$H^2(\mathbb{Z}/2; N)$ has to vanish. Indeed, we have an exact sequence:

$$H^1(\mathbb{Z}/2; N(c)) \to H^2(\mathbb{Z}/2; N_c) \to H^2(\mathbb{Z}/2; N) \cong \mathbb{Z}/2,$$

where the second morphism maps the generator of $H^2(\mathbb{Z}/2; N_c)$ to the class $[v]$. The first map being $\mu \otimes \mathbb{Z}/2$ which has rank 1, the class of $[v]$ vanishes. We find that the real locus of its neighbourhood is the product of a Möbius band:

$$\mathbb{R} \times^{\mathrm{id}_{S^1}} S^1 = \{(z; \zeta) \in \mathbb{C} \times S^1 \mid \zeta \bar{z} = z\},$$

with $\mathbb{R}^\times$. The circle action is provided by the following formula:

$$S^1 \stackrel{\sim}{\to} (\mathbb{R} \times^{\mathrm{id}_{S^1}} S^1) \times \mathbb{R}^\times \quad \xi \cdot (z; \zeta; x) = (\xi z; \xi^2 \zeta; x). \qquad (5.1)$$

Thus, a typical circular orbit $\omega$ in this toric orbit of type $(1; 1)_0$ is given by $\{(0; \zeta; 1) : \zeta \in S^1\}$. One of its equivariant circular neighbourhoods is provided by the following equivariant embedding:

$$\Lambda \quad \to \quad (\mathbb{R} \times^{\mathrm{id}_{S^1}} S^1) \times \mathbb{R}^\times \\ [z; \zeta] \quad \mapsto \quad \left( \frac{\mathfrak{S}(z)\zeta}{1-|z|^2}; \zeta^2; e^{\mathfrak{R}(z)/1-|z|^2} \right).$$

We note that the vanishing of the cohomology class of the primitive generator of an invariant ray is a way to distinguish toric orbits of type $(1; 1)_1$ from toric orbits of $(1; 1)_0$. The same analysis (with the same notation $c$ and $v$) for a toric orbit of type $(1, 1)_1$ yields the model:

$$(\mathbb{A}_{\mathbb{R}}^1 \times_{\mathbb{R}}^{\mu} \mathrm{SO}_{2, \mathbb{R}}^0) \times_{\mathbb{R}} \mathrm{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_{m, \mathbb{C}} = \mathbb{A}_{\mathbb{R}}^1 \times_{\mathbb{R}} \mathrm{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_{m, \mathbb{C}},$$

where $\mu : 0 \to \mathbb{Z}$. We have the following exact sequence:

$$0 = H^1(\mathbb{Z}/2; N(c)) \to H^2(\mathbb{Z}/2; N_c) \to H^2(\mathbb{Z}/2; N),$$

so $[v]$ does not vanish in this case.

Let us now assume that we are given a toric orbit of type $(0; 1)_0$ associated to an invariant cone $c$ generated by two primitive vectors $v_1$ and $v_2$. This toric orbit is actually made of a single circular orbit. We note that at most one of the classes $[v_1]$, $[v_2] \in H^2(\mathbb{Z}/2; N)$ can vanish for $\{v_1; v_2\}$ is a basis of $\ker(1 - \tau)$. This orbit is contained in a toric subvariety of type $(1; 1)_0$ if and only if one of the two classes vanishes. In both cases (one or no vanishing), the local model of toric equivariant neighbourhood is provided by:

$$\mathbb{A}_{\mathbb{R}}^2 \times_{\mathbb{R}}^{\mu} \mathrm{SO}_{2, \mathbb{R}},$$

where $\mu : \mathbb{Z} \to \mathbb{Z}^2$ satisfies the exact sequence:

$$0 \longrightarrow H^1(\mathbb{Z}/2; N(c)) \xrightarrow{\mathrm{d}} H^2(\mathbb{Z}/2; N_c) \longrightarrow H^2(\mathbb{Z}/2; N) \\ \cong \underset{\mathbb{Z}/2}{\stackrel{\sim}{\longrightarrow}} \underset{\mu \otimes \mathbb{Z}/2}{\stackrel{\sim}{\longrightarrow}} (\mathbb{Z}/2)^2$$

Thus, if none of the class vanishes, $\mu$ can be taken to be the diagonal $\mu_{1,1} : k \mapsto (k; k)$ and otherwise $\mu_{0,1} : k \mapsto (0; k)$. In the first case, we have an equivariant diffeomorphism:

$$V_{2,1} \quad \to \quad \mathbb{R}^2 \times^{\mu_{1,1}} S^1 = \{(z_1; z_2; \zeta) \in \mathbb{C}^2 \times S^1 \mid \zeta \bar{z}_i = z_i, \forall i\} \\ [z; \zeta] \quad \mapsto \quad \left( \frac{\zeta \mathfrak{R}(z)}{1-|z|^2}; \frac{\zeta \mathfrak{S}(z)}{1-|z|^2}; \zeta^2 \right),$$

so that we have an exceptional orbit of invariant $(2; 1)$. In the other case, we have the following equivariant diffeomorphism:

$$\Lambda \quad \to \quad \mathbb{R}^2 \times^{\mu_{0,1}} S^1 = \{(x; z; \zeta) \in \mathbb{R} \times \mathbb{C} \times S^1 \mid \zeta \bar{z} = z\} \\ [z; \zeta] \quad \mapsto \quad \left( \frac{\mathfrak{R}(z)}{1-|z|^2}; \frac{\zeta \mathfrak{S}(z)}{1-|z|^2}; \zeta^2 \right).$$

Hence, we have a special exceptional orbit.

To summerise the situation we found that:
(i) The number $t$ of components of the surface of special exceptional orbits equals $e_{1,1}^0(X)$, the number of invariant rays such that the class of their primitive generator vanishes;

(ii) The number of curves of fixed points equals $e_{1,0}^0(X)$;

(iii) Every exceptional orbit has invariant (2; 1) and their number $u$ equals the number of cones made of two invariant rays such that none of the classes of their primitive generator vanish.

We want now to compute $u$. To do so, let us introduce the canonical fibre $F$ of $X$. Since $X$ has compact real locus, $F$ is complete. Recall that its fan is a complete fan of $\ker(1 - \tau)$ whose rays are: either invariant rays of the fan $C$ of $X$, or spanned by the sum of the two generators of a bidimensional cone of $C$ whose rays are exchanged by $\tau$. Thus, the class of the primitive generator of a new ray of $F$ vanishes by construction. Let us consider a circular enumeration $(c_k)_{k \in \mathbb{Z}/m}$ of the rays of $F$. It means that there is $m$ rays in the fan of $F$. This number equals the number of maximal toric subvariety of $X$ i.e. $e_{1,1}^1(X) + e_{1,1}^0(X) + e_{1,0}^0(X)$ or more concisely $e_{1,1}(X) + e_{1,0}(X)$. We accordingly denote by $h_k \in \{0; 1\}$ the number that vanishes if and only if the class of the primitive generator of the corresponding ray does. As with the fan of $X$, two rays of a bidimensional cone of $F$ cannot both have vanishing $h_k$'s. This means that for all $k \in \mathbb{Z}/m$, the maximum of $h_k$ and $h_{k-1}$ is always 1. Thus we have:

$$ 
\begin{aligned}
u + m &= \sum_{k \in \mathbb{Z}/m} \min(h_k; h_{k-1}) + 1 = \sum_{k \in \mathbb{Z}/m} \min(h_k; h_{k-1}) + \max(h_k; h_{k-1}) \\
&= \sum_{k \in \mathbb{Z}/m} h_k + h_{k-1} = \sum_{k \in \mathbb{Z}/m} 2h_k \\
&= 2e_{1,1}^1(X).
\end{aligned}
$$

Thus, we find that $u = e_{1,1}^1(X) - e_{1,1}^0(X) - e_{1,0}^0(X)$.

To finish the proof of the proposition, let us compute the genus of the quotient $X(\mathbb{R})/S^1$. Let us consider $\pi : \text{Bl}_W X \to X$ the resolution of the winding of $X$ introduced in Proposition 3.31. We will first establish that the toric morphism $\pi$ induces an homeomorphism between $\text{Bl}_W X(\mathbb{R})/S^1$ and $X(\mathbb{R})/S^1$. With all that precedes, we notice that $W(\mathbb{R})$ is exactly the fixed locus of the circle action on $X(\mathbb{R})$. Moreover, $\pi$ induces an homeomorphism:

$$ \pi/S^1 : (\text{Bl}_W X(\mathbb{R})/S^1) - (E(\mathbb{R})/S^1) \xrightarrow{\approx} (X(\mathbb{R})/S^1) - (W(\mathbb{R})/S^1), \quad (5.2) $$

where $E$ denotes the exceptional divisor. A component of $W$ is a real projective line. Following Proposition 2.15, $\pi$ is given, in a neighbourhood of such a component, by the following equivariant model:

$$ \pi : (\mathbb{R} \times_{\text{id}_z} S^1) \times \mathbb{P}^1(\mathbb{R}) \xrightarrow{(z; \zeta; p)} \mathbb{C} \times \mathbb{P}^1(\mathbb{R}) \xrightarrow{(z; p)} (z; p). $$

It gives rise to the following commutative square:

$$ 
\begin{tikzcd}
(z; \zeta; p) \arrow[r] \arrow[d] & (\mathbb{R} \times_{\text{id}_z} S^1) \times \mathbb{P}^1(\mathbb{R}) \arrow[r, "\pi"] \arrow[d] & \mathbb{C} \times \mathbb{P}^1(\mathbb{R}) \arrow[d] \\
(|z|; p) \arrow[r] & \mathbb{R}_+ \times \mathbb{P}^1(\mathbb{R}) \arrow[r, "\text{quot.}/S^1", "\cong"] & \mathbb{R}_+ \times \mathbb{P}^1(\mathbb{R})
\end{tikzcd}
\quad (z; p) $$

Hence $\pi/S^1$ is a local homeomorphism between compact spaces, i.e. a finite covering. Since $E(\mathbb{R})$ is the preimage of $W(\mathbb{R})$, and that both are invariant under the action of the circle, the homeomorphism (5.2) ensures that $\pi/S^1$ has degree 1 and is an homeomorphism. Now, the description of $\text{Bl}_W X(\mathbb{R})$ as $F(\mathbb{R}) \times^\Gamma S^1$, cf. (3.2), asserts that $\text{Bl}_W X(\mathbb{R})/S^1$ is homeomorphic to $F(\mathbb{R})/\Gamma$. Thereafter, we find that:

$$ 2\chi(F(\mathbb{R})/\Gamma) = \chi(F(\mathbb{R})) + \chi(F(\mathbb{R})^{\Gamma}). \quad (5.3) $$

See [Degtyarev and Kharlamov, 2000, Corollary A.1.3]. Every bidimensional cone of the fan of $F$ represents a fixed point of the action of $\Gamma$. Likewise, every ray of the fan of $F$ spanned by a primitive generator whose class vanishes in $H^2(\mathbb{Z}/2; N)$ corresponds to a circle entirely made of $\Gamma$-fixed points. Thus, $\chi(F(\mathbb{R})^\Gamma)$ equals the number of exceptional orbits:

$$ \chi(F(\mathbb{R})^{\Gamma}) = e_{1,1}^{1}(X) - e_{1,1}^{0}(X) - e_{1,0}^{0}(X) \quad (5.4) $$
Then, by Proposition 4.6 and Corollary 4.9, we have:

$$ \begin{aligned} \chi(F(\mathbb{R})) &= e[X](-2; 1) \\ &= 4e_{2,1}(X) - 2(e_{1,1}(X) + e_{1,0}(X)) + e_{0,1}(X) + e_{0,0}(X). \end{aligned} $$

We note that in the fan of $F$ there is $1 = e_{2,1}(X)$ point, $e_{1,1}(X) + e_{1,0}(X)$ rays, and $e_{0,1}(X) + e_{0,0}(X)$ bidimensional cones. This a consequence of $F$ being an equivariant torus embedding of a split torus and of the formula of Proposition 4.8. Since $F$ is complete, so is its fan and thus there is as many rays as bidimensional cones i.e. $e_{1,1}(X) + e_{1,0}(X) = e_{0,1}(X) + e_{0,0}(X)$, cf. (4.3). Therefore:

$$ \chi(F(\mathbb{R})) = 4 - e_{1,1}(X) - e_{1,0}(X). \qquad (5.5) $$

Using (5.3), (5.4), and (5.5) we find that:

$$ \chi(X(\mathbb{R})/S^1) = 2 - e_{1,1}^0(X) - e_{1,0}^0(X). $$

Since $e_{1,1}^0(X) + e_{1,0}^0(X)$ is the number of boundary components of $X(\mathbb{R})/S^1$, we find that the genus has to vanish. The quotient is a sphere with holes, hence orientable. We depicted an example in Figure 7. $\square$

<figure><img src="image_17.png" /><figcaption>(a) The shape of the fan of $X$: the “horizontal” plane corresponds to $\ker(1 - \tau)$. We indicated the cohomology classes of the generators of the invariant rays by 0s and 1s. Here $a_{1,0} = 7$, $a_{0,1} = 2$, $t = 1$, and $e = 3$.</figcaption></figure>

<caption>(b) The quotient $X(\mathbb{R})/S^1$: Two boundary components correspond to the two pairs of tridimensional cones, the other corresponds to the ray of cohomology class 0. The three cusps correspond to the three bidimensional cones made of two rays of class 1.</caption>

Figure 7: The fan of an improperly wound toric variety $X$ of type $(2, 1, )_1$ and the quotient of its real locus by the circle action.

**Lemma 5.10.** Let $\mathbb{G}_{m,\mathbb{R}}^2 \hookrightarrow X$ be a smooth and complete real equivariant torus embedding. There exists a $\gamma \in \{\pm 1\}^2$ that has only a finite number of fixed points in $X(\mathbb{R})$ if and only if $X$ is a Hirzebruch surface of even parameter.

*Proof.* Let $\mathbb{G}_{m,\mathbb{R}}^2 \hookrightarrow X$ be a smooth and complete real equivariant torus embedding. Let $N$ denote the cocharacter lattice of $\mathbb{G}_{m,\mathbb{R}}^2$ and $\gamma \in N/2N$ be a non-trivial 2-torsion element. The fixed point set of $\gamma$ in $X(\mathbb{R})$ is the union of the real loci of the toric orbits $O(c)$ of $X$ such that the 2-torsion of their isotropy group contains $\gamma$. Hence, such that $\gamma$ belongs to $N_c/2N_c$. Naturally, every toric fixed point belongs to $X(\mathbb{R})^\gamma$. Let $D$ be a toric divisor whose ray is directed by a primitive vector $v$. The real locus $D(\mathbb{R})$ belongs to $X(\mathbb{R})^\gamma$ if and only if $v = \gamma (\mod 2N)$. Therefore, no primitive generator of the fan of $X$ must have the same “parity” as $\gamma$ if the fixed point set is to be finite. Let us consider $e := e_{1,0}(X)$ the number of toric divisors of $X$. Since $X$ is complete, it is at least equal to 3. We will show that if it is different from 4 then one primitive generators of the rays of the fan of $X$ will be of the same parity as $\gamma$.

$e = 3$ In this case, the smoothness forces $X$ to be $\mathbb{P}_{\mathbb{R}}^2$. The three primitive generators of the rays of its fan satisfy $v_1 + v_2 + v_3 = 0$. The same relation must hold in $N/2N$. Since none of the vector is divisible by 2, the relation imposes that $\{[v_1]; [v_2]; [v_3]\} = N/2N \setminus \{0\}$. Hence, one of them has the same parity as $\gamma$;
$e \ge 5$ Let $\{v_k\}_{k \in \mathbb{Z}/e}$ be a circular enumeration of the primitive generators of the rays of the fan of $X$. The classification of smooth equivariant embeddings of split tori imposes that, for some $k \in \mathbb{Z}/e$, we have $v_k = v_{k-1}+v_{k+1}$, cf. [Fulton, 1993, §2.5, third exercise] or [Cox et al., 2011, proof of Lemma 10.4.2]. With the same argument as in the previous case, one of the three vectors $v_{k-1}$, $v_k$, $v_{k+1}$ must have the same parity as $\gamma$.

Hence, if $X(\mathbb{R})^\gamma$ is finite, $e = 4$. Following again the classification of smooth equivariant embeddings of split tori, we know that $X$ is the Hirzebruch surface $F_{|b|}$ where the integer $b \in \mathbb{Z}$ satisfies $v_2 + v_4 = bv_1$ (under the assumption that the enumeration of the vectors satisfies $v_3 + v_1 = 0$, cf. [Fulton, 1993, §1.1]). We need to show that if $X(\mathbb{R})^\gamma$ is finite then $b$ is even. Since $\{v_1; v_2\}$ and $\{v_1; v_4\}$ are bases of $N$, $v_1 \ne v_2 (\text{mod } 2N)$ and $v_1 \ne v_4 (\text{mod } 2N)$. Therefore, $v_4 = v_2 (\text{mod } 2N)$ since none of $v_1, v_2, v_4$ is of the parity of $\gamma$. It follows that $bv_1$ belongs to $2N$, hence $b$ is even. Reciprocally, if $X$ is a Hirzebruch surface $F_{2m}$ then its fan is spanned by the primitive vectors $\partial x, \partial y, -\partial x, -\partial y + 2m\partial x$ in $\mathbb{Z}^2$ (up to isomorphism). Thus $\gamma = \partial x + \partial y (\text{mod } 2\mathbb{Z}^2)$ satisfies the fixed point property. $\square$

**Lemma 5.11.** Let $T \hookrightarrow X$ be a real equivariant torus embedding of type (2; 1)$_1$ with compact real locus and smooth topological core. The polynomial $e^*[X]$ is uniquely determined by the numbers $e_{1,1}^0(X)$, $e_{1,1}^1(X)$, and $e_{1,0}^0(X)$.

*Proof.* We have:

$$e^*[X] = e_{2,1}^1(X)xz + e_{1,1}^1(X)z + e_{1,1}^0(X)xy + e_{1,0}^0(X)x + e_{0,1}^0(X)y + e_{0,0}^0(X). \quad (5.6)$$

There is only one open orbit, so $e_{2,1}^1(X) = 1$. Moreover, Proposition 4.12 yields the following relations:

$$\left\{ \begin{array}{l} e_{1,1}^1(X) + e_{1,1}^0(X) + e_{1,0}^0(X) = e_{0,1}^0(X) + e_{0,0}^0(X) \\ e_{0,0}^0(X) = 2e_{1,0}^0(X). \end{array} \right. \qquad (5.7)$$

Thus, one can express $e^*[X]$ in terms of $e_{1,1}^0(X)$, $e_{1,1}^1(X)$, and $e_{1,0}^0(X)$. $\square$

**Theorem 5.12.** Let $T \hookrightarrow X, Y$ be two real equivariant torus embeddings of type (2; 1)$_1$ with compact real loci and smooth topological cores. If $e^*[X] = e^*[Y]$, then $X(\mathbb{R})$ is homeomorphic to $Y(\mathbb{R})$. If $X(\mathbb{R})$ is homeomorphic to $Y(\mathbb{R})$, then $e^*[X] = e^*[Y]$ except when their real loci are homeomorphic to $\mathbb{P}^2(\mathbb{R}) \times S^1$, in which case, their $e^*$-polynomials can either be $xz + 2z + xy + 3y$ or $xz + 2z + xy + x + 2y + 2$.

*Proof.* Let $T \hookrightarrow X$ be a real equivariant torus embedding of type (2; 1)$_1$ with compact real locus and smooth topological core. The only possible exceptional circular orbits of $X(\mathbb{R})$ are of unoriented type (2; 1). Thus, for any choice of orientation of the quotient $X(\mathbb{R})/S^1$, the Orlik-Raymond invariant of $X(\mathbb{R})$ will be of the following form:

$$\{ b; (0; 0; e_{1,1}^0(X); e_{1,0}^0(X)); \underbrace{(2; 1); \cdots; (2; 1)}_{u \text{ times}} \},$$

where $u = e_{1,1}^1(X) - e_{1,1}^0(X) - e_{1,0}^0(X)$ and $b$ is an integer, cf. Proposition 5.9.

Let us first assume that $e_{1,0}^0(X) + e_{1,1}^0(X)$ vanishes. In this case, $X$ is properly wound. Indeed, the properness of the winding is equivalent to the vanishing of $e_{1,1}^0(X)$. Thus, $X(\mathbb{R})$ is homeomorphic to $F(\mathbb{R}) \times_\Gamma S^1$, cf. (3.2). In the proof of Proposition 5.9, we determined that the fixed locus $F(\mathbb{R})^\Gamma$ is made of $e_{1,0}^0(X) + e_{1,1}^0(X)$ circles, and of $e_{1,1}^1(X) - e_{1,1}^0(X) - e_{1,0}^0(X)$ isolated points. Therefore, under the hypothesis $e_{1,0}^0(X) + e_{1,1}^0(X) = 0$, this fixed point set is finite, and Lemma 5.10 guarantees that $F$ is a Hirzebruch surface of even parameter. Thus, $F(\mathbb{R})$ is homeomorphic to $S^1 \times S^1$ and $\Gamma$ acts via an involution that have four fixed points. [Dugger, 2019, Corollary 5.8] states that there is only one such involution, up to conjugation by a homeomorphism, namely the pillow case involution $(\zeta; \xi) \mapsto (\tilde{\zeta}; \tilde{\xi})$. Thus, the real locus of $X$ is homeomorphic to the fibre product of two Klein bottles $Kl$:

$$X(\mathbb{R}) \approx \text{Kl} \times_{S^1} \text{Kl}. \qquad (5.8)$$

We deduce from the ninth case of [Orlik and Raymond, 1968, Theorem 4], that the Orlik-Raymond invariant of $X(\mathbb{R})$ will either be:

$$\{b_0; (0; 0; 0; 0); (2; 1); (2; 1); (2; 1); (2; 1)\} \text{ or } \{-b_0 - 4; (0; 0; 0; 0); (2; 1); (2; 1); (2; 1); (2; 1)\},$$
for some integer $b_0$. The ambiguity comes from the two possible orientations of the quotient. Furthermore, the same theorem asserts that no other Orlik-Raymond threefold will yield this particular threefold. Thus, given $X$, (5.8) holds if and only if $e^*[X] = xz + 4z + 4y$.

We assume now that $e_{1,0}^0(X) + e_{1,1}^0(X)$ is positive. In this case, $b$ has to vanish. Thus, regardless of a chosen orientation of the quotient, the Orlik-Raymond invariant is given by:

$$ \left\{ 0; \underbrace{(0; 0; e_{1,1}^0(X); e_{1,0}^0(X))}_{u \text{ times}}; \underbrace{(2; 1); \dots; (2; 1)} \right\}, $$

where $u = e_{1,1}^1(X) - e_{1,1}^0(X) - e_{1,0}^0(X)$. At this point, we have proved that $e^*[X]$ determines, up to a choice of orientation of the quotient, the Orlik-Raymond invariant of $S^1 \rightsquigarrow X(\mathbb{R})$. [Orlik and Raymond, 1968, Theorem 2] ensures that if $e^*[X] = e^*[Y]$ then $X(\mathbb{R})$ is homeomorphic to $Y(\mathbb{R})$.

To prove the converse statement, we need to show that if $e^*[X] \neq e^*[Y]$ then $X(\mathbb{R})$ is not homeomorphic to $Y(\mathbb{R})$ except when $e^*[X] = xz + 2z + xy + 3y$ and $e^*[Y] = xz + 2z + xy + x + 2y + 2$ in which case, both real loci are homeomorphic to $\mathbb{P}^2(\mathbb{R}) \times S^1$. We have already showed that if $e_{1,1}^0(X) + e_{1,0}^0(X)$ vanishes then necessarily $e^*[X] = xz + 4z + 4y$, and $e^*[Y] \neq e^*[X]$ implies that $Y(\mathbb{R})$ is not homeomorphic to $X(\mathbb{R})$. Let us now assume that $e_{1,0}^0(X) + e_{1,1}^0(X)$ is positive. If $e_{1,1}^0(X)$ vanishes, then $e_{1,0}^0(X)$ is necessarily positive, and [Orlik and Raymond, 1968, Theorem 5] asserts that we have the following alternative:

(i) $e^*[X] = xz + z + xy + 2y$ in which case $X(\mathbb{R})$ is homeomorphic to $S^1 \times \mathbb{Z}/2 S^2$ where $\mathbb{Z}/2$ acts on both factors by the antipodal involution (This is not actually possible for real equivariant torus embeddings of type $(2; 1)_1$ but we will address this in Proposition 5.14);

(ii) $e^*[X] = xz + 2z + xy + 3y$ in which case $X(\mathbb{R})$ is homeomorphic to $\mathbb{P}^2(\mathbb{R}) \times S^1$;

(iii) $e^*[X] = xz + 2z + 2xy + 4y$ in which case $X(\mathbb{R})$ is homeomorphic to $\mathrm{Kl} \times S^1$;

(iv) $e^*[X]$ does not belong to the previous list. In this case, $X(\mathbb{R})$ only admits the given circle action, i.e. if $e^*[Y] \neq e^*[X]$ then $Y(\mathbb{R})$ is not homeomorphic to $X(\mathbb{R})$.

If $e_{1,1}^0(X)$ is positive then [Orlik and Raymond, 1968, Theorem 3] tells us that $X(\mathbb{R})$ is homeomorphic to the connected sum:

$$ X(\mathbb{R}) \approx (e_{1,1}^0(X) - 1) \cdot (S^2 \times S^1) + e_{1,0}^0(X) \cdot (\mathbb{P}^2(\mathbb{R}) \times S^1) + u \cdot \mathbb{P}^3(\mathbb{R}), \quad (5.9) $$

where $u = e_{1,1}^1(X) - e_{1,1}^0(X) - e_{1,0}^0(X)$. This is the prime decomposition of $X(\mathbb{R})$. The uniqueness of such decomposition, cf. [Hempel, 1976], ensures that the right hand side of (5.9) determines $e^*[X]$ under the provision that $e_{1,1}^0(X)$ is positive. Furthermore, the only threefold arising as the right hand side of (5.9) that was already listed is $\mathbb{P}^2(\mathbb{R}) \times S^1$. The $e^*$-polynomials that leads to this real locus is $e^*[X] = xz + 2z + xy + x + 2y + 2$. We showed that the only couple of polynomials leading to the same manifold are $xz + 2z + xy + x + 2y + 2$ and $xz + 2z + xy + 3y$. $\square$

As a biproduct of the proof of Theorem 5.12 we find the following proposition.

**Proposition 5.13.** Let $T \rightsquigarrow X$ be a real equivariant torus embedding of type $(2; 1)_1$ with compact real locus and smooth topological core. If $X$ is properly wound then $X(\mathbb{R})$ is a prime threefold. If, on the contrary, $X$ is not properly wound, then $e_{1,1}^0(X)$ is positive and the prime decomposition of $X(\mathbb{R})$ is the following:

$$ X(\mathbb{R}) \approx (e_{1,1}^0(X) - 1) \cdot (S^2 \times S^1) + e_{1,0}^0(X) \cdot (\mathbb{P}^2(\mathbb{R}) \times S^1) + (e_{1,1}^1(X) - e_{1,1}^0(X) - e_{1,0}^0(X)) \cdot \mathbb{P}^3(\mathbb{R}). $$

**Proposition 5.14.** Let $e = e_{2,1}^1 xz + e_{1,1}^1 z + e_{1,1}^0 xy + e_{1,0}^0 xy + e_{0,1}^0 y + e_{0,0}^0 y$ be a polynomial with non-negative integral coefficients. There is a real equivariant torus embedding of type $(2; 1)_1$ with compact real locus and smooth topological core whose $e^*$-polynomial is given by $e$ if and only if:

$$ \left\{ \begin{array}{ll} e_{2,1}^1 = 1 & e_{1,1}^1 \ge e_{1,1}^0 + e_{1,0}^0 \\ e_{0,0}^0 = 2e_{1,0}^0 & e_{0,1}^0 + e_{0,0}^0 \ge 3 \\ e_{1,1}^1 + e_{1,1}^0 + e_{1,0}^0 = e_{0,1}^0 + e_{0,0}^0 & (e_{1,1}^0 + e_{1,0}^0 = 0) \Rightarrow (e_{1,1}^1 = 4). \end{array} \right. \qquad (5.10) $$

*Proof.* Propositions 4.12 and 5.9 ensure the necessity of the statement. For the sufficiency, let us consider the lattice $N := \mathbb{Z}[\tau] \oplus \mathbb{Z}[1] = \langle \partial x; \partial y; \partial z \rangle$, where $\tau$ acts as $\tau \partial x = \partial y$, $\tau \partial z = \partial z$. We need to exhibit a smooth equivariant fan $C$ of $N$ that contains $\langle \partial x + \partial y; \partial z \rangle$ in its support, and that is made of:
(i) $e_{2,1}^1 = 1$ vertex;

(ii) $e_{1,0}^0$ pairs of exchanged rays;

(iii) $e_{1,1}^1$ invariant rays spanned by a primitive vector with a non vanishing class in $H^2(\mathbb{Z}/2; N)$;

(iv) $e_{1,1}^0$ invariant rays spanned by a primitive vector with a trivial class in $H^2(\mathbb{Z}/2; N)$;

(v) $e_{1,0}^0$ bidimensional cones on a pair of exchanged rays;

(vi) $e_{0,1}^0$ bidimensional cones on a pair of invariant rays;

(vii) $e_{0,0}^0$ tridimensional cones.

In the basis $\{\partial x + \partial y; \partial z\}$ of the invariant subspace of $N$, the integer vectors with vanishing cohomology class are exactly those with an even $\partial z$-coordinate. We denote by $L$ the sublattice $\langle \partial x + \partial y; 2\partial z \rangle$ of $\langle \partial x + \partial y; \partial z \rangle$. If $e_{1,1}^0 + e_{1,0}^0$ vanishes then, the polynomial $e$ is given by $xz + 4z + 4y$. In this case, the fan of Figure 8 satisfies our requirements. From now on, we will assume that $e_{1,1}^0 + e_{1,0}^0$ is positive. We will start by constructing a complete fan $C_2$ in $\langle \partial x + \partial y; \partial z \rangle$ that we will lift to a fan of $N$. This will be the fan of the canonical fibre. We consider the following procedure:

<figure><img src="image_18.png" /><figcaption>Figure 8: A fan whose associated $e^*$-polynomial is given by $xz + 4z + 4y$.</figcaption></figure>

1st Step: We consider the complete fan $C_0$ of $\langle \partial x + \partial y; \partial z \rangle$ made of the three rays $\mathbb{R}_+(\partial z)$, $\mathbb{R}_+(\partial x + \partial y)$, and $\mathbb{R}_+(-\partial x - \partial y - \partial z)$. Exactly one of them is spanned by a primitive vector contained in $L$;

2nd Step: We remark that the number $e_{1,1}^1$ is at least equal to 2. Indeed, $2e_{1,1}^1 \ge e_{1,1}^1 + (e_{1,1}^0 + e_{1,0}^0) \ge 3$. Since $e_{1,1}^1$ is an integer, it is at least equal to 2. Then, we construct the fan $C_1$. If the number $e_{1,1}^1$ is 2, we set $C_1 = C_2$, otherwise we subdivide $C_0$ into $C_1$ by adding the rays spanned by the vectors $\partial z + i(\partial x + \partial y)$, for all integers $1 \le i \le e_{1,1}^1 - 2$. This is depicted in Figure 9a. The fan $C_1$ has $1 + e_{1,1}^1$ rays, only one of which is spanned by a primitive vector contained in $L$;

3rd Step: We assumed $e_{1,1}^0 + e_{1,0}^0$ positive. If it equals 1 we set $C_2$ to be $C_1$. Otherwise, we subdivide $C_1$ by adding the rays spanned by the vectors $-(\partial x + \partial y)$ and $(2\partial z + (2i - 1)(\partial x + \partial y))$ for all $1 \le i \le e_{1,1}^0 + e_{1,0}^0 - 2$ (if $e_{1,1}^0 + e_{1,0}^0 = 2$ we only add the first ray). This procedure is illustrated in Figure 9b. Every new ray lies exactly between two rays spanned by a primitive vector with non vanishing cohomology class for $e_{1,1}^1 \ge e_{1,1}^0 + e_{1,0}^0$. We can also note that $C_2$ is smooth and complete. It has $e_{1,1}^1$ rays spanned by a primitive vector of non-vanishing class and $e_{1,1}^0 + e_{1,0}^0$ rays spanned by a primitive vector of vanishing class.

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /><figcaption>Figure 9: The construction of $C_2$ from $C_0$. The added rays at each step are dashed and the sublattice $L$ is represented by white dots.</figcaption></figure>

To obtain the fan $C$, we will replace the first $e_{1,0}^0$ rays spanned by:

$$ (\partial x + \partial y), \quad -(\partial x + \partial y), \quad 2\partial z + (2i - 1)(\partial x + \partial y), \quad \forall 1 \le i \le e_{1,1}^0 + e_{1,0}^0 - 2, $$
by the bidimensional cones of the following list:

$$ \langle \partial x; \partial y \rangle_{\mathbb{R}_+}, \quad \langle -\partial x; -\partial y \rangle_{\mathbb{R}_+}, \quad \langle \partial z + (2i - 1)\partial x; \partial z + (2i - 1)\partial y \rangle_{\mathbb{R}_+}, \quad \forall 1 \le i \le e_{1,1}^0 + e_{1,0}^0 - 2. $$

Then, we replace the bidimensional cones of $C_2$ that were adjacent to the removed rays by tridimensional cones using the following procedure: if the ray $\rho$ is replaced by a bidimensional cone $c$ then a cone of the form $\rho + \rho'$ becomes $c + \rho'$. It is illustrated in Figure 10. In the end, we have the desired fan $C$.

<figure><img src="image_21.png" /><figcaption>Figure 10: Procedure to Obtain Orbits of Type $(1,0)_0$.</figcaption></figure>

<figure><img src="image_22.png" /></figure>

**Remark 5.15.** We can deduce from Propositions 1.41 and 5.14 that every threefold of the form $h \cdot (S^2 \times S^1) + k \cdot (\mathbb{P}^2(\mathbb{R}) \times S^1) + l \cdot \mathbb{P}^3(\mathbb{R})$ where $h, k, l \ge 0$ are three integers whose sum is positive, can be realised as the real locus of a complete smooth equivariant torus embedding of type $(2; 1)_1$.

Table 6 summarises the discussion of this last section. It displays the types of topology a smooth and compact toric threefold under the action of a non-split torus can have.

<table><caption>Table 6: Topological Types of some Real Toric Threefolds. The upper left box is purposefully marked with a question mark as we did not determine these topological types.</caption><thead><tr><th>(p; q)<sub>r</sub></th><th>(3;0)</th><th>(2;1)</th><th>(1;2)</th><th>(0;3)</th></tr></thead><tbody><tr><td>0</td><td>?</td><td>((h &middot; P<sup>2</sup>(R)) &middot; S<sup>1</sup>)<sub>h&ge;1</sub> ; (S<sup>1</sup>)<sup>3</sup> ; &empty;</td><td>(S<sup>1</sup>)<sup>3</sup> ; &empty;</td><td>(S<sup>1</sup>)<sup>3</sup> ; &empty;</td></tr><tr><td>1</td><td>n.a.</td><td>((h &middot; P<sup>2</sup>(R)) &rarr; X &rarr; S<sup>1</sup>)<sub>h&ge;1</sub><br/>(S<sup>1</sup> &middot; S<sup>1</sup>) &rarr; X &rarr; S<sup>1</sup><br/>h &middot; (S<sup>2</sup> &middot; S<sup>1</sup>)<br/>+k &middot; (P<sup>2</sup>(R) &middot; S<sup>1</sup>)<br/>+l &middot; P<sup>3</sup>(R)<br/>h,k,l&ge;0<br/>h+k+l&ge;1</td><td>(h &middot; P<sup>2</sup>(R) &middot; S<sup>1</sup>)<sub>2&ge;h&ge;0</sub><br/>(L(2k; l))<sub>gcd(2k;l)=1</sub><br/>&empty;</td><td>n.a.</td></tr></tbody></table>

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