Homotopy types of ${{\textrm{Spin}}}^{{\textrm{c}}}(n)$-gauge groups over $S^4$

Abstract

The gauge group of a principal G-bundle P over a space X is the group of G-equivariant homeomorphisms of P that cover the identity on X. We consider the gauge groups of bundles over $S^4$ with ${{\textrm{Spin}}}^{{\textrm{c}}}(n)$, the complex spin group, as structure group and show how the study of their homotopy types reduces to that of ${{\textrm{Spin}}}(n)$-gauge groups over $S^4$. We then advance on what is known by providing a partial classification for ${{\textrm{Spin}}}(7)$- and ${{\textrm{Spin}}}(8)$-gauge groups over $S^4$.

1 Introduction

Let G be a topological group and X a space. The gauge group ${\mathscr {G}}(P)$ of a principal G-bundle P over X is defined as the group of G-equivariant bundle automorphisms of P which cover the identity on X. A detailed introduction to the topology of gauge groups of bundles can be found in [24, 42]. The study of gauge groups is important for the classification of principal bundles, as well as understanding moduli spaces of connections on principal bundles [7, 50, 52]. As is well known, Donaldson [12] discovered a deep link between the gauge groups of certain $\textrm{SU}(2)$-bundles and the differential topology of 4-manifolds.

Key properties of gauge groups are invariant under continuous deformation and so studying their homotopy theory is important. Having fixed a topological group G and a space X, an interesting problem is that of classifying the possible homotopy types of the gauge groups ${\mathscr {G}}(P)$ of principal G-bundles P over X.

Crabb and Sutherland showed [8, Theorem 1.1] that if G is a compact, connected, Lie group and X is a connected, finite CW-complex, then the number of distinct homotopy types of ${\mathscr {G}}(P)$, as $P\rightarrow X$ ranges over all principal G-bundles over X, is finite. In fact, since isomorphic G-bundles give rise to homeomorphic gauge groups, it will suffice to the let $P\rightarrow X$ range over the set of isomorphism classes of principal G-bundles over X.

Explicit classification results have been obtained, especially for the case of gauge groups of bundles with low rank, compact, Lie groups as structure groups and $X=S^4$ as base space. In particular, the first such result was obtained by Kono [31] in 1991. Using the fact that isomorphism classes of principal $\textrm{SU}(2)$-bundles over $S^4$ are classified by $k\in {\mathbb {Z}}\cong \pi _3(\textrm{SU}(2))$ and denoting by ${\mathscr {G}}_k$ the gauge group of the principal $\textrm{SU}(2)$-bundle $P_k\rightarrow S^4$ corresponding to the integer k, Kono showed that there is a homotopy equivalence ${\mathscr {G}}_k\simeq {\mathscr {G}}_l$ if and only if $(12,k)=(12,l)$, where (m, n) denotes the greatest common divisor of m and n. Since 12 has six divisors, it follows that there are precisely six homotopy types of $\textrm{SU}(2)$-gauge groups over $S^4$.

Results formally similar to that of Kono have been obtained for principal bundles over $S^4$ with different structure groups, among others, by: Hamanaka and Kono [17] for $\textrm{SU}(3)$-gauge groups; Theriault [53, 54] for $\textrm{SU}(n)$-gauge groups, as well as [48] for $\textrm{Sp}(2)$-gauge groups; Cutler [9, 10] for $\textrm{Sp}(3)$-gauge groups and $\textrm{U}(n)$-gauge groups; Kishimoto and Kono [27] for $\textrm{Sp}(n)$-gauge groups; Kishimoto, Theriault and Tsutaya [30] for $G_2$-gauge groups; Kamiyama, Kishimoto, Kono and Tsukuda [25] for $\textrm{SO}(3)$-gauge groups; Kishimoto, Membrillo-Solis and Theriault [29] for $\textrm{SO}(4)$-gauge groups; Hasui, Kishimoto, Kono and Sato [20] for $\textrm{PU}(3)$- and $\textrm{PSp}(2)$-gauge groups; and Hasui, Kishimoto, So and Theriault [21] for bundles with exceptional Lie groups as structure groups.

There are also several classification results for gauge groups of bundles with base spaces other than $S^4$ [2, 16, 18, 20, 22, 23, 28, 32, 33, 35, 36, 39,40,41, 43, 45, 46, 51, 55, 57].

The complex spin group $\textrm{Spin}^{{\textrm{c}}}(n)$ was first introduced in 1964 in a paper of Atiyah, Bott and Shapiro [4]. There has been an increasing interest in the $\textrm{Spin}^{{\textrm{c}}}(n)$ groups ever since the publication of the Seiberg–Witten equations for 4-manifolds [58], whose formulation requires the existence of $\textrm{Spin}^{{\textrm{c}}}(n)$-structures, and more recently for the role they play in string theory [6, 13, 44].

In this paper we examine $\textrm{Spin}^{{\textrm{c}}}(n)$-gauge groups over $S^4$. We begin by recalling some basic properties of the complex spin group $\textrm{Spin}^{{\textrm{c}}}(n)$ and showing that it can be expressed as a product of a circle and the real spin group $\textrm{Spin}(n)$. For $n\geqslant 6$, we show that this decomposition is reflected in the corresponding gauge groups.

Theorem 1.1

For $n\geqslant 6$ and any $k\in {\mathbb {Z}}$, we have

The homotopy theory of $\textrm{Spin}^{{\textrm{c}}}(n)$-gauge groups over $S^4$ therefore reduces to that of the corresponding $\textrm{Spin}(n)$-gauge groups. We advance on what is known on $\textrm{Spin}(n)$-gauge groups by providing a partial classification for $\textrm{Spin}(7)$- and $\textrm{Spin}(8)$-gauge groups over $S^4$.

Theorem 1.2

• (a) If $(168,k)=(168,l)$, there is a homotopy equivalence

  $$\begin{aligned} {\mathscr {G}}_k(\textrm{Spin}(7))\simeq {\mathscr {G}}_l(\textrm{Spin}(7)) \end{aligned}$$

  after localising rationally or at any prime.

• (b) If ${\mathscr {G}}_k(\textrm{Spin}(7))\simeq {\mathscr {G}}_l(\textrm{Spin}(7))$ then $(84,k)=(84,l)$.

We note that the discrepancy by a factor of 2 between parts (a) and (b) is due to the same discrepancy for $G_2$-gauge groups.

Theorem 1.3

• (a) If $(168,k)=(168,l)$, there is a homotopy equivalence

  $$\begin{aligned} {\mathscr {G}}_k(\textrm{Spin}(8))\simeq {\mathscr {G}}_l(\textrm{Spin}(8)) \end{aligned}$$

  after localising rationally or at any prime.

• (b) If ${\mathscr {G}}_k(\textrm{Spin}(8))\simeq {\mathscr {G}}_l(\textrm{Spin}(8))$ then $(28,k)=(28,l)$.

For the $\textrm{Spin}(8)$ case, in addition to the same 2-primary indeterminacy appearing in the $\textrm{Spin}(7)$ case, there are also known [26, 49] difficulties at the prime 3 due to the non-vanishing of $\pi _{10}(\textrm{Spin}(8))_{(3)}$.

2 $\textrm{Spin}^{{\textrm{c}}}(n)$ groups

For $n\geqslant 1$, the complex spin group $\textrm{Spin}^{{\textrm{c}}}(n)$ is defined as the quotient

where denotes the central subgroup of order 2. The group $\textrm{Spin}^{{\textrm{c}}}(n)$ is a special case of the more general notion of $\textrm{Spin}^k(n)$ group introduced in [1].

The first low rank $\textrm{Spin}^{{\textrm{c}}}(n)$ groups can be identified as follows:

• $\textrm{Spin}^{{\textrm{c}}}(1)\cong \textrm{U}(1)\simeq S^1$;

• ;

• ;

• .

The group $\textrm{Spin}^{{\textrm{c}}}(n)$ fits into a commutative diagram

where q is the quotient map, $\lambda :\textrm{Spin}(n)\rightarrow \textrm{SO}(n)$ denotes the double covering map of the group $\textrm{SO}(n)$ by $\textrm{Spin}(n)$ and denotes the degree 2 map. Furthermore, we observe that the map

is a double covering of by $\textrm{Spin}^{{\textrm{c}}}(n)$.

3 Method of classification

A principal bundle isomorphism determines a homeomorphism of gauge groups induced by conjugation [42]. We therefore begin by considering isomorphism classes of principal $\textrm{Spin}^{{\textrm{c}}}(n)$-bundles over $S^4$. These are classified by the free homotopy classes of maps . Since $\textrm{Spin}^{{\textrm{c}}}(n)$ is connected, $\textrm{B}\textrm{Spin}^{{\textrm{c}}}(n)$ is simply-connected and hence there are isomorphisms

Remark 3.1

Note that for $n=3$ we have $\textrm{Spin}^{{\textrm{c}}}(3)\cong \textrm{U}(2)$, and the homotopy types of $\textrm{U}(2)$-gauge groups over $S^4$ have been studied by Cutler in [10].

For $n\geqslant 5$, let ${\mathscr {G}}_k$ denote the gauge group of the $\textrm{Spin}^{{\textrm{c}}}(n)$-bundle $P_k\rightarrow S^4$ classified by $k\in {\mathbb {Z}}$. By [3, 15], there is a homotopy equivalence

the latter space being the k-th component of , meaning the connected component containing the map classifying $P_k\rightarrow S^4$.

There is an evaluation fibration

where ${{\,\textrm{ev}\,}}$ evaluates a map at the basepoint of $S^4$ and the fibre is the k-th component of the pointed mapping space . This fibration extends to a homotopy fibration sequence

Furthermore, by [47] there is, for each $k\in {\mathbb {Z}}$, a homotopy equivalence

The space on the right-hand side is homotopy equivalent to by the pointed exponential law, and is more commonly denoted as $\Omega ^3_0\textrm{Spin}^{{\textrm{c}}}(n)$. We therefore have the following homotopy fibration sequence:

$$\begin{aligned} {\mathscr {G}}_k\longrightarrow \textrm{Spin}^{{\textrm{c}}}(n)\xrightarrow {\ \partial _k\ }\Omega _0^3\textrm{Spin}^{{\textrm{c}}}(n)\longrightarrow \textrm{B}{\mathscr {G}}_k\longrightarrow \textrm{B}\textrm{Spin}^{{\textrm{c}}}(n), \end{aligned}$$

which exhibits the gauge group ${\mathscr {G}}_k$ as the homotopy fibre of the map $\partial _k$. This is a key observation, as it implies that the homotopy theory of the gauge groups ${\mathscr {G}}_k$ depends on the maps $\partial _k$.

Lemma 3.2

(Lang [34, Theorem 2.6]) The adjoint of $\partial _k:\textrm{Spin}^{{\textrm{c}}}(n)\rightarrow \Omega _0^3\textrm{Spin}^{{\textrm{c}}}(n)$ is homotopic to the Samelson product , where $\epsilon \in \pi _3(\textrm{Spin}^{{\textrm{c}}}(n))$ is a generator and 1 denotes the identity map on $\textrm{Spin}^{{\textrm{c}}}(n)$.

As the Samelson product is bilinear, we have $\langle k\epsilon ,1\rangle \simeq k\langle \epsilon ,1\rangle$, and hence, taking adjoints once more, $\partial _{k}\simeq k \partial _{1}$.

Lemma 3.3

(Theriault [48, Lemma 3.1]) Let X be a connected CW-complex and let Y be an H-space with a homotopy inverse. Suppose that $f\in [X,Y]$ has finite order and let $m\in {\mathbb {N}}$ be such that $mf\simeq *$. Then, for any integers $k,l\in {\mathbb {Z}}$ such that $(m,k)=(m,l)$, the homotopy fibres of kf and lf are homotopy equivalent when localised rationally or at any prime.

Remark 3.4

The lemma of Theriault is the local analogue of a lemma used by Hamanaka and Kono in their study [17] of $\textrm{SU}(3)$-gauge groups over $S^4$.

Part (a) of Theorems 1.2 and 1.3 will follow as applications of Lemma 3.3, whereas for part (b) we will need to determine suitable homotopy invariants of the gauge groups.

4 $\textrm{Spin}^{{\textrm{c}}}(n)$-gauge groups

We begin with a decomposition of $\textrm{Spin}^{{\textrm{c}}}(n)$ as a product of spaces which will be reflected in an analogous decomposition of $\textrm{Spin}^{{\textrm{c}}}(n)$-gauge groups.

From the definition of $\textrm{Spin}^{{\textrm{c}}}(n)$, we can construct the commutative diagram

There is, therefore, an exact sequence

$$\begin{aligned} 1\longrightarrow \textrm{Spin}(n)\longrightarrow \textrm{Spin}^{{\textrm{c}}}(n)\longrightarrow S^1 \longrightarrow 1, \end{aligned}$$

and hence a fibration

A section for ($\star$) can be obtained as follows:

Hence ($\star$) splits, and we have a homeomorphism

We are now ready to show that the decomposition

for $n\geqslant 6$ holds as stated in Theorem 1.1.

Proof of Theorem 1.1

Let $\varrho$ and g denote the maps in the fibration

$$\begin{aligned} \textrm{Spin}(n) \xrightarrow {\ \varrho \ }\textrm{Spin}^{{\textrm{c}}}(n) \xrightarrow {\ g \ } S^1, \end{aligned}$$

and let denote a section of g.

As $\pi _4(\textrm{Spin}^{{\textrm{c}}}(n))\cong 0$ for $n\geqslant 6$, there is a lift in the diagram

Define the map b to be the composite

$$\begin{aligned} {\mathscr {G}}_k(\textrm{Spin}^{{\textrm{c}}}(n)) \longrightarrow \textrm{Spin}^{{\textrm{c}}}(n) \xrightarrow {\ g \ } S^1. \end{aligned}$$

Since, in particular, s is a right homotopy inverse for g, the map a is a right homotopy inverse for b. Therefore we have , where $F_b$ denotes the homotopy fibre of b.

As the map $\varrho :\textrm{Spin}(n)\rightarrow \textrm{Spin}^{{\textrm{c}}}(n)$ is a group homomorphism, it classifies to a map

Since $\varrho$ induces an isomorphism in $\pi _3$, it respects path-components in and for any $k\in {\mathbb {Z}}$. We therefore have a diagram of fibration sequences

Furthermore, observe that for all $k\in {\mathbb {Z}}$ we have

and, similarly, . Since $\varrho$ induces isomorphisms on $\pi _m$ for $m\geqslant 2$, it follows that $(\textrm{B}\varrho )_*$ induces isomorphisms

for all m and is therefore a homotopy equivalence by Whitehead’s theorem.

We can extend the fibration diagram (1) to the left as

where $\partial _k'$ denotes the boundary map associated to $\textrm{Spin}(n)$-gauge groups over $S^4$.

Since $(\textrm{B}\varrho )_*$ is a homotopy equivalence, the leftmost square is a homotopy pull-back. Since we know that there is a fibration

$$\begin{aligned} \textrm{Spin}(n) \xrightarrow {\ \varrho \ }\textrm{Spin}^{{\textrm{c}}}(n) \xrightarrow {\ g \ } S^1, \end{aligned}$$

it follows that we also have a fibration

$$\begin{aligned} {\mathscr {G}}_k(\textrm{Spin}(n)) \xrightarrow {\ {\mathscr {G}}_k(\varrho ) \ }{\mathscr {G}}_k(\textrm{Spin}^{{\textrm{c}}}(n)) \xrightarrow {\ b \ } S^1. \end{aligned}$$

In particular, the space ${\mathscr {G}}_k(\textrm{Spin}(n))$ is seen to be the homotopy fibre $F_b$ of the map $b:{\mathscr {G}}_k(\textrm{Spin}^{{\textrm{c}}}(n))\rightarrow S^1$ and hence we have

$\square$

Remark 4.1

Alternatively, the referee suggested the following approach to a proof of Lemma 1.1. Since the map $\textrm{B}\varrho$ induces an isomorphism , to any principal $\textrm{Spin}^{{\textrm{c}}}(n)$-bundle P over $S^4$ we can associate a principal $\textrm{Spin}(n)$-bundle $P'$ over $S^4$ such that . There exists then a fibrewise exact sequence of adjoint bundles

Recalling [3, Section 2] that gauge groups can be defined as spaces of sections of adjoint bundles, we obtain a diagram of fibration sequences

Showing, as we have done, that ${\mathscr {G}}(P)\rightarrow S^1$ admits a homotopy section then leads to the statement of Lemma 1.1.

In light of Theorem 1.1, the homotopy theory of $\textrm{Spin}^{{\textrm{c}}}(n)$-gauge groups over $S^4$ for $n\geqslant 6$ is completely determined by that of $\textrm{Spin}(n)$-gauge groups over $S^4$.

Remark 4.2

By a result of Cutler [10], there is a decomposition

of $\textrm{U}(2)$-gauge groups over $S^4$ whenever k is even. Given that $\textrm{Spin}^{{\textrm{c}}}(3)\cong \textrm{U}(2)$ and $\textrm{Spin}(3)\cong \textrm{SU}(2)$, the statement of Theorem 1.1 still holds true when $n=2$ provided that k is even. Cutler also shows that for odd k, so Theorem 1.1 does not hold for $n=2$.

5 $\textrm{Spin}(n)$-gauge groups

We now shift our focus to principal $\textrm{Spin}(n)$-bundles over $S^4$ and the classification of their gauge groups. In the interest of completeness, we recall that, for $n\leqslant 6$, the following exceptional isomorphisms hold.

Table 1 The exceptional isomorphisms

The cases $n=1,2$ are trivial. Indeed, as $\pi _3(\textrm{O}(1))\cong \pi _3(\textrm{U}(1))\cong 0$, there is only one isomorphism class of $\textrm{O}(1)$- and $\textrm{U}(1)$-bundles over $S^4$ (namely, that of the trivial bundle), and hence there is only one possible homotopy type for the corresponding gauge groups. The case $n=3$ was studied by Kono in [31]. The case $n=4$ can be reduced to the $n=3$ case by [5, Theorem 5]. The case $n=5$ was studied by Theriault in [48]. Finally, the case $n=6$ was studied by Cutler and Theriault in [11].

We shall now explore the $n=7$ case. Recall that we have a fibration sequence

$$\begin{aligned} {\mathscr {G}}_k(\textrm{Spin}(7))\longrightarrow \textrm{Spin}(7)\xrightarrow {\ k\partial _1\ } \Omega ^3_0\textrm{Spin}(7). \end{aligned}$$

Lemma 5.1

Localised away from the prime 2, the boundary map

$$\begin{aligned} \textrm{Spin}(7)\xrightarrow {\ \partial _1\ } \Omega ^3_0\textrm{Spin}(7) \end{aligned}$$

has order 21.

Proof

Harris [19] showed that $\textrm{Spin}(2m+1)\simeq _{(p)}\textrm{Sp}(m)$ for odd primes p. This result was later improved by Friedlander [14] to a p-local homotopy equivalence of the corresponding classifying spaces. Then, in particular, localising at an odd prime p, we have a commutative diagram

where $\partial _1':\textrm{Sp}(3)\rightarrow \Omega _0^3\textrm{Sp}(3)$ denotes the boundary map associated to $\textrm{Sp}(3)$-gauge groups over $S^4$ studied in [9]. Hence the result follows from the calculation in [9, Theorem 1.2] where it is shown that $\partial _1'$ has order 21 after localising away from the prime 2.$\square$

Lemma 5.2

Let $F\rightarrow X\rightarrow Y$ be a homotopy fibration, where F is an H-space, and let $\partial :\Omega Y \rightarrow F$ be the homotopy fibration connecting map. Let $\alpha :A\rightarrow \Omega Y$ and $\beta :B\rightarrow \Omega Y$ be maps such that

• (1) is a homotopy equivalence, where $\mu$ is the loop multiplication on $\Omega Y$;

• (2) is nullhomotopic.

Then the orders of $\partial$ and coincide.

Proof

Let denote the canonical homotopy action of the loopspace $\Omega Y$ onto the homotopy fibre F, and let . Consider the diagram

The left portion of the diagram commutes by the assumption that , while the right and bottom portions commute by properties of the canonical action $\theta$. Therefore

and hence the orders of $\partial$ and coincide.$\square$

Lemma 5.3

Localised at the prime 2, the order of the boundary map

$$\begin{aligned} \textrm{Spin}(7)\xrightarrow {\ \partial _1\ } \Omega ^3_0\textrm{Spin}(7) \end{aligned}$$

is at most 8.

Proof

The strategy here will be to show that $\partial _8$ is nullhomotopic. This will suffice as we have $\partial _8\simeq 8\partial _1$ by Lemma 3.2.

By a result of Mimura [37, Proposition 9.1], the fibration

$$\begin{aligned} G_2\xrightarrow {\ \alpha \ } \textrm{Spin}(7) \longrightarrow S^7 \end{aligned}$$

splits at the prime 2. Let denote a right homotopy inverse for $\textrm{Spin}(7)\rightarrow S^7$. Then the composite

is a 2-local homotopy equivalence.

Observe that we have since $\pi _{10}(\textrm{Spin}(7))\cong {\mathbb {Z}}/8{\mathbb {Z}}$ and . Therefore, by Lemma 5.2, the order of $\partial _8$ equals the order of .

As $\alpha$ is a group homomorphism, there is a diagram of evaluation fibrations

Since $\partial _8'\simeq 8\partial '_1\simeq *$ by [30, Theorem 1.1], we must have $\partial _8\simeq 8\partial _1\simeq *$.$\square$

Proof of Theorem 1.2 (a)

Lemmas 5.1 and 5.3 imply that $168\partial _1\simeq *$, so the result follows from Lemma 3.3. $\square$

We now move on to consider $\textrm{Spin}(8)$-gauge groups.

Lemma 5.4

Localised at the prime 2 (resp. 3), the order of the boundary map

$$\begin{aligned} \textrm{Spin}(8)\xrightarrow {\ \partial _1\ } \Omega ^3_0\textrm{Spin}(8) \end{aligned}$$

is at most 8 (resp. 3).

Proof

There is a fibration

$$\begin{aligned} \textrm{Spin}(7)\xrightarrow {\ \alpha \ } \textrm{Spin}(8)\longrightarrow S^7 \end{aligned}$$

which admits a section , and hence splits integrally. Therefore, we have a homeomorphism

Integrally, we have

(see, e.g. the table in [38]). Hence the same argument presented in the proof of Lemma 5.3 shows that $8\partial _1\simeq *$ and $3\partial _1\simeq *$ after localising at $p=2$ and $p=3$, respectively. $\square$

Lemma 5.5

Let $p\ne 3$ be an odd prime. Then the p-primary order of the boundary map $\partial _1:\textrm{Spin}(8)\rightarrow \Omega ^3_0\textrm{Spin}(8)$ is bounded from above by that of $\partial _1:\textrm{Spin}(7)\rightarrow \Omega ^3_0\textrm{Spin}(7)$.

Proof

As , any map is nullhomotopic after localisation at an odd prime p different from 3. Thus, decomposing $\textrm{Spin}(8)$ as and arguing as in the proof of Lemma 5.3 yields the statement. $\square$

Proof of Theorem 1.3 (a)

Lemmas 5.4 and 5.5 imply that $168\partial _1\simeq *$, so the result follows from Lemma 3.3. $\square$

6 Homotopy invariants of $\textrm{Spin}(n)$-gauge groups

Lemma 6.1

If ${\mathscr {G}}_k(\textrm{Spin}(7))\simeq {\mathscr {G}}_l(\textrm{Spin}(7))$, then $(21,k)=(21,l)$.

Proof

As in the proof of Lemma 5.1, localising at an odd prime, we have an equivalence . We therefore have a diagram of homotopy fibrations

where $\partial _k':\textrm{Sp}(3)\rightarrow \Omega _0^3\textrm{Sp}(3)$ denotes the boundary map studied in [9]. Thus, by the five lemma, we have

$$\begin{aligned} \pi _{11}(\textrm{B}{\mathscr {G}}_k(\textrm{Spin}(7)))\cong \pi _{11}(\textrm{B}{\mathscr {G}}_k(\textrm{Sp}(3))). \end{aligned}$$

Hence the result now follows from the calculations in [9, Theorem 1.1] where it is shown that, integrally,

$\square$

In their study of the homotopy types of $G_2$-gauge groups over $S^4$ in [30], Kishimoto, Theriault and Tsutaya constructed a space $C_k$ for which

$$\begin{aligned} H^*(C_k)\cong H^*({\mathscr {G}}_k(G_2)) \end{aligned}$$

in mod 2 cohomology in dimensions 1 through 6. The cohomology of $C_k$ is then shown to be as follows.

Lemma 6.2

([30, Lemma 8.3]) We have

• if $(4,k)=1$ then $C_k\simeq S^3$, so $H^*(C_k)\cong H^*(S^3)$;

• if $(4,k)=2$ or $(4,k)=4$ then , where $P^n(p)$ denotes the nth dimensional mod p Moore space;

• if $(4,k)=2$ then $\textrm{Sq}^2$ is non-trivial on the degree 4 generator in $H^*(C_k)$;

• if $(4,k)=4$ then $\textrm{Sq}^2$ is trivial on the degree 4 generator in $H^*(C_k)$.

We make use of the same spaces $C_k$ as follows.

Lemma 6.3

If ${\mathscr {G}}_k(\textrm{Spin}(7))\simeq {\mathscr {G}}_l(\textrm{Spin}(7))$, then we have $(4,k)=(4,l)$.

Proof

As in the proof of Lemma 5.3, recall that we have a 2-local homotopy equivalence

Since the map $\alpha :G_2\rightarrow \textrm{Spin}(7)$ is a homomorphism, we have a commutative diagram

Furthermore, as $\pi _7(\Omega _0^3G_2)\cong \pi _{10}(G_2)\cong 0$, we have

and thus there is a commutative diagram

for some $\gamma$ representing a class in $\pi _7(\Omega ^3S^7)\cong \pi _{10}(S^7)\cong {\mathbb {Z}}/8{\mathbb {Z}}$.

We therefore have a commutative diagram

which induces a map of fibres $\phi :M\rightarrow {\mathscr {G}}_k(\textrm{Spin}(7))$, where M denotes the homotopy fibre of the map .

Since the lowest dimensional cell in appears in dimension 10, the canonical map is a homotopy equivalence in dimensions less than 9. It thus follows that M is homotopy equivalent to the homotopy fibre of in dimensions up to 8. Since the homotopy fibre of is just the product , the zig-zag of maps

induces isomorphisms in mod-2 cohomology in dimensions 1 through 6, and therefore we have

From the fibration sequence

$$\begin{aligned} \Omega ^4S^7 \longrightarrow F_k \longrightarrow S^7 \end{aligned}$$

we see that $H^*(F_k)\cong H^*(\Omega ^4S^7)$ in dimensions 1 through 6 for dimensional reasons, and hence we have

$$\begin{aligned} H^*(F_k)\cong {\mathbb {Z}}/2{\mathbb {Z}}[y_3,y_6] ,\quad *\leqslant 6, \end{aligned}$$

where $|y_i|=i$, which, in turn, yields

Since $H^*(F_k)$ does not contribute any generators in degree 4 to $H^*({\mathscr {G}}_k(\textrm{Spin}(7)))$, the result now follows from Lemma 6.2. Indeed, the presence of a degree 4 generator allows us to distinguish between the $(4,k)=1$ case and the $2\mid k$ cases, whereas the vanishing of the Steenrod square $Sq^2$ on the degree 4 generator in $H^*({\mathscr {G}}_k(\textrm{Spin}(7)))$ coming from $H^*(C_k)$ can be used to distinguish between the $(4,k)=2$ and $(4,k)=4$ cases. $\square$

Proof of Theorem 1.2 (b)

Combine Lemmas 6.1 and 6.3. $\square$

Lemma 6.4

If ${\mathscr {G}}_k(\textrm{Spin}(8))\simeq {\mathscr {G}}_l(\textrm{Spin}(8))$, then $(4,k)=(4,l)$.

Proof

As in the proof of Lemma 5.3, the splitting of $G_2\rightarrow \textrm{Spin}(7)\rightarrow S^7$ at the prime 2 implies that there is a 2-local homotopy equivalence

Since the fibration $\textrm{Spin}(7)\rightarrow \textrm{Spin}(8)\rightarrow S^7$ also splits after localising at any prime, there is a decomposition

where $\iota :\textrm{Spin}(7)\rightarrow \textrm{Spin}(8)$ is the inclusion homomorphism and $\gamma$ is a homotopy inverse for the map $\textrm{Spin}(8)\rightarrow S^7$.

Since the map is a homomorphism, we have a commutative diagram

Furthermore, as $\pi _7(\Omega _0^3G_2)\cong \pi _{10}(G_2)\cong 0$, we have

and thus there are commutative diagrams

for some $\delta ,\delta '$ representing classes in . We therefore have a commutative diagram

Arguing as in the proof of Lemma 6.3, we conclude that

Observing that does not contribute any generators in degree 4 to $H^*({\mathscr {G}}_k(\textrm{Spin}(8)))$ and arguing as in the proof of Lemma 6.3 yields the statement. $\square$

Lemma 6.5

If ${\mathscr {G}}_k(\textrm{Spin}(8))\simeq {\mathscr {G}}_l(\textrm{Spin}(8))$, then $(7,k)=(7,l)$.

Proof

Localising at $p=7$, we have

Applying the functor $\pi _{11}$ and noting that

$$\begin{aligned} \pi _{10}(S^7)\cong \pi _{11}(S^7)\cong \pi _{14}(S^7)\cong 0 \end{aligned}$$

(see, e.g. [56]), we find that the evaluation fibration

reduces to the exact sequence

$$\begin{aligned} \pi _{11}(G_2) \longrightarrow \pi _{11}(\Omega ^3_0 G_2)\longrightarrow \pi _{11}(\textrm{B}{\mathscr {G}}_k(\textrm{Spin}(8)))\longrightarrow 0. \end{aligned}$$

Hence the result follows from [30]. $\square$

Proof of Theorem 1.3 (b)

Combine Lemmas 6.4 and 6.5. $\square$