Abstract
The gauge group of a principal Gbundle P over a space X is the group of Gequivariant 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 \(S^4\).
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1 Introduction
Let G be a topological group and X a space. The gauge group \({\mathscr {G}}(P)\) of a principal Gbundle P over X is defined as the group of Gequivariant 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 4manifolds.
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 Gbundles 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 CWcomplex, then the number of distinct homotopy types of \({\mathscr {G}}(P)\), as \(P\rightarrow X\) ranges over all principal Gbundles over X, is finite. In fact, since isomorphic Gbundles give rise to homeomorphic gauge groups, it will suffice to the let \(P\rightarrow X\) range over the set of isomorphism classes of principal Gbundles 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, MembrilloSolis 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 4manifolds [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 2primary indeterminacy appearing in the \(\textrm{Spin}(7)\) case, there are also known [26, 49] difficulties at the prime 3 due to the nonvanishing 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 simplyconnected 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 kth 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 kth 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 righthand 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:
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 CWcomplex and let Y be an Hspace 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
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
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
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 pathcomponents 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 pullback. Since we know that there is a fibration
it follows that we also have a fibration
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.
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
Lemma 5.1
Localised away from the prime 2, the boundary map
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 plocal 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 Hspace, 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
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
splits at the prime 2. Let denote a right homotopy inverse for \(\textrm{Spin}(7)\rightarrow S^7\). Then the composite
is a 2local 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
is at most 8 (resp. 3).
Proof
There is a fibration
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 pprimary 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
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
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 nontrivial 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 2local 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 zigzag of maps
induces isomorphisms in mod2 cohomology in dimensions 1 through 6, and therefore we have
From the fibration sequence
we see that \(H^*(F_k)\cong H^*(\Omega ^4S^7)\) in dimensions 1 through 6 for dimensional reasons, and hence we have
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 2local 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
(see, e.g. [56]), we find that the evaluation fibration
reduces to the exact sequence
Hence the result follows from [30]. \(\square \)
Proof of Theorem 1.3 (b)
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Rea, S. Homotopy types of \({{\textrm{Spin}}}^{{\textrm{c}}}(n)\)gauge groups over \(S^4\). European Journal of Mathematics 9, 48 (2023). https://doi.org/10.1007/s4087902300636x
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DOI: https://doi.org/10.1007/s4087902300636x