Products in spin^c-cobordism

A spin^c-structure on a compact smooth $n$-dimensional manifold $M$ is a reduction of its structure group from $O(n)$ to $Spin^{c}(n)$. We find the following perspective illuminating: a compact smooth manifold is

• •

  orientable if its first Stiefel–Whitney class $w_{1}$ vanishes

• •

  and admits a spin structure if its first two Stiefel–Whitney classes, $w_{1}$ and $w_{2}$, both vanish.

A spin^c-structure is intermediate between an orientation and a spin structure. Specifically, a compact smooth manifold $M$ admits a spin^c structure if its first Stiefel–Whitney class $w_{1}$ vanishes, and its second Stiefel–Whitney class $w_{2}$ is a reduction of an integral class. That is, $w_{2}\in H^{2}(M;\mathbb{F}_{2})$ is in the image of the reduction-of-coefficients map $H^{2}(M;\mathbb{Z})\rightarrow H^{2}(M;\mathbb{F}_{2})$. For these and many other relevant facts, consult Stong’s book [24].

The spin^c-cobordism ring, written $\Omega^{Spin^{c}}_{*}$, is the ring of spin^c-cobordism classes of compact smooth spin^c-manifolds. The addition is given by disjoint union of manifolds, while the multiplication is Cartesian product. There are several reasons to care about spin^c-cobordism: aside from its applications to mathematical physics, e.g. [7] and [28], spin^c-cobordism is of particular interest because it is one of the complex-oriented cobordism theories, and consequently there exists a one-dimensional group law on $\Omega^{Spin^{c}}_{*}$ which describes how the first Chern class in spin^c-cobordism behaves on a tensor product of complex line bundles. See [1] or [12] for these classical ideas, whose consequences for complex cobordism (as in [20]) have been enormous, but whose consequences for spin^c-cobordism have apparently never been fully explored¹¹1In future work, the authors hope to apply the results about the ring structure of the spin^c-cobordism ring obtained in this paper to the problem of describing the formal group law on the spin^c-cobordism ring in formal-group-law theoretic terms, similar to what Quillen did for complex and unoriented cobordism in [16],[17], what Baker–Morava did for $2$-inverted symplectic bordism in [6], and what Buchstaber did for symplectic bordism [9]. It seems impossible to get much understanding of the formal group law of spin^c-cobordism without first coming to some understanding of the structure of its coefficient ring $\Omega^{Spin^{c}}_{*}$, which is the goal of this paper..

Since spin^c-cobordism is an example of a “$(B,f)$-cobordism theory” in the sense of Thom, the general results of [25] ensure that there exists a spectrum $MSpin^{c}$ such that $\pi_{*}(MSpin^{c})\cong\Omega^{Spin^{c}}_{*}$. The homotopy type of $MSpin^{c}$ is understood as follows.

Away from $2$:

The map $\pi:BSpin^{c}\longrightarrow BSO\times K(\mathbb{Z},2)$ is an odd-primary homotopy equivalence and induces an isomorphism $\Omega^{Spin^{c}}_{*}[\frac{1}{2}]\cong\Omega^{SO}_{*}(K(\mathbb{Z},2))[\frac{1}{2}]$, and consequently $MSpin^{c}[\frac{1}{2}]\cong MSO[\frac{1}{2}]\wedge\mathbb{C}P^{\infty}$. Since $MSpin^{c}$ is complex-oriented, we have an isomorphism of graded rings $\pi_{*}(MSpin^{c}[\frac{1}{2}])\cong\pi_{*}(MSO)[\frac{1}{2}][[x]]$ with $x\in\pi_{2}(MSpin^{c})$. The ring $\pi_{*}(MSO)[\frac{1}{2}]\cong\pi_{*}(MSp)[\frac{1}{2}]\cong\pi_{*}(MSpin)[\frac{1}{2}]$ is understandable in various ways; see Wall [26] or Baker–Morava [6] for different approaches.

At $2$:

In 1966, Anderson, Brown, and Peterson [2],[3] proved that $MSpin^{c}$ splits $2$-locally as a wedge of suspensions of the connective complex $K$-theory spectrum $ku$ and the mod $2$ Eilenberg-Mac Lane spectrum $H\mathbb{F}_{2}$:

(1)

$\displaystyle MSpin^{c}_{(2)}$

$\displaystyle\simeq Z\vee\coprod_{J}\Sigma^{4\left|J\right|}ku_{(2)}$

where the coproduct (i.e., wedge sum) is taken over all partitions (i.e., unordered finite tuples of positive integers) $J$, and $\left|J\right|$ denotes the sum of the entries of $J$.

Not much is known about the summand $Z$ in (1), other than that

• •:

  it is a coproduct of suspensions of copies of $H\mathbb{F}_{2}$,

• •:

  and from a Poincaré series [2], it is known how to solve inductively for the number of copies of $\Sigma^{n}H\mathbb{F}_{2}$ in $Z$, for each $n$.

In that sense, $Z$ is understood additively.

This purely additive understanding of $Z$, and consequently of $2$-local $\Omega^{Spin^{c}}_{*}$, is not entirely satisfying. To see the problem, consider the following table, which we reproduce from Bahri–Gilkey [5]:

The $\mathbb{F}_{2}$-linear dimension of $\pi_{n}Z$, as recorded in table 1, is equivalently the number of copies of $\Sigma^{n}H\mathbb{F}_{2}$ in $2$-local $MSpin^{c}$, and equivalently the $\mathbb{F}_{2}$-rank of the $2$-torsion subgroup of $\Omega^{Spin^{c}}_{n}$. Hence this table is telling us about the $2$-torsion in the spin^c-cobordism ring. One has the sense that some deep pattern is present in the distribution of the $2$-torsion, but whatever it is, it cannot be seen clearly from these $\mathbb{F}_{2}$-ranks, nor from the Poincaré series used to inductively compute them.

However, since $\pi_{*}(Z)$ is precisely the $2$-torsion in $\Omega^{Spin^{c}}_{*}$, $\pi_{*}(Z)$ is not only a summand but also an ideal in $\Omega^{Spin^{c}}_{*}$. One wants to understand $\pi_{*}(Z)$ multiplicatively, i.e., one wants to be able to describe the ring structure on $\Omega^{Spin^{c}}_{*}$, including its $2$-torsion elements. A reasonably clear description of $\Omega^{Spin^{c}}_{*}$ as a ring would yield a far more illuminating understanding of $\pi_{*}(Z)$ than the inductive formula for its $\mathbb{F}_{2}$-rank in each degree, which is presently all we have.

Fifty years after the additive structure of $MSpin^{c}$ was described by Anderson–Brown–Peterson, the problem of calculating the ring structure of $\Omega^{Spin^{c}}_{*}$ remains open. The purpose of this paper is to make progress towards a solution to this problem, restricting to the $2$-local case, which is the most difficult.

A traditional notation for the unoriented bordism ring $\Omega^{O}_{*}\cong MO_{*}$ is $\mathfrak{N}_{*}$. In the 1968 book [24, pg. 351], Stong asks:

Open question: Can one determine these images nicely as subrings of $\mathfrak{N}_{*}$?

By “these images,” Stong refers to the images of the natural maps $\Omega^{Spin}_{*}\rightarrow\mathfrak{N}_{*}$ and $\Omega^{Spin^{c}}_{*}\rightarrow\mathfrak{N}_{*}$. Our approach to understanding the mod $2$ spin^c-cobordism ring begins by answering Stong’s open question in a range of degrees. We use the Anderson–Brown–Peterson splitting [2], product structure in the Adams spectral sequences, and Thom’s determination of $\mathfrak{N}_{*}$ using symmetric polynomials [25] to develop a method for calculating the image of the map $\Omega^{Spin^{c}}_{*}\rightarrow\mathfrak{N}_{*}$ through degree $d$, for any fixed choice of integer $d$. Our method gives a presentation for $\Omega^{Spin^{c}}/(2,\beta)$ through degree $d$, since the map $\Omega^{Spin^{c}}/(2,\beta)\longrightarrow\mathfrak{N}_{*}$ is injective. We carry out computer calculation using our method to obtain our first main theorem:

With Theorem A in hand, the patterns in table 1 become completely clear: in each degree in this range, one can see why the $\mathbb{F}_{2}$-linear dimension of the $2$-torsion subgroup of $\Omega^{Spin^{c}}_{*}$ takes the particular value it takes, as follows. Since Anderson–Brown–Peterson proved that the $2$-torsion coincides with the $\beta$-torsion in $\Omega^{Spin^{c}}_{*}$, in degrees $n\leq 33$ the $2$-torsion in $\Omega^{Spin^{c}}_{n}$ is simply the $\mathbb{F}_{2}$-linear combinations of the monomials in the generators $Z_{i},T_{i}$ such that at least one of the factors is $\beta$-torsion, i.e., at least one of the factors is either a generator $Z_{i}$ with $i\equiv 2\mod 4$, or a generator $T_{i}$. Here is the same table as table 1, but augmented with an $\mathbb{F}_{2}$-linear basis in each degree, using the multiplicative structure from Theorem A. We start in degree $10$ since there is no nontrivial $2$-torsion in $\Omega^{Spin^{c}}_{*}$ below degree $10$.

One can also read off the product structure on the $2$-torsion in $\Omega^{Spin^{c}}_{*}$ in degrees $\leq 33$ from this table, since it is given by multiplication of monomials along with the relations from Theorem A.

It is evident from Theorem A that, in degrees $\leq 33$, $\Omega^{Spin^{c}}_{*}$ has a subring generated by elements $Z_{4},Z_{8},Z_{12},Z_{16},\dots$ and by elements $Z_{2i}$ with $i$ odd and not one less than a power of $2$, subject to the relations $2Z_{2i}=0=\beta Z_{2i}$ for all odd $i$. We are able to show that this pattern extends into all degrees, and goes some way to describing the ideal $\pi_{*}(Z)$ of $2$-torsion elements of $\Omega^{Spin^{c}}_{*}$ in multiplicative terms:

Theorem B describes the multiplicative structure of some, but not all, of the $2$-torsion in $\Omega^{Spin^{c}}_{*}$. For example, in degrees $\leq 33$, it accounts for precisely those monomials in table 2 which are not divisible by the elements $T_{i}$. In particular, the lowest-degree $2$-torsion element of $\Omega^{Spin^{c}}_{*}$ which is not described by Theorem 4.3 is $T_{24}\in\Omega^{Spin^{c}}_{24}$.

We also calculate the image of the map $\Omega^{Spin}_{*}\rightarrow\mathfrak{N}_{*}$ through degree 31 in Proposition 3.2 via a similar method to the one used to calculate the image of the map $\Omega^{Spin^{c}}_{*}\rightarrow\mathfrak{N}_{*}$ in Theorem A. There is a noteworthy geometric consequence of Proposition 3.2. In the 1965 paper [14], Milnor asks this question:

Problem. Does there exist a spin manifold $\Sigma$ of dimension $24$ so that $s_{6}(p_{1},\dots,p_{6})[\Sigma]\equiv 1\mod 2$?

Here $s_{6}$ is a certain symmetric polynomial, and $p_{1},\dots,p_{6}$ are Pontryagin classes. The reason for Milnor’s question is that, in [14], Milnor proves that, for a compact smooth manifold $M$ of dimension $\leq 23$, the following conditions are equivalent:

1. (1)

   $M$ is unorientedly cobordant to a spin manifold.

2. (2)

   The Stiefel–Whitney numbers of $M$ involving $w_{1}$ and $w_{2}$ are all zero.

3. (3)

   $M$ is unorientedly cobordant to $N\times N$, with $N$ an orientable compact manifold.

Milnor points out that, if there exists a compact spin manifold $\Sigma$ whose Pontryagin number $s_{6}(p_{1},\dots,p_{6})[\Sigma]$ is odd, then these conditions would fail to be equivalent in dimension $24$. Anderson–Brown–Peterson [2],[3] established that, as a consequence of their splitting of $2$-local $MSpin$, there does indeed exist such a compact spin manifold $\Sigma$. However, it seems that no explicit description of that $24$-dimensional compact spin manifold has been given in the literature (or anywhere else, as far as we know).

In Theorem 3.6, we give an explicit formula for the unoriented bordism class of such a compact spin manifold $\Sigma$, as a disjoint union of products of real projective spaces and squares of Dold manifolds. We refer the reader to the Theorem 3.6 for a statement of that formula, which is lengthy. The formula is obtained using our calculation of the image of the map $\Omega^{Spin}_{24}\rightarrow\mathfrak{N}_{24}$ and the manifold representatives calculated in Proposition 3.5.

Thom’s famous calculation [25] established that the unoriented bordism ring $\Omega^{O}_{*}=\mathfrak{N}_{*}$ is isomorphic to a polynomial algebra over $\mathbb{F}_{2}$. A theorem of Stong [23, Proposition 14] shows that the spin^c cobordism ring, reduced modulo torsion and then reduced modulo $2$, is also isomorphic to a polynomial $\mathbb{F}_{2}$-algebra.

By contrast, the spin^c cobordism ring cannot itself be isomorphic to a polynomial algebra, since by [2], it has $2$-torsion but is not an $\mathbb{F}_{2}$-algebra, hence it has nontrivial zero divisors. Similarly, since the mod $2$ spin^c-cobordism ring has nontrivial $\beta$-torsion, it cannot be isomorphic to a polynomial $\mathbb{F}_{2}$-algebra.

It follows as a trivial consequence of Theorem A that the mod $(2,\beta)$ spin^c-cobordism ring still cannot be a polynomial $\mathbb{F}_{2}$-algebra. One can, with a bit of calculation, deduce the same fact from the additive structure of $2$-local $MSpin^{c}$, by verifying that the Poincaré series of the mod $(2,\beta)$ spin^c-cobordism ring is not the Poincaré series of any polynomial algebra. This avoids the use of our multiplicative methods. The advantage of our multiplicative methods is that we are able to prove that $\Omega^{Spin^{c}}_{*}/(2,\beta)$ is instead uniformly $F$-isomorphic to a polynomial algebra.

As far as we know, the terms “$F$-isomorphism” (perhaps better known as “inseparable isogeny”) and “uniform $F$-isomorphism” originated with Quillen [19]:

The notion of $F$-isomorphism is applied only to algebras over a field of positive characteristic, so we had better reduce modulo $2$ in order to apply this idea to the spin^c-cobordism ring. We get a positive result:

An $F$-isomorphism induces a homeomorphism on prime ideal spectra, so Theorem C yields a description of all prime ideals in the mod $2$ spin^c cobordism ring. That is, we have

• •

  Given a ring $R$ and symbols $x_{1},\dots,x_{n}$, we write $R\{x_{1},\dots,x_{n}\}$ for the free $R$-module with basis $x_{1},\dots,x_{n}$.

• •

  We write $\beta$ for the Bott element in $\pi_{2}(ku)$, and also for its corresponding element $\beta=[\mathbb{C}P^{1}]\in\Omega^{Spin^{c}}_{2}$ under the Anderson–Brown–Peterson splitting of $2$-local $MSpin^{c}$.

The first author would like to thank Bob Bruner for many helpful conversations related to this work, and the Simons Foundation for providing the license for a copy of Magma [8] used in calculations. The first author was partially supported by the electronic Computational Homotopy Theory (eCHT) research community, funded by National Science Foundation Research Training Group in the Mathematical Sciences grant 2135884.

There is an exact sequence of Lie groups

$\displaystyle 1\longrightarrow U(1)\longrightarrow Spin^{c}(n)\longrightarrow SO(n)\longrightarrow 1$

that gives rise to the fiber sequence

(3)

$\displaystyle BU(1)\longrightarrow BSpin^{c}\longrightarrow BSO.$

Using this fibration, Harada and Kono [11] computed the mod 2 cohomology of the space $BSpin^{c}$:

The triviality of the ideal $I$ in the cohomology of $BSpin^{c}$ is a consequence of the first $d_{2}$ differential in the Serre spectral sequence associated to the fiber sequence (3). It is not practical to write down a presentation for the $\mathbb{F}_{2}$-algebra $H^{*}(BSpin^{c};\mathbb{F}_{2})$ which is more explicit than (4), since the difficulty of calculating iterated Steenrod squares applied to the Stiefel–Whitney class $w_{3}$ grows rapidly as the number of Steenrod squares grows. For example, $\operatorname{{\rm Sq}}^{8}(\operatorname{{\rm Sq}}^{4}(\operatorname{{\rm Sq}}^{2}(w_{3}))$ has 38 monomials when expressed as a polynomial in the Stiefel–Whitney classes.

There is also an exact sequence

$\displaystyle 1\longrightarrow\mathbb{Z}/2\mathbb{Z}\longrightarrow Spin(n)\longrightarrow SO(n)\longrightarrow 1$

which gives rise to the fiber sequence

$\displaystyle B\mathbb{Z}/2\mathbb{Z}\longrightarrow BSpin(n)\longrightarrow BSO(n).$

Using this, Quillen calculated:

By the Thom isomorphism, we have that $H^{*}(MSpin^{c};\mathbb{F}_{2})\cong H^{*}(BSpin^{c};\mathbb{F}_{2})\{U\}$ and $H^{*}(MSpin;\mathbb{F}_{2})\cong H^{*}(BSpin;\mathbb{F}_{2})\{U\}$ as graded $\mathbb{F}_{2}$-vector spaces. Since
$H^{*}(BSpin^{c};\mathbb{F}_{2})$ and $H^{*}(BSpin;\mathbb{F}_{2})$ are quotients of $H^{*}(BO;\mathbb{F}_{2})\cong\mathbb{F}_{2}[w_{1},w_{2},w_{3},\dots]$, the action of Steenrod squares on $H^{*}(BSpin^{c};\mathbb{F}_{2})$ and $H^{*}(BSpin;\mathbb{F}_{2})$ is determined by the Wu formula $\operatorname{{\rm Sq}}^{i}w_{j}=\sum_{k=0}^{i}\binom{j+k-i-1}{k}w_{i-k}w_{j+k}$ and the Cartan formula. This, together with the formula $\operatorname{{\rm Sq}}^{n}U=w^{n}U$ for the action of Steenrod squares on the Thom class $U$, determines the action of the Steenrod squares on the cohomology $H^{*}(MSpin^{c};\mathbb{F}_{2})$ of the spin^c-bordism spectrum $MSpin^{c}$.

The following definitions are classical (see e.g. chapter 1 of [13]):

The monomial symmetric polynomials form a $\mathbb{Z}$-linear basis for the ring of symmetric polynomials. The set of all finite products of elementary symmetric polynomials also famously (by Newton) forms a $\mathbb{Z}$-linear basis for the ring of symmetric polynomials. Consequently, for each $\lambda$, there exists a unique polynomial $P_{\lambda}(X_{1},\dots,X_{n})$ such that

$\displaystyle P_{\lambda}\left(e_{1}(X_{1},\dots,X_{n}),e_{2}(X_{1},\dots,X_{n}),\dots,e_{n}(X_{1},\dots,X_{n})\right)$

$\displaystyle=m_{\lambda}(X_{1},\dots,X_{n}).$

For more details about the polynomials $P_{\lambda}(X_{1},\dots,X_{n})$, see the material on the transition matrix $M(m,e)$ and its inverse $M(e,m)$ in section 1.6 of [13], particularly (6.7)(i).

See [24], particularly pages 71 and 96 and surrounding material, for a nice exposition of the following result, which dates back to Thom [25]: let $\Lambda$ be the set of unordered finite-length tuples of positive integers, each of which is not equal to $2^{a}-1$ for any integer $a$. Such integers are called “non-dyadic,” and such partitions are called “non-dyadic partitions.” For each $\lambda\in\Lambda$, write $\left|\lambda\right|$ for the length of $\lambda$, and write $\left|\left|\lambda\right|\right|$ for the sum of the elements of $\lambda$. Consider the polynomial

	$\displaystyle P_{\lambda}(w_{1},\dots,w_{\left|\lambda\right|})$	$\displaystyle\in H^{*}(BO;\mathbb{F}_{2})$
		$\displaystyle\cong\mathbb{F}_{2}[w_{1},w_{2},\dots]$

in the Stiefel–Whitney classes $w_{1},w_{2},\dots$. Then (see page 96 of [24], or pages 301-302 of [26]) the set

$$\{P_{\lambda}(w_{1},\dots,w_{\left|\lambda\right|})U):\lambda\in\Lambda\}$$

is a homogeneous $A$-linear basis for the graded free $A$-module $H^{*}(MO;\mathbb{F}_{2})$, where $U\in H^{0}(MO;\mathbb{F}_{2})$ denotes the Thom class, and $A$ is the mod $2$ Steenrod algebra. Consequently, for each nonnegative integer $n$, $\pi_{n}(MO)$ is the $\mathbb{F}_{2}$-linear dual of the $\mathbb{F}_{2}$-vector space with basis the set

(5)

$$\left\{P_{\lambda}(w_{1},\dots,w_{\left|\lambda\right|})U:\lambda\in\Lambda,\left|\left|\lambda\right|\right|=n\right\}.$$

Given two tuples $(a_{1},\dots,a_{m})$ and $(b_{1},\dots,b_{n})$, we have their concatenation

$\displaystyle(a_{1},\dots,a_{m})\coprod(b_{1},\dots,b_{n})$

$\displaystyle:=(a_{1},\dots,a_{m},b_{1},\dots,b_{n}).$

The coproduct on $H^{*}(MO;\mathbb{F}_{2})$ is then given by

$\displaystyle\Delta(P_{\lambda}(w_{1},\dots,w_{\left|\lambda\right|})U)$

$\displaystyle=\sum_{\lambda^{\prime},\lambda^{\prime\prime}\in\Lambda:\ \lambda=\lambda^{\prime}\coprod\lambda^{\prime\prime}}P_{\lambda^{\prime}}(w_{1},\dots,w_{\left|\lambda^{\prime}\right|})U\otimes P_{\lambda^{\prime\prime}}(w_{1},\dots,w_{\left|\lambda^{\prime\prime}\right|})U.$

Consequently, if we write $\{Y_{\lambda}\}$ for the basis of $\pi_{*}(MO)$ dual to the basis (5), then $Y_{\lambda}Y_{\lambda^{\prime}}=Y_{\lambda\coprod\lambda^{\prime}}$, and $\mathfrak{N}_{*}\cong\pi_{*}(MO)\cong\mathbb{F}_{2}[Y_{(2)},Y_{(4)},Y_{(5)},Y_{(6)},Y_{(8)},\dots]$, with $Y_{(i)}$ in degree $i$. We will sometimes write $Y_{i}$ as an abbreviation for Thom’s generator $Y_{(i)}$ of $\mathfrak{N}_{*}$.

The first stages of the Whitehead tower for the orthogonal group are:

$\displaystyle BString=BO\langle 8\rangle\longrightarrow BSpin=BO\langle 4\rangle\longrightarrow BSO=BO\langle 2\rangle\longrightarrow BO.$

While $BSpin^{c}$ does not fit into this sequence via a connective cover, the map $BSpin\longrightarrow BSO$ factors through $BSpin^{c}$. There is a commutative diagram whose rows and columns are fiber sequences:

On the level of spectra, we have maps

(6)

$\displaystyle MString\longrightarrow MSpin\longrightarrow MSpin^{c}\longrightarrow MSO\longrightarrow MO.$

The maps induced in homotopy give the maps of respective cobordism rings. We will specifically consider the images of $MSpin^{c}_{*}$ and $MSpin_{*}$ in $MO_{*}$.

By the Anderson–Brown–Peterson splitting of $2$-local $MSpin^{c}$, the cohomology $H^{*}(MSpin^{c};\mathbb{F}_{2})$ splits as a direct sum of suspensions of $H^{*}(ku;\mathbb{F}_{2})\cong A//E(1)$ and of $H^{*}(H\mathbb{F}_{2};\mathbb{F}_{2})\cong A$. Here we are using the standard notation $A$ for the mod $2$ Steenrod algebra, and $A//E(1)$ for its quotient $A\otimes_{E(1)}\mathbb{F}_{2}$, where $E(1)$ is the subalgebra of $A$ generated by $\operatorname{{\rm Sq}}^{1}$ and by $Q_{1}=[\operatorname{{\rm Sq}}^{1},\operatorname{{\rm Sq}}^{2}]$. Hence the $s=0$-line in the $2$-primary Adams spectral sequence for $MSpin^{c}$,

(7)		$\displaystyle E_{2}^{s,t}\cong\operatorname{{\rm Ext}}_{A}^{s,t}\left(H^{*}(MSpin^{c};\mathbb{F}_{2}),\mathbb{F}_{2}\right)$	$\displaystyle\Rightarrow\pi_{t-s}(MSpin^{c})^{\wedge}_{2}$
	$\displaystyle d_{r}:E_{r}^{s,t}$	$\displaystyle\rightarrow E_{r}^{s+r,t+r-1}$

is a direct sum of suspensions of $\mathbb{F}_{2}$, with one summand $\Sigma^{t}\mathbb{F}_{2}$ for each summand $\Sigma^{t}ku_{(2)}$ in $MSpin^{c}_{(2)}$, and also with one summand $\Sigma^{t}\mathbb{F}_{2}$ for each summand $\Sigma^{t}H\mathbb{F}_{2}$ in $MSpin^{c}_{(2)}$.

Anderson–Brown–Peterson prove in [3] that all differentials in the Adams spectral sequence (7) are zero. Consequently the $s=0$-line $\hom_{A}(H^{*}(MSpin^{c};\mathbb{F}_{2}),\mathbb{F}_{2})$ is the reduction of $\pi_{*}(MSpin^{c})$ modulo the ideal generated by $2$ and by the Bott element $\beta\in\pi_{2}(ku)$.

This means we can calculate $\pi_{*}(MSpin^{c})/(2,\beta)$ simply by calculating$\hom_{A}(H^{*}(MSpin^{c};\mathbb{F}_{2}),\mathbb{F}_{2})$, i.e., the $A_{*}$-comodule primitives $\mathbb{F}_{2}\Box_{A_{*}}H_{*}(MSpin^{c};\mathbb{F}_{2})$. The advantage of thinking in terms of comodule primitives is that the Adams spectral sequence respects ring structure: if we calculate the homology $H_{*}(MSpin^{c};\mathbb{F}_{2})$ as a ring, then by simply restricting to the comodule primitives in $H_{*}(MSpin^{c};\mathbb{F}_{2})$, we have calculated $\Omega^{Spin^{c}}_{*}/(2,\beta)$.

The same remarks apply mutatis mutandis for the spin bordism spectrum $MSpin$, the oriented bordism spectrum $MSO$ or for the unoriented bordism spectrum $MO$ in place of $MSpin^{c}$. The Anderson–Brown–Peterson splitting for $MSpin$ is as a wedge of suspensions of $ko$, $ko\langle 2\rangle$, and $H\mathbb{F}_{2}$. The analogue of the Anderson–Brown–Peterson splitting for $MO$ is Thom’s splitting of $MO$ as a wedge of suspensions of $H\mathbb{F}_{2}$, while $MSO_{(2)}$ splits as a wedge of suspensions of $H\mathbb{Z}_{(2)}$ and $H\mathbb{F}_{2}$; as far as we know, the latter splitting was originally proven by Wall [26]. Since $H^{*}(MSpin^{c};\mathbb{F}_{2})$ is a quotient $A$-module of $H^{*}(MSO;\mathbb{F}_{2})$, which is in turn a quotient $A$-module of $H^{*}(MO;\mathbb{F}_{2})$, dualizing yields that $H_{*}(MSpin^{c};\mathbb{F}_{2})$ is a subcomodule of $H_{*}(MSO;\mathbb{F}_{2})$, which is in turn a subcomodule of $H_{*}(MO;\mathbb{F}_{2})$.

Hence our broad strategy for calculating $\Omega^{Spin^{c}}_{*}/(2,\beta)\cong MSpin^{c}_{*}/(2,\beta)$ and $\Omega^{Spin}_{*}/(2,\eta,\alpha,\beta)\cong MSpin_{*}/(2,\eta,\alpha,\beta)$, and the natural maps $MSpin_{*}/(2,\eta,\alpha,\beta)\rightarrow MSpin^{c}_{*}/(2,\beta)\rightarrow MO_{*}$, is to calculate the $A_{*}$-comodule primitives in $H_{*}(MSpin^{c};\mathbb{F}_{2})$ and in $H_{*}(MSpin;\mathbb{F}_{2})$, regarding each as $A_{*}$-subcomodule algebras of $H_{*}(MO;\mathbb{F}_{2})\cong\mathbb{F}_{2}[w_{1},w_{2},w_{3},\dots]\{U\}$. The resulting information will describe $\Omega^{Spin^{c}}_{*}/(2,\beta)$ as a subring of $\mathfrak{N}_{*}$. Details of this strategy are given in the description of the computational method in the proof of Proposition 3.1.