Introduction↩︎

A remarkable result of Conner and Floyd states that any smooth complex algebraic variety \(V_\mathbb{C}\), whose real points \(V_\mathbb{R}\) are also smooth, is cobordant to the product \(V_\mathbb{R} \times V_\mathbb{R}\) in the unoriented bordism group [1]. This result allowed Milnor to show the existence of a surjective ring homomorphism \(\mu: \Omega^U_{2*} \to \Omega^O_*\) from the unitary bordism ring to the unoriented bordism ring (reducing the dimension by a half), by taking the real points of the set of generators of the unitary bordism ring given by the Milnor hypersurfaces and the complex projective spaces [2]. Milnor deduced from this surjectivity that an unoriented bordism class contains a complex manifold if and only if it contains a square \(N \times N\) [2]. Since Milnor was the first to study the properties of the homomorphism \(\mu: \Omega^U_{2*} \to \Omega^O_*\), we decided to call this homomorphism the Milnor map.

Notably, the Milnor hypersurfaces that appear in the kernel of the Milnor map have the special property of possessing a free conjugation; namely, an involution that anticommutes with the almost complex structure of the complex manifold, and such that the action of the involution is fixed point free. Motivated by this, Stong organized this extra structure in the framework of bordism, defined the bordism group of unitary manifolds with free conjugations, and showed that the kernel of the Milnor map was precisely the image of the forgetful homomorphism of this bordism group into the unitary bordism group [3].

It is worth noting that the above results of Milnor and Stong, relied heavily on knowing an explicit set of generators for both the unitary bordism group and the unoriented bordism group. The Milnor hypersurfaces, along with their real counterparts, constitute the set of generators having enough extra symmetries which allow us to completely understand the behavior of the Milnor map.

In this work we put forward the idea of generalizing the construction of the Milnor map to the equivariant setup and we present preliminary results in this direction. We consider the torus group \(T^k\) and its real subgroup \(\mathbb{Z}_2^k\), along with the equivariant bordism groups \(\Omega^{U,T^k}_{2*}\), \(\Omega^{U,\mathbb{Z}_2^k}_{2*}\) and \(\Omega^{O,\mathbb{Z}_2^k}_{*}\), which are the unitary equivariant bordism groups of \(T^k\) and \(\mathbb{Z}_2^k\), and the unoriented equivariant bordism group of \(\mathbb{Z}_2^k\), respectively. We setup our idea as a conjecture:

We present two results that support the veracity of the conjecture. First, we show that the conjecture holds for the Milnor maps \(\mu^T\) in Thm. 6 and \(\mu^{\mathbb{Z}_2}\) in Cor. 2 by constructing explicit complex projective generators for both the unitary bordism group of semi-free circle actions \(\overline{\Omega}^{U,T}_{2*}\) and the unitary bordism group of \(\mathbb{Z}_2\)-actions. Next, we show that the equivariant Milnor map \(\widehat{\mu}^{T^n} : \widehat{\Omega}_{2n}^{U, T^n} \rightarrow \widehat{\Omega}_{n}^{O, \mathbb{Z}_2^n}\) exists and is surjective in Thm. 7 once restricted to the equivariant bordisms of fully effective \(T^n\)- and \(\mathbb{Z}_2^n\)-actions, and further restricted to the manifolds of the appropriate dimension. Here we employ results from the third author and collaborators in the study of unitary torus manifolds.

If the equivariant Milnor map \(\mu^{\mathbb{Z}_2^k}\) exists, we expect the kernel to lie in the image of the \(\mathbb{Z}_2^k\)-equivariant bordism group of unitary manifolds with free conjugations. This bordism group is defined as the magnetic \(\mathbb{Z}_2^k\times \overline{\mathbb{Z}}_2\)-equivariant bordism group \(\mathbf{\Omega}^{\mathbb{Z}_2^k \times \overline{\mathbb{Z}}_2}_*(E\overline{\mathbb{Z}}_2)\) associated to the classifying space \(E\overline{\mathbb{Z}}_2\) for free \(\overline{\mathbb{Z}}_2\)-actions. Accordingly, assuming the existence of such equivariant Milnor map \(\mu^{\mathbb{Z}_2^k}\), we expect the sequence of homomorphisms \[\mathbf{\Omega}^{\mathbb{Z}_2^k \times \overline{\mathbb{Z}}_2}_{2*}(E\overline{\mathbb{Z}}_2) \stackrel{}{\longrightarrow} \Omega^{U,\mathbb{Z}_2^k}_{2*} \stackrel{\mu^{\mathbb{Z}_2^k}}{\longrightarrow} \Omega^{O,\mathbb{Z}_2^k}_{*} \to 0,\] to be exact, where the left-hand morphism is given by forgetting the free involution structure.

We have confirmed the existence of this exact sequence for the cases for \(k=0\) in Thm. 9 and \(k=1\) in Thm. 10 by constructing explicit equivariant unitary manifolds with free conjugations that generate the kernel of the Milnor map. We emphasize here that prior knowledge of the explicit manifold representatives of the kernel of the equivariant Milnor map is essential.

In order to obtain and present the stated results, we have organized this work into four sections.

In the first section, we extend to the equivariant setting some fundamental bordisms relating equivariant manifolds with involution to equivariant manifolds constructed out of the fixed points of an involution. We then present the Milnor hypersurfaces and we rework the proof due to Milnor, of the surjectivity of the Milnor map.

In the second section, we present a set of generators for the unitary bordism group of semi-free circle actions, and similarly, for the unoriented \(\mathbb{Z}_2\)-equivariant bordism groups. As previously noted, knowing the explicit sets of generators allowed us to prove the surjectivity of the equivariant Milnor map for the circle and the cyclic group of order two.

In the third section, we provide a summary of results for quasitoric manifolds which, once suitably organized, enabled us to show the surjectivity of the equivariant Milnor map \(\widehat{\mu}^{T^n}\) when restricted to the unitary bordisms of fully effective actions of the torus \(T^n\). Here the surjectivity is only proved for bordism groups of \(T^n\)-equivariant unitary manifolds of dimension \(2n\).

In the fourth and last section we introduce the magnetic unitary bordism groups and we highlight some of its properties. We then restrict the magnetic unitary bordism groups to those of free conjugations and construct a set of generators for this bordism group. We show that the kernel of the Milnor map is precisely the image of the forgetful map of the magnetic unitary bordism of free conjugations, and we construct a concrete set of manifolds realizing this kernel. We generalize this proof to the \(\mathbb{Z}_2\)-equivariant Milnor map, also showing a concrete set of manifolds realizing this kernel.

1 Equivariant bordisms and fixed points↩︎

1.1 Fundamental cobordisms↩︎

Here we will present a series of bordisms that relate an equivariant manifold with the normal bundle of the fixed points.

We will be interested in two types of manifolds. Unitary manifolds endowed with a compatible action of the torus \(T^k\), and unoriented manifolds endowed with an action of the group \(\mathbb{Z}_2^k\). Denote the corresponding bordism rings \(\Omega^{U, T^k}_*\) and \(\Omega^{O,\mathbb{Z}_2^k}_*\) respectively (see [4] for the definitions).

The following cobordisms are fundamental:

In the case that the \(\widehat{T}\)-action were free on the whole cobordism, we would obtain a \(T^k\)-equivariant cobordism between \(M\) and the quotient manifold: \[\begin{align} S(E\oplus \mathbb{C})/\widehat{T}. \end{align}\] By unwinding the definition of this quotient space, we are led to the concept of a twisted projective space, which we describe in the following definition.

Unfortunately, this projective space \(\mathbb{P}^t_{\mathbb{C}}(E^+ \oplus E^- \oplus \mathbb{C}^r)\) does not inherit the structure of a manifold unless the group \(\widehat{T}\) acts freely on the sphere bundle \(S(E^+ \oplus E^-)\). This happens precisely when the action of \(\widehat{T}\) is semi-free.

The unoriented version of Lem. 1 is the following:

Now, we want to apply the previous lemma to the case on which we have an almost complex manifold \((M,J)\) endowed with an involution \(\tau : M \to M\) which behaves like complex conjugation. Recall that an almost complex manifold is a manifold \(M\) endowed with and endomorphism \(J :TM \to TM\) such that \(J^2=-1\).

1.2 Milnor map↩︎

The previous result of Prop. 1 generalizes Conner and Floyd’s result [1] to the equivariant setup. Applied to the context of smooth complex algebraic varieties \(V_\mathbb{C}\) endowed with a unitary action of \(T^k\), we can say that the real points \(V_\mathbb{R}\) are endowed with an action of \(\mathbb{Z}_2^k \subset T^k\), and, if the real points \(V_\mathbb{R}\) are smooth, the manifold \(V_\mathbb{C}\) is cobordant to \(V_\mathbb{R} \times V_\mathbb{R}\) in the \(\mathbb{Z}_2^k \times \mathbb{Z}_2\)-equivariant unoriented bordism ring. By forgetting the conjugation involution, we have that \([V_\mathbb{C}]_2=[V_\mathbb{R} \times V_\mathbb{R}]_2\) in \(\Omega^{O, \mathbb{Z}_2^k}_*\).

In the non-equivariant setting, a smooth complex algebraic variety \(V_\mathbb{C}\) whose real points \(V_\mathbb{R}\) are smooth, is cobordant to the product \(V_\mathbb{R} \times V_\mathbb{R}\). Motivated by this remarkable result, Milnor noted that the image of the forgetful homomorphism \[\begin{align} F: \Omega^{U}_* \to \Omega^{O}_*, \;[M] \mapsto [M]_2 \end{align}\] from the unitary bordism ring \(\Omega^{U}_*\) to the unoriented bordism ring \(\Omega^{O}_*\), is precisely the subring generated by all squares \([N \times N]_2 \in \Omega^{O}_{2*}\) [2].

Since a (redundant) set of generators of the unitary bordism ring may be taken to be the complex manifolds \(\mathbb{C}P^n\) and the hypersurfaces: \[\begin{align} \label{Milnor95hypersurfaces95definition} H_\mathbb{C}(m,n) : = \{\big([\mathbf{w}], [\mathbf{z}]\big) \in \mathbb{P}_\mathbb{C}^m \times \mathbb{P}_\mathbb{C}^n \;| \;w_0z_0 + \cdots +w_mz_m=0 \} \end{align}\tag{1}\] for \(m \leq n\), and their real points \(\mathbb{R}P^n\) and \(H_\mathbb{R}(m,n)\) respectively (fixed point sets of the complex conjugation involutions) are also a (redundant) set of generators for the unoriented bordism ring, Milnor was able to show that there is a surjective ring homomorphism: \[\begin{align} \mu: \Omega_{2*}^U \to \Omega_{*}^O \end{align}\] defined by the real points of the complex manifolds that generate the unitary bordism ring: \[\begin{align} [\mathbb{P}_\mathbb{C}^m] \mapsto [\mathbb{P}_\mathbb{R}^m]_2, \;&\;[H_\mathbb{C}(m,n)] \mapsto [H_{\mathbb{R}}(m,n)]_2. \end{align}\]

Milnor further showed an explicit set of generators that realizes this map. Let us recall this proof.

Note that the forgetful homomorphism becomes the square of the Milnor map \(\mu\), thus producing the relation: \[\begin{align} [V_\mathbb{C}]_2=[V_\mathbb{R} \times V_\mathbb{R}]_2 \in \Omega^{O}_{2*}. \end{align}\]

An equivariant version of this homomorphism is one of the aims of this work. To define this map, we need to generalize to the equivariant setup certain properties used in the definition of the Milnor map. In particular we have:

• Both bordism rings \(\Omega^U_*\) and \(\Omega^O_*\) are free polynomial algebras. Namely, \(\mathbb{Z}[y_{2j} | j \in \mathbb{N}^{>0}]\) and \(\mathbb{Z}_2[x_k | k \in \mathbb{N}^{>0}, k \neq 2^s-1]\) respectively.

• The generators of the unitary bordism ring may be taken to be complex manifolds. Then the Milnor map is simply defined as the real points of those manifolds.

• The real points of the complex manifolds that generate the unitary bordism group also generate the unoriented bordism ring.

These three properties have to be accommodated to the equivariant setup.

2 Semi-free actions of the circle and unitary involutions↩︎

Consider the equivariant geometric bordism groups \(\Omega^{U,T}_*\) and \(\Omega_*^{O,\mathbb{Z}_2}\) of equivariant unitary manifolds with circle actions and equivariant unoriented manifolds with involutions [4]. One can define a homomorphism \(\Omega^{U,T}_{2*} \to \Omega_*^{O,\mathbb{Z}_2}\), using the explicit set of generators for the module \(\Omega^{U,T}_{2*}\) presented in [8], [9] that satisfy the properties listed above.

Instead, we will show that it is enough to consider a small class of \(T\)-equivariant manifolds in order to show the surjectivity to \(\Omega_*^{O,\mathbb{Z}_2}\). The manifolds of interest are \(T\)-equivariant unitary manifolds where the action is semi-free (see Def. 2).

2.1 Unitary bordism of semi-free circle actions↩︎

This bordism ring has been extensively studied and it has served as one first candidate to completely determine the ring structure of the equivariant bordism ring [10].

We claim that the generalization of the Milnor map restricted to semi-free actions \[\begin{align} \mu : \overline{\Omega}^{U,T}_{2*} \to {\Omega}^{O,\mathbb{Z}_2}_* \end{align}\] is surjective. Let us see how this map is constructed.

First note that both bordism rings \(\overline{\Omega}^{U,T}_{2*}\) and \({\Omega}^{O,\mathbb{Z}_2}_*\) are free modules over the rings \(\Omega^U_*\) and \(\Omega^O_*\) respectively [4], [11], [12]. This takes care of condition (a) since we will define the equivariant Milnor map using the generators as modules rather than rings.

To determine a set of generators for both bordism groups \(\overline{\Omega}^{U,T}_{2*}\) and \({\Omega}^{O,\mathbb{Z}_2}_*\) we will make use of Conner–Floyd’s long exact sequence for pairs of families of subgroups [13]. Let \(\mathcal{A}\) denote the family of all subgroups and \(\{\{1\}\}\) the family with only the trivial subgroup, both for \(T\) and \(\mathbb{Z}_2\), and take the associated long exact sequences: \[\begin{align} \label{short-exact-sequence-relative-semi-free} 0 \to \overline{\Omega}^{U,T}_{2*} \to \overline{\Omega}^{U,T}_{2*}\{\mathcal{A}, \{1\}\} \overset{\partial}{\to} \overline{\Omega}^{U,T}_{2*-1}\{\{1\}\} \to 0\\ 0 \to {\Omega}^{O,\mathbb{Z}_2}_{*} \to {\Omega}^{O,\mathbb{Z}_2}_{*}\{\mathcal{A}, \{1\}\} \overset{\partial}{\to} {\Omega}^{O,\mathbb{Z}_2}_{*-1}\{\{1\}\} \to 0 \end{align}\tag{3}\] which in both cases split as short exact sequences. This is because we can find sections for the boundary maps by the following argument. A manifold \([E]\) in \(\overline{\Omega}^{U,T}_{2*-1}\{\{1\}\}\) has a free action of \(T\) and therefore it is a principal \(T\)-bundle. Take the associated complex line bundle \(E \times_T \mathbb{C}\) and note that the ball bundle \(B(E \times_T \mathbb{C})\) of vectors of norm less than or equal to \(1\) defines a bordism class in \(\overline{\Omega}^{U,T}_{2*}\{\mathcal{A}, \{1\}\}\). Thus, we have the sections: \[\begin{align} \overline{\Omega}^{U,T}_{2*-1}\{ \{1\}\} & \to \overline{\Omega}^{U,T}_{2*}\{\mathcal{A}, \{1\}\} & {\Omega}^{O,\mathbb{Z}_2}_{*-1}\{ \{1\}\} & \to {\Omega}^{O,\mathbb{Z}_2}_{*}\{\mathcal{A}, \{1\}\} \\ [E]& \mapsto [B(E \times_T \mathbb{C)]} & [H] & \mapsto [B(H \times_{\mathbb{Z}_2} \mathbb{R})] \end{align}\] where in both cases \(\partial B(E \times_T \mathbb{C)} \cong E\) and \(\partial B(H \times_{\mathbb{Z}_2} \mathbb{R}) \cong H\).

Now we will construct projection maps \[\begin{align} \overline{\Omega}^{U,T}_{2*}\{\mathcal{A}, \{1\}\} &\to \overline{\Omega}^{U,T}_{2*} & {\Omega}^{O,\mathbb{Z}_2}_{*}\{\mathcal{A}, \{1\}\} &\to {\Omega}^{O,\mathbb{Z}_2}_{*} \end{align}\] using the fact that the relative bordism groups of the pair of families \(\{\mathcal{A}, \{1\}\}\) can be written in terms of non-equivariant bordism groups.

Take a manifold with boundary \((N, \partial N)\) defining a class in \(\overline{\Omega}^{U,T}_{2*}\{\mathcal{A}, \{1\}\}\); this is a unitary manifold \(N\) with a compatible action of \(T\) such that \(T\) acts freely on \(\partial N\). Let \(F=N^T\) be the fixed point set of the action, and write it as a union of connected components \(F = \cup_m F_m\) where \(n_m = \mathrm{dim}_\mathbb{R}(F_m)\). Clearly, each connected component \(F_m\) is a closed submanifold embedded in the interior of \(N\) since \(\partial F=F\cap \partial N\) is empty. Denote \(E_m \to F_m\) the normal bundles of \(F_m\) in \(N\) and note that the circle action splits each \(E_m\) into the Whitney sum \(E_m^+\oplus E_m^-\) where \(T\) acts on \(E_m^+\) by multiplication and on \(E_m^-\) by multiplication of the inverse. Denote the rank of these vector bundles by \(n_m^\pm\mathrel{\vcenter{:}}= \mathrm{rank}_\mathbb{C}E_m^\pm\).

There is a cobordism \[\begin{align} (N,\partial N) \sim \bigcup_m \big(B(E_m^+ \oplus E_m^-),S(E_m^+ \oplus E_m^-)\big) \end{align}\] which allows us to make the identification \[\begin{align} \overline{\Omega}^{U,T}_{2n}\{\mathcal{A}, \{1\}\} \cong &\bigoplus_{k+k^++k^-=n} \Omega^{U}_{2k}(BU(k^+)\times BU(k^-)) \\ [(N,\partial N)] \mapsto & \sum_{m} [F_m \to BU(n_m^+)\times BU(n_m^-)] \end{align}\] where the maps \(F_m \to BU(n_m^+)\times BU(n_m^-)\) classify the vector bundles \(E_m^+ \oplus E_m^-\).

We define the homomorphism \[\begin{align} \Omega^{U}_{2k}(BU(k^+)\times BU(k^-)) &\to \overline{\Omega}^{U,T}_{2n} \\ [F \to BU(k^+) \times BU(k^-)] &\mapsto [\mathbb{P}^t_\mathbb{C}(E^+ \oplus E^- \oplus \mathbb{C})] \end{align}\] where \(E^+ \oplus E^-\) is the vector bundle that the map \(F \to BU(k^*) \times BU(k^-)\) classifies, and \(\mathbb{P}^t_\mathbb{C}(E^+ \oplus E^- \oplus \mathbb{C})\) is the twisted projective space of Def. 1 and appearing in Lem. 2.

The twisted projective space \(\mathbb{P}^t_\mathbb{C}(E^+ \oplus E^- \oplus \mathbb{C})\) inherits a \(T\)-action which is given by the following equation in homogeneous coordinates: \[\begin{align} t \cdot [v^+:v^-:z] = [tv^+:t^{-1}v^-:z]. \end{align}\]

The \(T\)-action on the twisted projective space is semi-free and its \(T\)-fixed points are: \[\begin{align} \mathbb{P}^t_\mathbb{C}(E^+ \oplus E^- \oplus \mathbb{C})^T = F \sqcup \mathbb{P}^t_\mathbb{C}(E^+ \oplus E^- ), \end{align}\] with normal bundles isomorphic to the following bundles: \[\begin{align} E^+ \oplus E^- &\to F & \mathbb{C} \times \mathbb{P}^t_\mathbb{C}(E^+ \oplus E^- ) & \to \mathbb{P}^t_\mathbb{C}(E^+ \oplus E^- ). \end{align}\] The \(T\)-action on the first bundle matches the original \(T\)-action on \(E^+ \oplus E^-\), while on the second the \(T\)-action is given by the following equation: \[\begin{align} t \cdot (z, [v^+:v^-]) = (t^{-1}z,[v^+:v^-]), \end{align}\] where \(t \in T\), \(z \in \mathbb{C}\) and \([v^+:v^-] \in \mathbb{P}^t_\mathbb{C}(E^+ \oplus E^- )\). We obtain the following result which is essentially in [8].

Lemma 5 allows us to generate elements of the semi-free bordism group with the twisted projective spaces. Therefore, we now need to find a good set of generators for the non-equivariant bordism groups \(\Omega^{U}_{2k}(BU(k^+)\times BU(k^-))\). Fortunately this has already been addressed by Conner and Floyd and presented in [14].

The canonical map \(BU(1)^n \to BU(n)\) induces a map in bordism \(\Omega_*^U(BU(1)^n) \to \Omega_*^U(BU(n))\). From this map the bordism group \(\Omega_*^U(BU(n))\) is a free \(\Omega^{U}_*\)-module with basis: \[\begin{align} \{ \alpha_{s_1s_2 \dots s_n} | 0\leq s_1 \leq s_2 \leq \cdots \leq s_n \} \end{align}\] where \(\alpha_{s_1s_2 \dots s_n}\) represents the classifying map of the product of the tautological line bundles \[\begin{align} \label{definition95bundle95over95CP} \xymatrix{ \gamma_{s_1} \times \cdots \times \gamma_{s_n} \ar[d]& \\ \mathbb{P}_\mathbb{C}^{s_1} \times \cdots \times \mathbb{P}_\mathbb{C}^{s_n} \ar[r] & BU(n). } \end{align}\tag{4}\]

Therefore \(\Omega^{U}_{2k}(BU(k^+)\times BU(k^-))\) is a free \(\Omega^{U}_*\)-module with basis the classifying maps of the bundles: \[\begin{align} \xymatrix{ \gamma_{s_1}^+ \times \cdots \times \gamma_{s_{k^+}}^+ \times \gamma_{r_1}^- \times \cdots \times \gamma_{r_{k^-}}^- \ar[d]& \\ \mathbb{P}_\mathbb{C}^{s_1} \times \cdots \times \mathbb{P}_\mathbb{C}^{s_{k^+}} \times \mathbb{P}_\mathbb{C}^{r_1} \times \cdots \times \mathbb{P}_\mathbb{C}^{r_{k_-}}\ar[r] & BU(k^+) \times BU(k^-). } \end{align}\]

Therefore we obtain the following result:

Note that the twisted projective spaces of Prop. 4 define projective fibrations over projective spaces: \[\begin{align} \xymatrix{ \mathbb{P}_\mathbb{C}(\mathbb{C}^{k^++1} \times \overline{\mathbb{C}}^{k^-}) \ar[r] &\mathbb{P}^t_\mathbb{C}(\gamma_{s_1}^+ \times \cdots \times \gamma_{s_{k^+}}^+ \times \gamma_{r_1}^- \times \cdots \times \gamma_{r_{k^-}}^- \oplus \mathbb{C}) \ar[d]\\ & \mathbb{P}_\mathbb{C}^{s_1} \times \cdots \times \mathbb{P}_\mathbb{C}^{s_{k^+}} \times \mathbb{P}_\mathbb{C}^{r_1} \times \cdots \times \mathbb{P}_\mathbb{C}^{r_{k^-}}. } \end{align}\]

2.2 \(\mathbb{Z}_2\)-equivariant unoriented bordism↩︎

The unoriented version of the previous procedure works without complications. We have the isomorphism: \[\begin{align} {\Omega}^{O,\mathbb{Z}_2}_{n}\{\mathcal{A}, \{1\}\} \cong &\bigoplus_{k+k'=n} \Omega^{O}_{k}(BO(k')) \\ [(N,\partial N)] \mapsto & \sum_{m} [F_m \to BO(n_m')] \end{align}\] where \(F=\cup_m F_m\) is the fixed point set of the \(\mathbb{Z}_2\)-action on \(N\), \(F_m\)’s are the connected components, and the map \(F_m \to BO(n_m')\) classifies the normal bundle of \(F_m\) in \(N\) whose rank is \(n_m'\).

We can construct closed \(\mathbb{Z}_2\)-equivariant manifolds from the information on the normal bundles using the following homomorphism: \[\begin{align} \Omega^{O}_{k}(BO(k')) &\to {\Omega}^{O,\mathbb{Z}/2}_{n} \\ [F \to BO(k')] &\mapsto [\mathbb{P}_\mathbb{R}(E \oplus \mathbb{R})], \end{align}\] with \(E \to F\) the real bundle classified by the map \(F \to BO(k')\), and \(\mathbb{P}_\mathbb{R}(E \oplus \mathbb{R})\) is the projectivization of the bundle \(E\oplus \mathbb{R}\) with \(\mathbb{Z}_2\)-action defined by the equation: \[\begin{align} t \cdot [\nu : x] = [tv:x] \end{align}\] where \(t =\pm 1\), \(\nu \in E\) and \(x \in \mathbb{R}\).

The fixed point sets of this projective space are: \[\begin{align} \mathbb{P}_\mathbb{R}(E \oplus \mathbb{R})^{\mathbb{Z}_2} = F \sqcup \mathbb{P}_\mathbb{R}(E) \end{align}\] with normal bundles isomorphic to the following bundles: \[\begin{align} E &\to F & \mathbb{R}\times \mathbb{P}_\mathbb{R}(E) \to \mathbb{P}_\mathbb{R}(E) \end{align}\] and with the \(\mathbb{Z}_2\)-actions on both bundles non-trivial.

Analogously as in Lem. 5, we have the following result:

We are now left, once again, with the task of finding a good set of generators for the bordism group \(\Omega^{O,\mathbb{Z}_2}_k(BO(k'))\). Thom proved the isomorphism [5]: \[\begin{align} H_*(BO(n); \mathbb{Z}_2) \otimes_{\mathbb{Z}_2} \Omega^{O}_* \cong \Omega^O_*(BO(n)) \end{align}\] and the canonical map \(BO(1)^{n} \to BO(n)\) induces an isomorphism of groups \[\begin{align} H_*(BO(1)^{n}; \mathbb{Z}_2)^{\mathfrak{S}_{n}} \cong H_*(BO(n);\mathbb{Z}_2) \end{align}\] where the symmetric group \(\mathfrak{S}_{n}\) acts by permutation. This isomorphism follows from the isomorphism in cohomology: \[\begin{align} H^*(BO(n); \mathbb{Z}_2) \cong \mathbb{Z}_2[\omega_1, \dots, \omega_n] \cong \mathbb{Z}_2[x_1, \cdots, x_n]^{\mathfrak{S}_n} \cong H^*(BO(1)^n; \mathbb{Z}_2)^{\mathfrak{S}_n}. \end{align}\]

Since the homology of \(BO(1) = \mathbb{P}_\mathbb{R}^\infty\) is generated by the canonical maps \(\mathbb{P}_\mathbb{R}^s \to \mathbb{P}_\mathbb{R}^\infty\), we see that \(\Omega_*^O(BO(n))\) is a free \(\Omega^{O}_*\)-module with basis: \[\begin{align} \{ \xi_{u_1u_2 \dots u_n} | 0\leq u_1 \leq u_2 \leq \cdots \leq u_n \} \end{align}\] where \(\xi_{u_1u_2 \dots u_n}\) represents the classifying map of the product of the tautological line bundles \[\begin{align} \xymatrix{ \eta_{u_1} \times \cdots \times \eta_{u_n} \ar[d]& \\ \mathbb{P}_\mathbb{R}^{u_1} \times \cdots \times \mathbb{P}_\mathbb{R}^{u_n} \ar[r] & BO(n). } \end{align}\]

Analogously to Prop. 4 we obtain the following result:

Now that we have explicit bases for the bordism groups \(\overline{\Omega}^{U,T}_{2*}\) and \({\Omega}^{O,\mathbb{Z}_2}_{*}\) comprised by complex and real algebraic varieties presented in Prop. 4 and Prop. 5 respectively, we just need to check that the set of real points of the complex algebraic varieties that generate the bordism group of semi-free actions spans the basis of real algebraic varieties of the \(\mathbb{Z}_2\)-equivariant unoriented bordism group.

The real points of the bundle \(\gamma_s \to \mathbb{P}_\mathbb{C}^s\) is the bundle \(\eta_s \to \mathbb{P}_\mathbb{R}^s\), and therefore the real points of the twisted projective bundle: \[\begin{align} \mathbb{P}^t_\mathbb{C}(\gamma_{s_1}^+ \times \cdots \times \gamma_{s_{k^+}}^+ \times \gamma_{r_1}^- \times \cdots \times \gamma_{r_{k^-}}^- \oplus \mathbb{C}) \end{align}\] is the real projective bundle: \[\begin{align} \mathbb{P}_\mathbb{R}(\eta_{s_1} \times \cdots \times \eta_{s_{k^+}} \times \eta_{r_1} \times \cdots \times \eta_{r_{k^-}} \oplus \mathbb{R}) \end{align}\] where the induced \(\mathbb{Z}_2\)-action reduces to the action: \[\begin{align} t \cdot [\nu_1: \dots : \nu_{{k+}}:\nu'_1: \cdots : \nu'_{k^-}:x] = [t\nu_1: \dots : t\nu_{{k+}}:t\nu'_1: \cdots : t\nu'_{k^-}:x] \end{align}\] for \(t = \pm 1\).

2.3 Equivariant Milnor map↩︎

We now have all the necessary ingredients to deduce the existence of the Milnor map for semi-free actions based on Prop. 4 and Prop. 5:

We apply Prop. 1 to the previous result and we obtain a generalization of Milnor’s result [2]:

One notices that there is an intermediate bordism ring between the bordism ring of unitary semi-free actions and the \(\mathbb{Z}_2\)-equivariant unoriented bordism ring. This is the \(\mathbb{Z}_2\)-equivariant unitary bordism \(\Omega^{U, \mathbb{Z}_2}_*\) of unitary involutions. Moreover, this ring sits in the middle of the ring homomorphisms: \[\begin{align} \overline{\Omega}^{U,T}_{2*} \to {\Omega}^{U,\mathbb{Z}_2}_{2*} \to {\Omega}^{O,\mathbb{Z}_2}_{2*}, \end{align}\] where the first map restricts the action to the subgroup \(\mathbb{Z}_2 \subset T\) and the second map forgets the unitary structure.

A similar argument as the one given in Prop. 4 permits to show that any element of the the bordism ring \({\Omega}^{U,\mathbb{Z}_2}_{2*}\) may be written as a finite sum of projective spaces \[\begin{align} [M_0] \times [\mathbb{P}_\mathbb{C}(\gamma_{s_1} \times \cdots \times \gamma_{s_{k}} \oplus \mathbb{C})] \end{align}\] where \([M_0] \in \Omega^{U}_{2*}\) is a unitary manifold with trivial \(\mathbb{Z}_2\)-action, and the projective spaces are defined as in Eqn. 4 .

Since the \(\mathbb{Z}_2\)-equivariant Milnor map \(\mu^{\mathbb{Z}_2}\) is compatible with the Milnor map on the Grassmannians: \[\begin{align} \mu^k : \Omega^{U}_{2*}(BU(k)) \to \Omega_*^O(BO(k)), \end{align}\] then we obtain the following result.

This generalizes the calculation of the kernel of the Milnor map presented in Lem. 4. Here, it is worth noting that the Milnor hypersurfaces \(H_{\mathbb{C}}(2^{s-1},2^{s-1})\) are taken as unitary manifolds with trivial \(\mathbb{Z}_2\)-action.

3 Actions with isolated fixed points↩︎

As seen, a key point for the existence of an equivariant Milnor map from \(\Omega_{2*}^{U, T^k}\) to \(\Omega_*^{O, \mathbb{Z}_2^k}\) depends upon whether there is an involution on a representative of each class in \(\Omega_{2*}^{U, T^k}\) such that its fixed point set can admit a \(\mathbb{Z}_2^k\)-action, representing a class in \(\Omega_*^{O, \mathbb{Z}_2^k}\).

The seminal work of Davis and Januszkiewicz [7] provides the possibility for further work. Davis and Januszkiewicz introduced and studied two kinds of equivariant manifolds, quasitoric manifolds and small covers, both of which are topological versions of toric varieties. These are characterized as follows.

Let \(d = 1\) if \(\mathbb{F} = \mathbb{R}\), and \(d = 2\) if \(\mathbb{F} = \mathbb{C}\). A \(G_{\mathbb{F}}^n\)-manifold \(M^{dn}\) over a simple \(n\)-polytope \(P^n\) is a smooth closed manifold (which is required to be oriented if \(d = 2\)) with a locally standard \(G_{\mathbb{F}}^n\)-action such that its orbit space is homeomorphic to \(P^n\). Here, \(G_{\mathbb{F}}^n\) is \(\mathbb{Z}_2^n\) if \(\mathbb{F} = \mathbb{R}\), and \(T^n\) if \(\mathbb{F} = \mathbb{C}\), and a locally standard action means that the \(G_{\mathbb{F}}^n\)-action is locally isomorphic to a standard faithful representation of \(G_{\mathbb{F}}^n\) on \(\mathbb{F}^n\). Such a \(G_{\mathbb{F}}^n\)-manifold is called a small cover if \(\mathbb{F} = \mathbb{R}\) and a quasitoric manifold if \(\mathbb{F} = \mathbb{C}\). It was shown in [15] that every quasitoric manifold always admits a unitary structure, which depends on a choice of its omniorientations.

A \(G_{\mathbb{F}}^n\)-manifold \(\pi: M^{dn} \to P^n\) determines a characteristic function \(\lambda_{\mathbb{F}}: \mathcal{F}(P^n) \to \text{Hom}(G_{\mathbb{F}}, G_{\mathbb{F}}^n)\), satisfying that for each vertex \(v = F_{i_1} \cap \dots \cap F_{i_n}\) of \(P^n\), the elements \(\lambda_{\mathbb{F}}(F_{i_1}), \dots, \lambda_{\mathbb{F}}(F_{i_n})\) form a basis of \(\text{Hom}(G_{\mathbb{F}}, G_{\mathbb{F}}^n)\), where \(\mathcal{F}(P^n) = \{F_1, \dots, F_m\}\) is the set of all facets of \(P^n\). Note that \(\text{Hom}(G_{\mathbb{F}}, G_{\mathbb{F}}^n)\) is isomorphic to \(\mathbb{Z}_2^n\) if \(\mathbb{F} = \mathbb{R}\) and \(\mathbb{Z}^n\) if \(\mathbb{F} = \mathbb{C}\).

Davis and Januszkiewicz [7] tell us that one can use the principal \(G_{\mathbb{F}}^n\)-bundle \(P^n\times G_{\mathbb{F}}^n\) over \(P^n\) to reconstruct \(M^{dn}\) via \(\lambda_{\mathbb{F}}\). First \(\lambda_{\mathbb{F}}\) gives the following equivalence relation \(\sim_{\lambda_\mathbb{F}}\) on \(P^n\times G_{\mathbb{F}}^n\) \[\label{equiv} (x, g)\sim_{\lambda_{\mathbb{F}}} (y, h)\Longleftrightarrow \begin{cases} x=y, g=h & \text{ if } x\in \text{\rm int}(P^n)\\ x=y, g^{-1}h\in G_F &\text{ if } x\in \text{\rm int}F\subset \partial P^n \end{cases}\tag{6}\] where \(G_F\) is explained as follows: for each point \(x\in \partial P^n\), there exists a unique face \(F\) of \(P^n\) such that \(x\) is in its relative interior. If \(\dim F=k\), then there are \(n-k\) facets, say \(F_{i_1}, ..., F_{i_{n-k}}\), such that \(F=F_{i_1}\cap\cdots \cap F_{i_{n-k}}\), and furthermore, \(\lambda_{\mathbb{F}}(F_{i_1}), ..., \lambda_{\mathbb{F}}(F_{i_{n-k}})\) determine a subgroup of rank \(n-k\) in \(G_{\mathbb{F}}^n\), denoted by \(G_F\). Then the reconstruction of \(M^{dn}\) is exactly the quotient space \(P^n\times G_{\mathbb{F}}^n/\sim_{\lambda_{\mathbb{F}}}\), denoted by \(M(P^n, \lambda_{\mathbb{F}})\).

It is clear that the complex conjugation on \(T^n\subset \mathbb{C}^n\) induces an involution \(\tau\) on \(M(P^n, \lambda_\mathbb{C})\) such that the fixed point set \(M(P^n, \lambda_\mathbb{C})^\tau\) is just the small cover \(M(P^n, \lambda_\mathbb{R})\), where \(\lambda_\mathbb{R}\) is the mod 2 reduction of \(\lambda_\mathbb{C}\). On the other hand, a natural question arises: can every small cover be obtained in such a way? This is equivalent to the following lifting problem: Suppose that there exists a characteristic function \(\lambda_{\mathbb{R}}: \mathcal{F}(P^n) \to \mathbb{Z}_2^n\). Does there exist a characteristic function \(\lambda_{\mathbb{C}}: \mathcal{F}(P^n) \to \mathbb{Z}^n\) such that the following diagram commutes? \[\xymatrix{ & \mathbb{Z}^n \ar[d]^{\mod 2}\\ \mathcal{F}(P^n) \ar[ur]^{\lambda_{\mathbb{C}}}\ar[r]_{\lambda_{\mathbb{R}}} & \mathbb{Z}_2^n. }\] It is worth pointing out that some related work has been carried out in the papers [16]–[19], among others.

It was shown by the third author and collaborators [20], [21] the following results:

Note that for \(n=1\), \(\widehat{\Omega}_{2}^{U, T}\cong \mathbb{Z}\) is exactly generated by the class of \(\mathbb{C}P^1\) with the standard \(T\)-action, while \(\widehat{\Omega}_1^{O,\mathbb{Z}_2}\cong \{0\}\).

Furthermore, [21] demonstrates that the groups \(\widehat{\Omega}_{n}^{O, \mathbb{Z}_2^n}\) and \(\widehat{\Omega}_{2n}^{U, T^n}\) are generated by certain special types of small covers and omnioriented quasitoric manifolds, respectively.

Now with these preliminaries, we can define a group homomorphism \[\begin{align} \widehat{\mu}^{T^n} : \widehat{\Omega}_{2n}^{U, T^n} \longrightarrow \widehat{\Omega}_{n}^{O, \mathbb{Z}_2^n} \end{align}\] as follows. Choose a basis \(S\) of \(\widehat{\Omega}_{2n}^{U, T^n}\) as described in Lem. 9 (2). For any \([M(P^n, \lambda_\mathbb{C})]\in S\), set \(\widehat{\mu}^{T^n}([M(P^n, \lambda_\mathbb{C})]) = [M(P^n,\lambda_\mathbb{R})]_2\). From the corresponding fixed point data, which is a formal sum of all tangent representations at fixed points [20], [22], [23], the image of \([M(P^n, \lambda_\mathbb{C})]\) under the map \(\widehat{\mu}^{T^n}\) is independent of the choice of omnioriented quasitoric manifolds. Then extend \(\widehat{\mu}^{T^n}\) linearly to the whole group \(\widehat{\Omega}_{2n}^{U, T^n}\).

For an arbitrary element \(x = [M(P^n, \lambda_\mathbb{C})]\) in \(\widehat{\Omega}_{2n}^{U, T^n}\), write \(x = \sum a_i s_i\) with \(s_i\in S\) and \(a_i\in \mathbb{Z}\). Then \(\widehat{\mu}^{T^n}(x) = \sum a_i \widehat{\mu}^{T^n}(s_i)\). As can be seen from the fixed point data, this image is exactly \([M(P^n, \lambda_\mathbb{R})]_2\).

Obviously, if the lifting problem holds, then \(\widehat{\mu}^{T^n}\) would be surjective. However, the problem remains open except for some special cases. For example, \(|\mathcal{F}(P^n)| - n = 3, 4\) [17], [19]. Nevertheless, we can still prove that \(\widehat{\mu}^{T^n}\) is surjective for the following reason. If \(P^n\) is a product of some simplices, it is well-known that the lifting problem always holds (see Example 2.7 in [19]). Therefore, together with Lemma 9 (1), we have the following result:

Theorem 7 indicates that the lifting problem holds in the sense of equivariant bordism. This conclusion was already proved in Tan’s thesis [24]. We revisit it here in order to present another equivariant version of Milnor map and keep the exposition self-contained.

4 Magnetic unitary bordism↩︎

Almost complex manifolds and stably almost complex manifolds with compatible conjugation maps have been the subject of study in several authors [3], [25]–[29]. Here we propose a bordism theory which incorporates both, the stably unitary structure, and the compatible conjugation involution. This defines the bordism theory associated to magnetic equivariant vector bundles [30].

4.1 Magnetic equivariant vector bundles↩︎

A magnetic group is a group \(G\) endowed with a \(\mathbb{Z}_2\) grading defined by a surjective homomorphism \(\phi : G \to \overline{\mathbb{Z}}_2\). Here \(\overline{\mathbb{Z}}_2\) will simply be the group \({\mathbb{Z}}_2\), the line over it has the purpose of distinguishing from other \(\mathbb{Z}_2\) groups that may appear in \(G\). A magnetic representation (also known as a corepresentation in Wigner’s book [31]) is a complex vector space \(V\) where the group \(G\) acts such that the elements of \(G_0 = \mathrm{ker}(\phi)\) act complex linearly and the elements of \(G \backslash G_0\) act complex antilinearly. Morphisms of magnetic representations are defined by \(G\)-equivariant complex linear maps.

If \(G\) is a compact Lie group, the irreducible magnetic representations of \(G\) can be classified by its algebra of \(G\)-equivariant endomorphisms. This can also be seen by the type of complex irreducible representations that appear once the magnetic representation is restricted to the subgroup \(G_0\). If \(V\) is a magnetic irreducible representation and \(\mathrm{res}^G_{G_0}\) denotes the restriction to \(G_0\), then Wigner proved: \[\begin{align} \mathrm{End}_G(V) \cong\begin{cases} \mathbb{R} & \mathrm{res}^G_{G_0}V \in \mathrm{Irrep}(G_0)\\ \mathbb{H} & \mathrm{res}^G_{G_0}V \cong W \oplus W \\ \mathbb{C} & \mathrm{res}^G_{G_0}V \cong W \oplus \overline{W} \end{cases} \end{align}\] where \(W \in \mathrm{Irrep}(G_0)\) and \(\overline{W}\) is its conjugate representation (taking into account the complex conjugation and the conjugation action of \(\overline{\mathbb{Z}}_2\) in \(\mathrm{Irrep}(G_0)\)).

If \(X\) is a finite \(G\)-equivariant \(CW\)-complex, a magnetic \(G\)-equivariant vector bundle over \(X\) is a complex vector bundle \(E\) endowed with a compatible action of \(G\) such that it is complex linear on the fibers for \(G_0\) and complex antilinear for \(G \backslash G_0\).

The Grothendieck group of isomorphism classes of magnetic \(G\)-equivariant vector bundles over \(X\) defines the magnetic \(G\)-equivariant K-theory \(\mathbf{K}_G(X)\) and the suspensions \[\widetilde{\mathbf{K}}_G^{-q}(X) \mathrel{\vcenter{:}}= \widetilde{\mathbf{K}}_G(\Sigma^qX)\] and Bott periodicity \({\mathbf{K}}_G^{-q} \cong {\mathbf{K}}_G^{-q-8}\) permits to define a \(G\)-equivariant cohomology theory \({\mathbf{K}}_G^{*}\).

This K-theory comes with a natural forgetful map to complex \(G_0\)-equivariant K-theory \(\mathbf{K}_G^*(X) \to KU^*_{G_0}(X)\), landing in the \(\overline{\mathbb{Z}}_2\)-invariant part and becoming a rational isomorphism [32]: \[\begin{align} \mathbf{K}_G^*(X) \otimes \mathbb{Q} \overset{\cong}{\to} KU^*_{G_0}(X)^{\overline{\mathbb{Z}}_2} \otimes \mathbb{Q}. \end{align}\]

4.2 Magnetic equivariant bordism↩︎

The magnetic \(G\)-equivariant bordism is the bordism theory framed on stably magnetic \(G\)-equivariant bundles. An \(n\)-dimensional manifold \(M\) endowed with a \(G\)-action is said to be stably magnetic \(G\)-equivariant if the stabilization of the tangent bundle can be endowed with the structure of a magnetic \(G\)-equivariant vector bundle. This is an isomorphism of real \(G\)-equivariant bundles \[\begin{align} TM^n \oplus \theta \cong \xi^N_\mathbb{C} \end{align}\] where \(\xi_\mathbb{C}\) is a magnetic \(G\)-equivariant vector bundle over \(M\) of complex rank \(N\), \(\theta\) is a trivial real vector bundle of rank \(2N-n\) over \(M\) endowed with an involution of real vector bundles (or a \(\mathbb{Z}_2\)-action) \(\mu : \theta \to \theta\) compatible with the \(G\)-action on \(M\), taken as a \(G\)-equivariant real vector bundle through the homomorphism \(\phi\). This means that \(G\) acts on \(\theta \simeq M \times \mathbb{R}^{2N-m}\) by the equation \[\begin{align} g(m,\lambda) = \begin{cases} (gm, \mu(\lambda)) & \phi(g)=1\\ (gm,\lambda) & \phi(g)=0. \end{cases} \end{align}\] Two stable magnetic \(G\)-equivariant structures \(TM_i \oplus \theta_i \cong (\xi_i)_\mathbb{C}\), \(i =1,2\), are stably equivalent if there is a \(G\)-equivariant diffeomorphism \(f: M_1 \overset{\cong}{\to} M_2\) and an isomorphism of magnetic \(G\)-equivariant bundles \[\begin{align} TM_1 \oplus \theta_1 \oplus \mathbb{C}^{k_1} \cong (\xi_1)_\mathbb{C} \oplus \mathbb{C}^{k_1} \cong (\xi_2)_\mathbb{C} \oplus \mathbb{C}^{k_2} \cong TM_2 \oplus \theta_2 \oplus \mathbb{C}^{k_2} \end{align}\] for some \(k_1\) and \(k_2\), compatible with the differential \(df_*:TM_1 \to TM_2\). Here, the magnetic \(G\)-equivariant structure on the extra copies of \(\mathbb{C}\) is given by the homomorphism \(\phi : G \to \overline{\mathbb{Z}}_2\); namely complex conjugation via \(\phi\).

In terms of framings it is equivalent to find a lift: \[\begin{align} \xymatrix{ && \mathbb{B}_GU(N) \ar[d] \ar[r] &\mathbb{B}_GU \ar[d] \\ M \ar[r]_h \ar@{.>}[rru]^\xi& B_GO(n) \ar[r]_{\times \theta} & B_GO(2N) \ar[r] &B_GO } \end{align}\] where \(B_GO(n)\) is the classifying space for \(G\)-equivariant real bundles, \(h\) denotes the classifying map of the tangent bundle, and \(\mathbb{B}_GU(N)\) is the classifying space of magnetic \(G\)-equivariant vector bundles. This last space can be taken to be the Grassmannian of \(N\)-dimensional complex planes in a complete universe for magnetic \(G\)-equivariant representations. The spaces \(\mathbb{B}_GU\) and \(B_GO\) denote the limit of the stabilizations.

The manifold \(M\) bounds in this context if there is a magnetic \(G\)-equivariant manifold \(W\) such that \(\partial W=M\). In other words, there exists a lift \(W \to \mathbb{B}_GU\) of the tangent bundle \(W \to B_GO\) such that the restriction map \(\partial W\to \mathbb{B}_GU\) is \(G\)-homotopic to the map \(\xi\). Note that the restriction of a stably magnetic \(G\)-equivariant bundle on \(W\) to its boundary \(M\): \[\begin{align} (TW \oplus \theta )|_M \cong TM \oplus \mathbb{R} \oplus \theta|_M \end{align}\] is also a stably magnetic \(G\)-equivariant bundle. Note that the normal bundle of \(M\) in \(W\) is trivial and the elements in \(G \backslash G_0\) act trivially on the fibers. The only non-trivial action comes from the group \(\overline{\mathbb{Z}}_2\) and its action can only be trivial or the sign representation.

These bordism groups \(\mathbf{\Omega}^G_*(X)\) define a \(G\)-equivariant homology theory following the general structure defined by Conner and Floyd [1].

Note that the magnetic bordism groups come with natural forgetful maps: \[\begin{align} \Omega^{O,G}_* \leftarrow \mathbf{\Omega}^G_* \to \Omega^{U,G_0}_* \end{align}\] where the first forgets the unitary structure, and the second restricts the bordisms to a \(G_0\)-action.

4.3 Magnetic unitary bordism↩︎

We are particularly interested in the magnetic bordism groups \(\mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_*\) and \(\mathbf{\Omega}^{T^k \rtimes \overline{\mathbb{Z}}_2}_*\). Here, the magnetic structure of the groups is given by the projection of the groups to \(\overline{\mathbb{Z}}_2\), and the action of \(\overline{\mathbb{Z}}_2\) on the torus \(T^k\) is given by complex conjugation.

Before analyzing these bordism groups let us briefly mention some of the previous related results. Landweber was the first to study the \(\mathbb{Z}_2\)-equivariant homotopy of the unitary bordism spectrum \(MU\) defining the homotopy bordism groups [26]: \[\begin{align} MU_{p,q} = \pi_{p,q}(MU) = \lim_k \pi_{p+k,q+k}(MU(k)) =\lim_k [(B^{p+k,q+k}, \partial B^{p+k,q+k}), (MU(k),*)]_{\mathbb{Z}_2} \end{align}\] where \(B^{p,q}\) is the unit in \(\mathbb{R}^{p,q}\), its \(\mathbb{Z}_2\)-structure maps \((x,y)\) to \((-x,y)\), \(MU(k)\) is the Thom space of the canonical bundle over \(BU(k)\) with the natural conjugation map, and \([-,-]_{\mathbb{Z}_2}\) denotes the homotopy classes of \(\mathbb{Z}_2\)-equivariant maps. There are canonical maps: \[\begin{align} \Omega^{O}_q \leftarrow MU_{p,q} \to \Omega^U_{p+q} \end{align}\] where the first restricts to the \(\mathbb{Z}_2\) fixed points, and the second forgets the \(\mathbb{Z}_2\)-action. Landweber further shows that the following diagram is commutative: \[\begin{align} \xymatrix{ MU_{n,n} \ar@{->>}[r] \ar@{->>}[d] & \Omega^U_{2n} \ar@{->>}[ld]^\mu \\ \Omega^O_n } \end{align}\] where both the horizontal and the vertical maps are surjective and \(\mu\) denotes the Milnor map. Moreover, the horizontal map is a rational isomorphism and its kernel is 2-torsion.

Our interest lies in the geometric version of Landweber bordism groups. Allow us to make the following definition:

A magnetic unitary manifold consists of the following information:

• A manifold \(M\) endowed with an involution \(\tau : M \to M\).

• A stably almost complex structure \(J : TM \oplus \theta \to TM \oplus \theta\) where \(\theta\) is a trivial real bundle over \(M\).

• A lift of the involution \(\tau_* : TM \oplus \theta \to TM \oplus \theta\) such that \(\tau_* \circ J = - J \circ \tau_*\) and \(\tau_*^2=1.\)

Although the three maps \(\tau\), \(J\) and \(\tau_*\) are essential, we will identify the magnetic unitary class of this manifold with the pair \((M, \tau_*)\); the stably almost complex structure will be clear from the context,

The additive inverse \(-[(M, \tau_*)]\) of the bordism class \([(M, \tau_*)]\) consists of the same manifold \(M\) with involution \(\tau\), together with the stably almost complex structure \(J \oplus (-i) : TM \oplus \theta \oplus \mathbb{C} \to TM \oplus \theta \oplus \mathbb{C}\), and involution \(\tau_* \oplus \mathbb{K}\). This follows from taking the boundary of the interval \([-1,1]\) with constant stably almost complex structure \(T[-1,1] \times \mathbb{R} \cong [-1,1] \times \mathbb{C}\) and conjugation \(\mathbb{K}\). The boundary consists of two points with opposite complex structures \(\mathbb{C} \sqcup \overline{\mathbb{C}}=\{1\} \times \mathbb{C} \cup \{-1\} \times \overline{\mathbb{C}}\).

Noting that the generators of the unitary bordism ring can be taken to be complex manifolds, we obtain the following simple result:

To study the magnetic unitary bordism groups we will make use of the long exact sequence associated to the pair of families \(\mathcal{A}\) and \(\{\{1\}\}\) of subgroups of \(\overline{\mathbb{Z}}_2\), where the first one has all subgroups \(\mathcal{A} = \{ \overline{\mathbb{Z}}_2,\{1\}\}\), and the second one only the trivial one. This has been masterfully explained in [1] and defines the long exact sequence: \[\begin{align} \label{long95exact95sequence95restricted95unrestricted} \cdots \to \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_*\{\{1\}\} \overset{\iota}{\to} \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_* \overset{\Psi}{\to} \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_*\{\mathcal{A}, \{1\}\} \overset{\partial}{\to} \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{*-1}\{\{1\}\} \to \cdots \end{align}\tag{8}\] where \(\mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_*\{\{1\}\}\) consists of the magnetic unitary bordism group of manifolds whose conjugation is free, and \(\mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_*\{\mathcal{A}, \{1\}\}\) consists of the magnetic unitary bordism group of manifolds whose boundary has a free action of the conjugation. The map \(\iota\) simply sends the magnetic unitary manifolds with free actions to the unrestricted ones. The map \(\Psi\) takes a tubular neighborhood of the fixed points of the conjugation map and assigns the unit ball bundle; note that the action of the conjugation is free on the sphere bundle of the normal bundle of the fixed points of the conjugation.

4.4 Magnetic unitary bordism of free conjugations↩︎

Of particular interest are the magnetic unitary manifolds with free conjugation \(\mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{*}\{\{1\}\}\). Let us describe a series of constructions yielding non-zero stably magnetic manifolds with free conjugations in \(\mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{*}\{\{1\}\}\).

• Twin magnetic unitary manifolds. Take a smooth projective manifold \(M_{\mathbb{C}}\) defined by homogeneous polynomials with real coefficients (the generator of the unitary bordism ring \(\Omega^U_{2*}\) are of this type). Denote by \(\mathbb{K}:M_\mathbb{C} \to M_\mathbb{C}\) the complex conjugation map induced by complex conjugation in the homogeneous coordinates. Denote the pair \((M_\mathbb{C}, \mathbb{K})\).

  Take the disjoint union of two copies of \(M\) as complex manifold, \[\begin{align} M_\mathbb{C} \sqcup M_\mathbb{C} \mathrel{\vcenter{:}}= M \times \{-1\} \cup M \times \{1\}, \end{align}\] and endow this manifold with the conjugation action: \[\begin{align} \label{twin-magnetic-mainfold} \mathbb{K} \times(-1) : M_\mathbb{C} \sqcup M_\mathbb{C} &\to M_\mathbb{C} \sqcup M_\mathbb{C} \\ (m,t) & \mapsto (\mathbb{K}m, -t) \end{align}\tag{9}\] and note that the pair \((M_\mathbb{C}\sqcup M_\mathbb{C}, (\mathbb{K} \times(-1))_*)\) is a magnetic unitary manifold with free conjugation. Call it the twin magnetic unitary manifold.

  The composition of the homomorphisms of groups \[\begin{align} \Omega^{U}_{2*}& \longrightarrow \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{2*}\{\{1\}\} \overset{F_U \circ \iota}{\longrightarrow } \Omega^{U}_{2*}\\ [M_\mathbb{C}] \mapsto & [M_\mathbb{C}\sqcup M_\mathbb{C}, (\mathbb{K} \times(-1))_*] \mapsto 2[M_\mathbb{C}], \end{align}\] where the second map is the composition of the map \(\iota: \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{*}\{\{1\}\} \to \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{*}\) from magnetic unitary manifolds with free conjugations to magnetic unitary manifolds with unrestricted conjugation, and \(F_U\) the forgetful map of Eqn 7 , implying that the twin magnetic unitary manifolds \[\begin{align} \label{twin95magnetic95unitary95manifolds} [M_\mathbb{C}\sqcup M_\mathbb{C}, (\mathbb{K} \times(-1))_*] \in \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{2*}\{\{1\}\} \end{align}\tag{10}\] are all non-trivial.

• Opposite twin magnetic unitary manifolds. For a smooth projective manifold \(M_{\mathbb{C}}\) with complex conjugation \(\mathbb{K}: M_\mathbb{C} \to M_\mathbb{C}\) and almost complex structure \(J_M\) such that \(M_{\mathbb{C}}\) represents a generator of \(\Omega^U_{2n}\), we can also take the pair of opposite manifolds: \[\begin{align} \overline{M}_\mathbb{C} \sqcup {M}_\mathbb{C} \mathrel{\vcenter{:}}= \overline{M} \times \{-1\} \cup M \times \{1\}, \end{align}\] where \(\overline{M}\) means the same manifold \(M\) but with the opposite almost complex structure \(-J_M\).

  Endow this manifold with the conjugation action: \[\begin{align} \label{opposite-twin-magnetic-mainfold} \mathrm{id} \times(-1) : \overline{M}_\mathbb{C} \sqcup M_\mathbb{C} &\to \overline{M}_\mathbb{C} \sqcup M_\mathbb{C} \\ (m,t) & \mapsto (m, -t) \end{align}\tag{11}\] and note that the equation: \[\begin{align} (-J_M\times\{-1\}) = - (J_M \times \{1\}) \end{align}\] implies that the map \(\mathrm{id} \times(-1)\) is indeed a conjugation. The pair \((\overline{M}_\mathbb{C}\sqcup M_\mathbb{C}, (\mathrm{id} \times(-1))_*)\) is a magnetic unitary manifold with free conjugation. Call it the opposite twin magnetic unitary manifold.

  In this case the composition of the homomorphisms of groups \[\begin{align} \Omega^{U}_{2n}& \longrightarrow \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{2n}\{\{1\}\} \overset{F_U \circ \iota}{\longrightarrow } \Omega^{U}_{2n}\\ [M_\mathbb{C}] \mapsto & [\overline{M}_\mathbb{C}\sqcup M_\mathbb{C}, (\mathrm{id} \times(-1))_*] \mapsto (1+(-1)^n)[M_\mathbb{C}] \end{align}\] depends on the parity of \(n\). In particular, when the complex dimension of \(M\) is even, then opposite twin magnetic unitary manifold \((\overline{M}_\mathbb{C}\sqcup M_\mathbb{C}, (\mathrm{id} \times(-1))_*)\) is non-trivial in \(\mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{2n}\{\{1\}\}\).

• Milnor hypersurfaces with free conjugation. The Milnor hypersurfaces \(H_\mathbb{C}(m,m)\) defined in Eqn. 1 can be endowed with a free conjugation action as follows: \[\begin{align} \mathbb{K}_{\mathrm{swap}}: H_\mathbb{C}(m,m) \to H_\mathbb{C}(m,m) \\ \big([\mathbf{w}], [\mathbf{z}]\big) \mapsto \big([\overline{\mathbf{z}}], [\overline{\mathbf{w}}]\big). \end{align}\]

  Note that this conjugation is free, because any fixed point of the conjugation would imply that \([\mathbf{w}] = [\overline{\mathbf{z}}]\) and the solution of the equation \(z_0\overline{z_0} + \cdots z_m \overline{z_m}=0\) would be empty.

  Denote these magnetic unitary manifolds by the pair \((H_\mathbb{C}(m,m), {\mathbb{K}_{\mathrm{swap}}}_*)\) and differentiate them from the magnetic unitary manifolds \((H_\mathbb{C}(m,m), \mathbb{K}_*)\) where the conjugation is the complex conjugation and its fixed points are the real Milnor hypersurfaces: \[\begin{align} H_\mathbb{C}(m,m)^\mathbb{K}= H_\mathbb{R}(m,m). \end{align}\]

  The magnetic unitary manifolds \[\begin{align} \label{Milnor95hypersurfaces95free} [H_\mathbb{C}(2^s,2^s) , {\mathbb{K}_{\mathrm{swap}}}_*] \in \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{2(2^{s+1}-1)}\{\{1\}\} \end{align}\tag{12}\] define non-trivial elements since their images under the forgetful map \(F_U\) are the unitary manifolds \(H_\mathbb{C}(2^s,2^s)\) whose characteristic numbers are congruent with 2 modulo 4. Thus they are generators of the unitary bordism ring modulo 2.

• Spheres with antipodal stable conjugation. Take the trivialization of the stabilization of the tangent bundle of the spheres: \[\begin{align} TS^n \times \mathbb{R} &\overset{\cong}{\to} S^n \times \mathbb{R}^{n+1}\\ (v,w,r) & \mapsto (v,w+rv) \end{align}\] where \(v \in \mathbb{R}^{n+1}\) with \(|v|=1\) is a point in \(S^n\) and \(w \in \mathbb{R}^{n+1}\) with \(w \cdot v =0\) represents a point in \(T_vS^{n}\). Take the antipodal action \(I\) on \(\mathbb{R}^{n+1}\) with \(I(x)=-x\) and note that the induced action on the stable tangent bundle is: \[\begin{align} I: TS^n \times \mathbb{R} \to TS^n \times \mathbb{R}\\ (v,w,r) \mapsto (-v,-w,r), \end{align}\] namely, trivial on the normal bundle of the embedding \(S^n \hookrightarrow \mathbb{R}^{n+1}\).

  Take the stable almost complex structure \[\begin{align} J_{S^n_{\mathrm{st}}}: TS^n \times \mathbb{R} \times \mathbb{R}^{n+1} & \to TS^n \times \mathbb{R} \times \mathbb{R}^{n+1} \end{align}\] induced from the almost complex structure \[\begin{align} S^n \times \mathbb{R}^{n+1} \times \mathbb{R}^{n+1} &\to S^n \times \mathbb{R}^{n+1} \times \mathbb{R}^{n+1}\\ (v,x,y) & \mapsto (v,-y,x). \end{align}\]

  This almost complex structure can be written as follows: \[\begin{align} J_{S^n_{\mathrm{st}}}: TS^n \times \mathbb{R} \times \mathbb{R}^{n+1} & \to TS^n \times \mathbb{R} \times \mathbb{R}^{n+1} \\ (v,w,r,w_1+r_1v) & \mapsto (v,-w_1,-r_1,w+rv) \end{align}\] where \(v \in S^n\), \(w,w_1 \in T_vS^n\), \(r,r_1 \in \mathbb{R}\), and \(y=w_1 + r_1v \in \mathbb{R}^{n+1}\) is written in terms of the vector \(v\) and a complementary component in \(T_vS^n\).

  Define the lift of the antipodal map \(I\) as the map: \[\begin{align} {I}_* : TS^n \times \mathbb{R} \times \mathbb{R}^{n+1} & \to TS^n \times \mathbb{R} \times \mathbb{R}^{n+1} \\ (v,w,r,y) & \mapsto (-v,-w,r,y) \end{align}\] which in terms of a tangent and normal decompositions it looks like: \[\begin{align} {I}_* (v,w,r,w_1+r_1v) = (-v,-w,r,w_1-r_1(-v)), \end{align}\] and note that \({I}_* \circ J_{S^n_{\mathrm{st}}} =- J_{S^n_{\mathrm{st}}} \circ {I}_*\) because we have the following equations: \[\begin{align} {I}_* \circ J_{S^n_{\mathrm{st}}}(v,w,r,w_1+r_1v) & = {I}_*(v,-w_1,-r_1,w+rv) \\ & = (-v,w_1,-r_1,w+rv),\\ J_{S^n_{\mathrm{st}}} \circ {I}_* (v,w,r,w_1+r_1v) &= J_{S^n_{\mathrm{st}}} (-v,-w,r,w_1+r_1v)\\ &= J_{S^n_{\mathrm{st}}} (-v,-w,r,w_1-r_1(-v))\\ & = (-v,-w_1,r_1,-w+r(-v))\\ & = (-v,-w_1,r_1,-w-rv). \end{align}\]

  Further note that the involution \({I}_*\) makes the stable tangent bundle of the sphere into the vector bundle: \[\begin{align} TS^n \oplus \mathbb{R} \oplus \theta \end{align}\] where \(\theta = S^n \times \mathbb{R}^{n+1}\) and the involution on this bundle is simply \[\begin{align} \mu : S^n \times \mathbb{R}^{n+1} &\to S^n \times \mathbb{R}^{n+1}\\ (v,y)& \mapsto (-v, y). \end{align}\] We have therefore that \({I}_*=I \oplus \mu\).

The involution \({I}_*\) is therefore a conjugation of the stable almost complex structure \(J_{S^n_{\mathrm{st}}}\) of the sphere \(S^n\) covering the antipodal map \(I\). The pair \((S^n,{I}_*)\) thus defines an element in the magnetic unitary bordism group of free conjugations: \[\begin{align} [S^n,{I}_*] \in \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_n\{ \{1\}\}. \end{align}\]

These spheres with stably antipodal conjugation are non-trivial elements in the magnetic unitary bordism group of free conjugations because their images under the forgetful homomorphism \[\begin{align} \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_n\{ \{1\}\} & \to {\Omega}^{O,{\mathbb{Z}}_2}_n\{ \{1\}\}\\ [S^n, {I}_*] & \mapsto [S^n,I], \end{align}\] are non-trivial. Here we have that the canonical embeddings \(\mathbb{P}_\mathbb{R}^k \to B\mathbb{Z}_2\) are non-trivial elements in \({\Omega}^{O}_k(B\mathbb{Z}_2) \cong {\Omega}^{O,{\mathbb{Z}}_2}_k\{ \{1\}\}\).

Note furthermore, that this stably structure on the sphere can be seen as the restriction of a stably magnetic structure on the \((n+1)\)-dimensional unit ball \(B^{n+1}\). The involution on the ball maps \(v\) to \(-v\) and the conjugation map and the almost complex structure on the stable tangent structure are the following: \[\begin{align} TB^{n+1} \times \mathbb{R}^{n+1} &\to TB^{n+1} \times \mathbb{R}^{n+1} \\ (v, x,y) & \overset{\widetilde{I}}{\mapsto} (-v,-x,y) \\ (v, x,y) & \overset{J}{\mapsto} (v,-y,x), \end{align}\] for \(v \in B^{n+1}\), \(x, y \in \mathbb{R}^{n+1}\). Here we assume that \(TB^{n+1} = B^{n+1}\times \mathbb{R}^{n+1}\).

The map of stably vector bundles: \[\begin{align} TS^n \times \mathbb{R} \times \mathbb{R}^{n+1} &\to TB^{n+1} \times \mathbb{R}^{n+1}\\ (v,w,r,y)& \mapsto (v,w+rv,y) \end{align}\] is compatible with the almost complex structures and the conjugation maps. Therefore the magnetic unitary manifold with boundary \((B^{n+1}, \widetilde{I})\), which defines an element in \(\mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{n+1}\{\mathcal{A},\{1\}\}\), has the magnetic unitary sphere \((S^{n}, {I}_*)\) as its boundary: \[\begin{align} \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{n+1}\{\mathcal{A},\{1\}\} & \overset{\partial}{\to} \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{n}\{\{1\}\} \\ [B^{n+1}, \widetilde{I}] & \mapsto [S^{n},{I}_*]. \end{align}\]

Note that the map \[\begin{align} TB^{n+1} \times \mathbb{R}^{n+1} &\to TB^{n+1} \times \mathbb{R}^{n+1} \\ (v, x, y) & \mapsto (v,x,-y) \end{align}\] induces a unitary isomorphism between the almost complex structures \(J\) and \(-J\) which is compatible with the conjugation map \(\widetilde{I}\). So the triples \((B^{n+1}, \widetilde{I}, J)\) and \((B^{n+1}, \widetilde{I}, -J)\) are isomorphic.

4.5 Smith homomorphism↩︎

A very interesting map in the magnetic unitary bordism groups of free conjugations is the Smith homomorphism: \[\begin{align} \Delta: \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{*+1}\{\{1\}\} \to \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{*}\{\{1\}\}. \end{align}\] This is the generalization of the Smith homomorphism in the unoriented \(\mathbb{Z}_2\)-equivariant bordism groups with free involutions defined in [1].

Allow us describe this map. Take a unitary magnetic manifold \(M\) with free conjugation \(\tau\). Embed \(M\) \(\mathbb{Z}_2\)-equivariantly into a sphere \(S^n\) endowed with the antipodal action, i.e., \(M\hookrightarrow S^n\). Adjust the embedding such that the image of \(M\) is transverse to \(S^{n-1}\) in \(S^n\) and take the intersection of the image of \(M\) with \(S^{n-1}\) in \(S^n\). Denote this intersection by \(N \subset M\). Since \(S^{n-1}\) is also a \(\mathbb{Z}_2\)-space, then \(\mathbb{Z}_2\) acts freely on \(N\). The normal bundle of \(N\) in \(M\) is trivial. Moreover, the group \(\mathbb{Z}_2\) acts on this normal bundle via a fixed one-dimensional representation. Hence \(TM|_{N}\cong TN \oplus \mathbb{R}_\pm\) as a \(\mathbb{Z}_2\)-equivariant real vector bundle, and therefore any magnetic unitary structure on \(M\) restricts to one on \(N\).

The restriction of the isomorphism \(TM \oplus \theta \cong \xi_\mathbb{C}\) to \(N\) gives the isomorphism \(TN \oplus \mathbb{R}_{\pm} \oplus \theta|_N \cong \xi_\mathbb{C}|_N\) and therefore, \(N\) is endowed with a stable magnetic unitary structure. The assignment \(\Delta(M)=N\) is the Smith homomorphism.

Note that the sphere with antipodal conjugation \((S^{n+1},{I}_* )\) maps to \((S^{n},{I}_*)\) under the Smith homomorphism: \[\begin{align} \Delta(S^{n+1},{I}_*) = (S^{n},{I}_*), \end{align}\] and iterating this map we can obtain all spheres with antipodal stable conjugation.

These Smith homomorphisms are compatible with the ones in unoriented \(\mathbb{Z}_2\)-equivariant bordism fitting into the following commutative diagram: \[\begin{align} \xymatrix{ \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_k\{ \{1\}\} \ar[r]^\Delta \ar[d]& \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{k-1}\{ \{1\}\} \ar[r]^\Delta \ar[d] & \cdots \ar[r]^\Delta &\mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{0}\{ \{1\}\} \ar[d]\\ {\Omega}^{O,{\mathbb{Z}}_2}_k\{ \{1\}\} \ar[r]^\Delta & {\Omega}^{O,{\mathbb{Z}}_2}_{k-1}\{ \{1\}\} \ar[r]^\Delta & \cdots \ar[r]^\Delta & {\Omega}^{O,{\mathbb{Z}}_2}_0\{ \{1\}\} } \end{align}\] where the vertical maps forget the unitary structure and keep the conjugations as involutions.

Having described the Smith homomorphism, we are ready to present an exact sequence that incorporates both the Milnor map and the Smith homomorphism. This has been developed by Stong [3] in the context of AR bordism theory.

4.6 Exact sequence for the Milnor map↩︎

It turns out that the cokernel of the Smith homomorphism is isomorphic to the kernel of the Milnor map. This fact allows us to introduce the following exact sequence incorporating both the Smith homomorphism and the Milnor map.

We now present some diagrams involving the previously defined magnetic unitary manifolds. With respect to the unrestricted-restricted long exact sequence we have the assignments: \[\begin{align} \xymatrixrowsep{0mm} \xymatrix{\mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{*+1}\{\mathcal{A}, \{1\}\} \ar[r]^\partial & \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_*\{\{1\}\} \ar[r]^\iota & \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_* \ar[r]^\Psi & \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_*\{\mathcal{A}, \{1\}\} \\ [B^{2n}, \widetilde{I}] \ar@{|->}[r] & [S^{2n-1}, {I}_*] \ar@{|->}[r] & 0 & \\ & [M\sqcup M, (\mathbb{K} \times(-1))_*] \ar@{|->}[r] & [M\sqcup M, (\mathbb{K} \times(-1))_*] \ar@{|->}[r] & 0 \\ & [H_\mathbb{C}(2^s,2^s) , {\mathbb{K}_{\mathrm{swap}}}_*] \ar@{|->}[r]& [H_\mathbb{C}(2^s,2^s) ,{\mathbb{K}_{\mathrm{swap}}}_*] \ar@{|->}[r] & 0. } \end{align}\]

With respect to the exact sequence of Thm. 9 we have the assignments: \[\begin{align} \xymatrixrowsep{0mm} \xymatrix{ \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{*+1}\{ \{1\}\} \ar[r]^\Delta & \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{*}\{ \{1\}\} \ar[r]^{F_U \circ \iota} & \Omega^{U}_{*} \ar[r]^\mu & \Omega^{O}_* \\ [S^{n+1}, {I}_*] \ar@{|->}[r] & [S^{n}, {I}_*] \ar@{|->}[r] & 0 & \\ & [M\sqcup M, (\mathbb{K} \times(-1))_*] \ar@{|->}[r] & 2[M] \ar@{|->}[r] & 0\\ & [H_\mathbb{C}(2^s,2^s) , {\mathbb{K}_{\mathrm{swap}}}_*] \ar@{|->}[r]& [H_\mathbb{C}(2^s,2^s)] \ar@{|->}[r]& 0 \\ && [H_{\mathbb{C}}(2^k,2t2^k)] \ar@{|->}[r]& [H_{\mathbb{R}}(2^k,2t2^k)]\\ && [\mathbb{P}_\mathbb{C}^{2r}] \ar@{|->}[r]& [\mathbb{P}_\mathbb{R}^{2r}] } \end{align}\]

Note that the magnetic unitary manifolds with free conjugation \(2(H_\mathbb{C}(2^s,2^s) , {\mathbb{K}_{\mathrm{swap}}}_*)\) and \((H_\mathbb{C}(2^s,2^s) \sqcup H_\mathbb{C}(2^s,2^s), (\mathbb{K}\times (-1))_*)\) map under \(F_U \circ \iota\) to the same unitary manifold. So, in the case that they do not belong to the same bordism class in \(\mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{2(2^s-1)}\{ \{1\}\}\), there must exist a magnetic unitary manifold with free conjugation \(W\) such that \[\begin{align} \Delta(W) = 2(H_\mathbb{C}(2^s,2^s) , {\mathbb{K}_{\mathrm{swap}}}_*) - (H_\mathbb{C}(2^s,2^s) \sqcup H_\mathbb{C}(2^s,2^s), (\mathbb{K}\times (-1))_*). \end{align}\]

Let us consider \(0\)-dimensional magnetic bordism \(\mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{0}\). Note that a point with \(\mathbb{C}\) as stably magnetic bundle generate a copy of \(\mathbb{Z}\) in \(\mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{0}\). If we consider instead two points \(\{-1,1\}\) with free \(\mathbb{Z}_2\)-action and we take the class of \(\mathbb{C}_1 \sqcup \overline{\mathbb{C}}_{-1}\) for its stably complex vector bundle, we see that the involution \(\nu:\mathbb{C}_1 \to \overline{\mathbb{C}}_{-1}\), \(\nu(z)= z\) is a conjugation. This magnetic unitary manifold \(\mathbb{C}_1 \sqcup \overline{\mathbb{C}}_{-1}\) generates a \(2\)-torsion class in \(\mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{0}\{\{1\}\}\) which maps to zero in \(\mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{0}\). In this case we have: \[\begin{align} \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{0} & \cong \langle [\mathbb{C} , \mathbb{K}_*]\rangle \cong \mathbb{Z},\\ \mathbf{\Omega}^{\overline{\mathbb{Z}}_2}_{0} \{\{1\}\} & \cong \langle [\mathbb{C} \sqcup \mathbb{C}, {\mathbb{K}_{\mathrm{swap}}}_*], [\mathbb{C} \sqcup \overline{\mathbb{C}},\nu] \rangle \cong \mathbb{Z} \oplus \mathbb{Z}_2. \end{align}\]

4.7 Exact sequence for the \(\mathbb{Z}_2\)-equivariant Milnor map↩︎

The generalization of the sequences of Thm. 9 to the magnetic group \(\mathbb{Z}_2 \times \overline{\mathbb{Z}}_2\) is possible due to our calculation in Lem. 7 of the kernel of the \(\mathbb{Z}_2\)-equivariant Milnor map \(\mu^{\mathbb{Z}_2}\). Here the magnetic group \(\mathbb{Z}_2 \times \overline{\mathbb{Z}}_2\) means the group \(\mathbb{Z}_2 \times {\mathbb{Z}}_2\) with \(\phi\) the projection in the second \(\mathbb{Z}_2\).

Let \(\mathbf{\Omega}^{\mathbb{Z}_2\times\overline{\mathbb{Z}}_2}_{*}\) be the magnetic \(\mathbb{Z}_2\times\overline{\mathbb{Z}}_2\)-equivariant bordism groups, and consider the bordism groups where the conjugation (namely the action of the group \(\overline{\mathbb{Z}}_2\)) is free. For this we simply consider the family of subgroups \(\mathcal{F} = \{ \langle (1,0)\rangle, \langle (1,1)\rangle, \{(0,0)\} \}\) which does not contain the subgroup generated by \((0,1)\). The bordism groups \(\mathbf{\Omega}^{\mathbb{Z}_2\times\overline{\mathbb{Z}}_2}_{*}\{ \mathcal{F}\}\) of magnetic \(\mathbb{Z}_2\times\overline{\mathbb{Z}}_2\)-equivariant manifolds whose isotropy groups restrict to \(\mathcal{F}\) are precisely the equivariant bordism groups of free conjugations.

In terms of spaces we could take a model for the space \(E \mathcal{F}\). Consider \(V_+\) and \(V_-\) the real irreducible representations of \(\mathbb{Z}_2 \times \{0\}\) and consider the spheres \(S((V_+\times V_-)^{n})\) where the group \(\{0\} \times \overline{\mathbb{Z}}_2\) acts by the antipodal map (multiplication by \(-1\)). Taking the limit when \(n\) goes to infinity we end up with an infinite sphere \(S((V_+\times V_-)^{\infty})\) with an action of the group \(\mathbb{Z}_2\times\overline{\mathbb{Z}}_2\). This space is contractible, and the fixed point sets of the elements \((1,0)\) and \((1,1)\) are also contractible, being respectively the infinite spheres \(S((V_+)^{ \infty})\) and \(S((V_-)^{ \infty})\). Since the action of \(\{0\} \times \overline{\mathbb{Z}}_2\) is free, we see that a model for \(E \mathcal{F}\) is this infinite sphere \(S((V_+\times V_-)^{ \infty})\). Let us then take \(E \mathcal{F} = S((V_+\times V_-)^{ \infty})\), and thus we have the isomorphisms: \[\begin{align} \mathbf{\Omega}^{\mathbb{Z}_2\times\overline{\mathbb{Z}}_2}_{*}\{ \mathcal{F}\} \cong \mathbf{\Omega}^{\mathbb{Z}_2\times\overline{\mathbb{Z}}_2}_{*}(E \mathcal{F}) = \mathbf{\Omega}^{\mathbb{Z}_2\times\overline{\mathbb{Z}}_2}_{*}(S((V_+\times V_-)^{ \infty})). \end{align}\]

There are two Smith homomorphisms that can be defined in this setup: \[\begin{align} \overline{\Delta}_\pm: \mathbf{\Omega}^{\mathbb{Z}_2\times\overline{\mathbb{Z}}_2}_{*+1}\{ \mathcal{F}\} {\to} \mathbf{\Omega}^{\mathbb{Z}_2\times\overline{\mathbb{Z}}_2}_{*}\{ \mathcal{F}\}. \end{align}\] Take a magnetic \(\mathbb{Z}_2\times\overline{\mathbb{Z}}_2\)-equivariant manifold \(M\) with free conjugation in \(\mathbf{\Omega}^{\mathbb{Z}_2\times\overline{\mathbb{Z}}_2}_{*+1}\{ \mathcal{F}\}\), and consider its equivariant classifying map \(M \to E\mathcal{F}\). This map factors through a finite-dimensional sphere \(f: M \to S((V_+ \times V_-)^{ n})\) for some sufficiently large integer \(n\) and we want to find its transversal intersection with a \(\mathbb{Z}_2\times \{0\}\)-invariant sphere \(S^{2n-2}\). There are two distinctly different types of \(\mathbb{Z}_2\times \{0\}\)-invariant \((2n-2)\)-dimensional spheres in \(S((V_+ \times V_-)^{ n})\). We take either the sphere \(S(V_+^{n-1} \times V_-^n)\) or the sphere \(S(V_+^{n} \times V_-^{n-1})\).

The transversal intersection of \(M\) with the sphere \(S(V_+^{n} \times V_-^{n-1})\) will be denoted \(N_+\) while the transversal intersection with the sphere \(S(V_+^{n-1} \times V_-^n)\) will be denoted \(N_-\). The Smith homomorphisms are thus defined by the assignments: \[\begin{align} \overline{\Delta}_+(M) = N_+ \;\;\mathrm{and} \;\;\overline{\Delta}_-(M) = N_-. \end{align}\]

Consider the natural maps: \[\begin{align} \mathbf{\Omega}^{\mathbb{Z}_2\times\overline{\mathbb{Z}}_2}_{*}\{ \mathcal{F}\} \overset{\overline{\iota}}{\longrightarrow} \mathbf{\Omega}^{\mathbb{Z}_2\times\overline{\mathbb{Z}}_2}_{*} \overset{\overline{F}_U}{\longrightarrow} \Omega^{U, \mathbb{Z}_2}_*, \end{align}\] where the first map sends restricted bordism to unrestricted bordism, and the second forgets the conjugation. We allow ourselves to claim the following generalization of Thm. 9.

Acknowledgments↩︎

Zhi Lü thanks the Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS) for its financial support. This research was partially funded by SIMIS under grant number SIMIS-ID-2025-TP, and by NSFC through grant number 11971112. Bernardo Uribe acknowledges the financial support of the Institute of the Mathematical Sciences of the Americas in Miami, USA, of the Max Planck Institute for Mathematics in Bonn, Germany, of the International Center for Theoretical Physics in Trieste, Italy, through its associates program, of the Alexander Von Humboldt Foundation in Bonn, Germany, and of Fudan University in Shanghai, China.

Tools and computational resources disclosure↩︎

There was no use of any mathematical assistance AI in the elaboration of this work. All statements and proofs were devised by the authors themselves. Only grammar and spelling correction from the latex console was employed to ensure proper use of the English language.

References↩︎