjects onto
either $V_1$ or $V_2$.

\vskip 5pt

\noindent {\bf Example: manifolds pairs.}

\vskip 5pt

Taking $\hat X$ to be a compact $(n+1)$-dimensional manifold with non-empty boundary, let
$X$ be the boundary of $\hat X$, and let $U$ be any open $n$-disk in $X$. Note that both of
the groups $\check H^n(X)$ and $H_n(X)$ are free $\mathbb Z_2$-modules, generated by the
connected components of $X$ (each of which is a closed manifold). It is now immediate
that the map $\check H^n(X) \rightarrow \check H^n(X-U)$ has non-trivial kernel, as removing
$U$ changes one of the connected components of $X$ from a closed manifold to a compact
manifold with boundary. Similarly, this forces $J=\im\big(H_n(X-U)\rightarrow H_n(X)\big)$
to miss elements in $H_n(X)$: an explicit element $\alpha\notin J$ is given by the sum of the
fundamental classes of the connected components of $X$. Finally, we note that $\alpha \in K$
is the image of the relative fundamental class $\mu \in H_{n+1}(\hat X, X)$
of the manifold with boundary
$\hat X$. From the long-exact sequence of the pair $(\hat X,X)$, we immediately see that
$\alpha$ maps to zero in $H_n(\hat X)$, and hence $\alpha \in K$. This implies that $K\not
\subseteq J$, completing the verification of the homological hypotheses in our two theorems.

For a more concrete example, if $X=K^2$
denotes the Klein bottle, and if $f: K^2 \rightarrow \mathbb R^3$ is a continuous map which contains
an open set $U\subset Inj(f)$, then the first theorem implies that
$f(K^2)$ separates $\mathbb R^3$ into at least two connected
components. Our second theorem tells us that, if $\hat X= L^3$ is the ``solid Klein-bottle'', and
$F:L^3\rightarrow \mathbb R^3$ is an extension of $f$, then $F$ surjects onto one of the two components
$V_1,V_2$ incident to any prescribed open set $U\subset Inj(f)$.

The reader might also like to compare the example where $\hat X = S^1 \times [0,1]$ with
the non-example discussed at the beginning of this section. The distinction lies of course in
the choice of $X$, which in the present example is $\partial \hat X = S^1 \times \{0,1\}$, and in
the non-example, consisted solely of $S^1\times \{0\}$.

\vskip 5pt

\noindent {\bf Example: pseudo-manifold pairs.}

\vskip 5pt

Recall that an $n$-dimensional {\it pseudo-manifold} is a simplicial complex with the property
that every $(n-1)$-dimensional simplex is a face of {\it exactly two} $n$-simplices.
An $(n+1)$-dimensional pseudo-manifold with boundary is defined to be a simplical
pair $(\hat X,X)$ with
the property that: (1) $X$ is an $n$-dimensional pseudo-manifold, (2) every $n$-simplex
in $X$ is contained in a unique $(n+1)$-dimensional simplex, and (3) every $n$-simplex
in $\hat X - X$ is contained in exactly two $(n+1)$-simplices. From the homological viewpoint,
the important observation is that the constraint on the codimension one simplices
ensures that compact pseudo-manifolds still have a notion of a fundamental class (the sum of all
top-dimensional simplices). In particular, the arguments given earlier for manifolds easily
extend to the pseudo-manifold case.

The distinction with the manifold situation is that pseudo-manifolds are allowed to be
singular, but that the singularities are relatively ``small'', i.e. codimension at least two. The importance
of this class of topological spaces comes from the fact that every complex projective algebraic variety
is a pseudo-manifold (as the singularities will have complex codimension $\geq 1$).
Note that complex projective algebraic varieties can always be triangulated (see \cite{LW}). Examples
of compact pseudo-manifolds with boundary can be obtained by taking a complex projective variety
$V\subset \mathbb C P^n$, taking a suitable real codimension one, smooth submanifold
$M^{2n-1} \subset \mathbb C P^n$ which intersects $V$ non-trivially, and cutting $\mathbb C P^n$ open along $M^{2n-1}$. The portion of $V$ in any of the (at most two) connected components of the resulting
manifold with boundary will yield an example of a compact pseudo-manifold with boundary.

\vskip 5pt

\noindent {\bf Example: CW-complexes and universality.}

\vskip 5pt

We now proceed to consider the case where both $X$ and $\hat X$ are CW-complexes,
and where $U \subset X$ is a top-dimensional open cell $e ^n \subset X$. In this case, we
note that \v Cech cohomology coincides with singular cohomology.
Furthermore, it is easy to see from a Mayer-Vietoris sequence argument
that the map $\check H^n(X) \rightarrow \check H^n(X-e^n)$ has non-trivial kernel precisely if
the attaching map $S ^{n-1} \hookrightarrow X^{(n-1)}$ for the $n$-cell $e^n$ induces the zero
map on $(n-1)$-dimensional cohomology (where $X^{(n-1)}\subset X$ is the $(n-1)$-skeleton
of $X$). As such it is easy to construct CW-complexes satisfying
the homological conditions of our first theorem. Similarly, it is easy to extend such an
$n$-dimensional CW-complex $X$ to an
$(n+1)$-dimensional CW-complex $\hat X$ satisfying the homological conditions for
our Theorem 2. For example, one extension which always works is the case where $\hat X$
is taken to be the cone over $X$. To see this, first observe that $\hat X$ is contractible,
and hence that $H_n(\hat X)=0$. This forces $K= \ker \big(H_n(X) \rightarrow H_n(\hat X)\big)=
H_n(X)$. So as long as the map $H_n(X-u) \rightarrow H_n(X)$ is {\it not} surjective, the
pair $(\hat X, X)$ will satisfy the homological conditions of our Theorem 2. But observe that,
from the fact that $X$ is a CW-complex satisfying the conditions of our Theorem 1,
we have that $H^{n-1}(X-U) \rightarrow 0 \in H^{n-1}(S^{n-1})$.
Since we are working with coefficients in $\mathbb Z_2$, the Ext term in the universal coefficient theorem
automatically vanishes, and we can identify the cohomology groups above as the duals of the
corresponding homology group. This forces the cohomological
statement above to be equivalent to the dual homological statement:
$\mathbb Z_2 \cong H_{n-1}(S^{n-1}) \rightarrow 0 \in H_{n-1}(X-U)$. The Mayer-Vietoris
sequence now yields:
$$H_n(X-u) \rightarrow H_n(X) \rightarrow \mathbb Z_2 \rightarrow 0$$
confirming that the first map is {\it not} surjective. This completes the verification that taking
$\hat X$ to be the cone over $X$ always satisfies the homological conditions for Theorem 2.

\vskip 10pt

Let us conclude with an observation: the example of pseudo-manifolds
is, in some sense, a ``universal example'' amongst CW-complexes.
Indeed, let us illustrate what we mean by reconsidering the
situation where $(\hat X, X)$ are CW-complexes, and $U \subset X$ is the interior of an $n$-cell
in $X$.  Since $X$ is assumed to satisfy the hypotheses of Theorem 1, the analysis in the
previous paragraph gives rise to a homology class $\alpha \in H_n(X)$ which is {\it not} in the image
of $H_n(X-U)$. Now recall that, given a homology class $\alpha\in H_n(X; \mathbb Z_2)$,
in an arbitrary topological space $X$, there exists an $n$-dimensional
pseudo-manifold $Y$ and a continuous map $\phi: Y\rightarrow X$ with the property that
$\phi_*[Y] = \alpha$.

Now it is easy to see, from the constraints on $\alpha$, that the image $\phi (Y)$ must pass through
$U$. One can further modify $Y$ so as to ensure that the corresponding
$\phi: Y \rightarrow X$ has the property that $\phi: \phi ^{-1}(U) \rightarrow U$ is a
homeomorphism (this can be done by an argument similar to the one at the end of Section 2).
This discussion establishes the following:

\begin{lemma}[Universality of pseudo-manifold example] Let $X$ be an $n$-dimensional
CW-complex, $U$ an $n$-cell in $X$, such that the pair $(X,U)$ satisfies the hypotheses
of Theorem 1 for the map $f: X\rightarrow S^{n+1}$. Then there exists a pseudo-manifold
pair $(Y, U^\prime)$ and a map $\phi: (Y, U^\prime) \rightarrow (X,U)$, with the property
that the pair $(Y, U^\prime)$ satisfies the hypotheses of Theorem 1 for
the composite map $f\circ \phi : Y \rightarrow S^{n+1}$.
\end{lemma}

In particular, the image of $\phi(Y)$ under $f$ already separates $S^{n+1}$, showing that
Theorem 1 for CW-complexes is actually a consequence of Theorem 1 for pseudo-manifolds.
A similar analysis can be used to show that  Theorem 2 for CW-complex pairs can also be
deduced from the pseudo-manifold case; we leave the details to the interested reader.

\vskip 10pt

\noindent {\bf Example: spaces which are not CW-complexes.}

\vskip 5pt

Finally, we give an example which {\it cannot} be deduced from the pseudo-manifold case: take
$X$ to be the closed topologists sine curve. It is well known that the first singular cohomology group
is $H^1(X) = 0$, while the first \v Cech cohomology group is $\check H^1(X)\cong \mathbb Z _2$.
If $U\subset X$ is an open interval in the ``sine portion'' of $X$, then one can readily verify that
both $H^1(X-U) = 0$ and $\check H^1(X-U) = 0$. In particular, we see that the pair $(X,U)$
satisfy the homological conditions for our Theorem 1, so any continuous map $f: X\rightarrow S^2$
which is injective on $U$ will have image that separates. Of course, the fact that
$H_1(X) \cong H^1(X) = 0$ tells us that there is no chance of using pseudo-manifolds to detect
separation (in contrast to the situation with CW-complexes).
For $\hat X$, one can again take the cone over $X$; it is easy to verify that the homological
conditions for Theorem 2 are indeed satisfied.