1 Introduction

Let G be a simple, simply-connected, compact Lie group and let M be an orientable, simply-connected, closed 4-manifold. Then the isomorophism class of a principal G-bundle P over M is classified by its second Chern class \(k\in {\mathbb {Z}}\). In particular, if \(k=0\), then P is a trivial G-bundle. The associated gauge group \({\mathcal {G}}_k(M)\) is the topological group of G-equivariant automorphisms of P which fix M.

A simply-connected 4-manifold is spin if and only if its intersection form is even. In the case of simply-connected 4-manifolds, the spin condition is equivalent to all cup product squares being trivial in mod 2 cohomology. In this paper, we consider the homotopy types of gauge groups \({\mathcal {G}}_k(M)\), where M is a non-spin 4-manifold such as \({\mathbb {C}}{\mathbb {P}}^2\). When M is a spin 4-manifold, topologists have been studying the homotopy types of gauge groups over M extensively over the last twenty years. On the one hand, Theriault showed in [16] that there is a homotopy equivalence

$$\begin{aligned} {\mathcal {G}}_k(M)\simeq {\mathcal {G}}_k(S^4)\times \prod ^d_{i=1}\Omega ^2G, \end{aligned}$$

where d is the second Betti number of M. Therefore to study the homotopy type of \({\mathcal {G}}_k(M)\) it suffices to study \({\mathcal {G}}_k(S^4)\). On the other hand, many cases of homotopy types of \({\mathcal {G}}_k(S^4)\)’s are known. For examples, there are 6 distinct homotopy types of \({\mathcal {G}}_k(S^4)\)’s for \(G=SU(2)\) [11], and 8 distinct homotopy types for \(G=SU(3)\) [5]. When localized rationally or at any prime, there are 16 distinct homotopy types for \(G=SU(5)\) [19] and 8 distinct homotopy types for \(G=Sp(2)\) [17].

When M is a non-spin 4-manifold, the author in [14] showed that there is a homotopy equivalence

$$\begin{aligned} {\mathcal {G}}_k(M)\simeq {\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\times \prod ^{d-1}_{i=1}\Omega ^2G, \end{aligned}$$

so the homotopy type of \({\mathcal {G}}_k(M)\) depends on the special case \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\). Compared to the extensive work on \({\mathcal {G}}_k(S^4)\), only two cases of \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\) have been studied, which are the SU(2)- and SU(3)-cases [12, 18]. As a sequel to [14], this paper investigates the homotopy types of \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\)’s in order to explore gauge groups over non-spin 4-manifolds.

A common approach to classifying the homotopy types of gauge groups is as follows. Atiyah, Bott and Gottlieb [1, 3] showed that the classifying space \(B{\mathcal {G}}_k(M)\) is homotopy equivalent to the connected component \({\text {Map}}_k(M, BG)\) of the mapping space \({\text {Map}}(M, BG)\) containing the map \(k\alpha \circ q\), where \(q:M\rightarrow S^4\) is the quotient map and \(\alpha \) is a generator of \(\pi _4(BG)\cong {\mathbb {Z}}\). The evaluation map \(ev:B{\mathcal {G}}_k(M)\rightarrow BG\) induces a fibration sequence

$$\begin{aligned} {\mathcal {G}}_k(M)\longrightarrow G\overset{\partial _k}{\longrightarrow }{\text {Map}}^*_k(M, BG)\longrightarrow B{\mathcal {G}}_k(M)\overset{ev}{\longrightarrow }BG, \end{aligned}$$
(1)

where \(\partial _k:G\rightarrow {\text {Map}}^*_k(M, BG)\) is the boundary map. The action of \(\pi _4(BG)\cong {\mathbb {Z}}\) on \({\text {Map}}^*_k(M, BG)\) induces a homotopy equivalence \({\text {Map}}^*_k(M,BG)\simeq {\text {Map}}^*_0(M, BG)\). Denote the composition \(G\overset{\partial _k}{\longrightarrow }{\text {Map}}^*_k(M,BG)\simeq {\text {Map}}^*_0(M, BG)\) also by \(\partial _k\) for convenience. For \(M=S^4\), \({\text {Map}}^*_0(M, BG)\simeq \Omega ^3_0G\) is an H-group so \([G, \Omega ^3_0G]\) is a group. The order of \(\partial _1:G\rightarrow \Omega ^3_0G\) is important for distinguishing the homotopy types of \({\mathcal {G}}_k(S^4)\).

Theorem 1.1

(Theriault, [17]) Let m be the order of \(\partial _1\). If \((m, k)=(m, l)\), then \({\mathcal {G}}_k(S^4)\) is homotopy equivalent to \({\mathcal {G}}_l(S^4)\) when localized rationally or at any prime.

For most cases of G, the exact value of the order of \(\partial _1\) is difficult to compute. When \(G=SU(n)\), the exact value or a partial result of the order of \(\partial _1\) was worked out for certain cases. For any number \(a=p^rq\) where q is coprime to p, the p-component of a is \(p^r\) and is denoted by \(\nu _p(a)\).

Theorem 1.2

([2, 5, 9, 11, 19, 20]) Let G be SU(n) and let m be the order of \(\partial _1\). Then

  • \(m=12\) for \(n=2\)

  • \(m=24\) for \(n=3\)

  • \(m=120\) for \(n=5\)

  • \(m=60\) or 120 for \(n=4\)

  • \(\nu _p(m)=\nu _p(n(n^2-1))\) for \(n<(p-1)^2+1\).

In Theorem 1.1, the g.c.d condition \((m,k)=(m,l)\) gives a sufficient condition for the homotopy equivalence \({\mathcal {G}}_k(S^4)\simeq {\mathcal {G}}_l(S^4)\). Conversely, there is a partial necessary condition for certain cases of \(G=SU(n)\).

Theorem 1.3

(Hamanaka and Kono [5]; Kishimoto, Kono and Tsutaya [9]) Let G be SU(n) and let p be an odd prime. If \({\mathcal {G}}_k(S^4)\) is homotopy equivalent to \({\mathcal {G}}_l(S^4)\), then

  • \((n(n^2-1),k)=(n(n^2-1),l)\) for n odd,

  • \(\nu _p(n(n^2-1),k)=\nu _p(n(n^2-1),l)\) for n less than \((p-1)^2+1\).

In this paper we consider gauge groups over \({\mathbb {C}}{\mathbb {P}}^2\). Take \(M={\mathbb {C}}{\mathbb {P}}^2\) in (1) and denote the boundary map by \(\partial '_k:G\rightarrow {\text {Map}}^*_0({\mathbb {C}}{\mathbb {P}}^2, BG)\). Since \({\text {Map}}^*_0({\mathbb {C}}{\mathbb {P}}^2, BG)\) is not an H-space, \([G,{\text {Map}}^*_0({\mathbb {C}}{\mathbb {P}}^2,BG)]\) is not a group so the order of \(\partial '_k\) makes no sense. However, we can still define an “order” of \(\partial '_k\) [18], which will be described in Sect. 2. We show that the “order” of \(\partial '_1\) helps distinguish the homotopy type of \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\) as in Theorem 1.1.

Theorem 1.4

Let \(m'\) be the “order” of \(\partial '_1\). If \((m', k)=(m', l)\), then \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\) is homotopy equivalent to \({\mathcal {G}}_l({\mathbb {C}}{\mathbb {P}}^2)\) when localized rationally or at any prime.

We study the SU(n)-gauge groups over \({\mathbb {C}}{\mathbb {P}}^2\) and use unstable K-theory to give a lower bound on the “order” of \(\partial '_1\) that is in the spirit of Theorem 1.2.

Theorem 1.5

When G is SU(n), the “order” of \(\partial '_1\) is at least \(\frac{1}{2}n(n^2-1)\) for n odd, and \(n(n^2-1)\) for n even.

Localized rationally or at an odd prime, we have \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\simeq {\mathcal {G}}_k(S^4)\times \Omega ^{2} G\) [16]. The homotopy types of \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\) are then completely determined by that of \({\mathcal {G}}_k(S^4)\), which have been investigated in many cases when the localizing prime is relatively large [6, 7, 9, 10, 20]. A large part of the remaining cases can be understood by studying the 2-localized order of \(\partial '_1\), on which Theorem 1.5 gives bounds for the SU(n) case. For example, combining Theorem 1.5 with Lemma 2.2 implies the order of \(\partial '_1\) is either 120 or 60 for \(G=SU(5)\). Furthermore, when \(G=SU(4)\) since the order of \(\partial _1\) is either 120 or 60, the order of \(\partial '_1\) is either 60 or 120.

Finally we prove a necessary condition for the homotopy equivalence \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\simeq {\mathcal {G}}_l({\mathbb {C}}{\mathbb {P}}^2)\) similar to Theorem 1.3.

Theorem 1.6

Let G be SU(n). If \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\) is homotopy equivalent to \({\mathcal {G}}_l({\mathbb {C}}{\mathbb {P}}^2)\), then

  • \((\frac{1}{2}n(n^2-1),k)=(\frac{1}{2}n(n^2-1),l)\) for n odd,

  • \((n(n^2-1),k)=(n(n^2-1),l)\) for n even.

The author would like to thank his supervisor, Professor Stephen Theriault, for his guidance in writing this paper, and thank the Referee for his careful reading and useful comments.

2 Some facts about boundary map \(\partial '_1\)

Take M to be \(S^4\) and \({\mathbb {C}}{\mathbb {P}}^2\) respectively in fibration (1) to obtain fibration sequences

$$\begin{aligned}&{\mathcal {G}}_k(S^4)\longrightarrow G\overset{\partial _k}{\longrightarrow }\Omega ^3_0G\longrightarrow B{\mathcal {G}}_k(S^4)\overset{ev}{\longrightarrow }BG \end{aligned}$$
(2)
$$\begin{aligned}&{\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\longrightarrow G\overset{\partial '_k}{\longrightarrow }{\text {Map}}^*_0({\mathbb {C}}{\mathbb {P}}^2,BG)\longrightarrow B{\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\overset{ev}{\longrightarrow }BG. \end{aligned}$$
(3)

There is also a cofibration sequence

$$\begin{aligned} S^3\overset{\eta }{\longrightarrow }S^2\longrightarrow {\mathbb {C}}{\mathbb {P}}^2\overset{q}{\longrightarrow }S^4, \end{aligned}$$
(4)

where \(\eta \) is Hopf map and q is the quotient map. Due to the naturality of \(q^*\), we combine fibrations (2) and (3) to obtain a commutative diagram of fibration sequences

(5)

It is known, [13], that \(\partial _k\) is triple adjoint to Samelson product

$$\begin{aligned} \langle {k\imath ,{\mathbb {1}}}\rangle :S^3\wedge G\overset{k\imath \wedge {\mathbb {1}}}{\longrightarrow }G\wedge G\overset{\langle {{\mathbb {1}},{\mathbb {1}}}\rangle }{\longrightarrow }G, \end{aligned}$$

where \(\imath :S^3\rightarrow SU(n)\) is the inclusion of the bottom cell and \(\langle {{\mathbb {1}},{\mathbb {1}}}\rangle \) is the Samelson product of the identity on G with itself. The order of \(\partial _k\) is its multiplicative order in the group \([G, \Omega ^3_0G]\).

Unlike \(\Omega ^3_0G\), \({\text {Map}}^*_0({\mathbb {C}}{\mathbb {P}}^2, BG)\) is not an H-space, so \(\partial '_k\) has no order. In [18], Theriault defined the “order” of \(\partial '_k\) to be the smallest number \(m'\) such that the composition

$$\begin{aligned} G\overset{\partial _k}{\longrightarrow }\Omega ^3_0G\overset{m'}{\longrightarrow }\Omega ^3_0G\overset{q^*}{\longrightarrow }{\text {Map}}^*_0({\mathbb {C}}{\mathbb {P}}^2, BG) \end{aligned}$$

is null homotopic. In the following, we interpret the “order” of \(\partial '_k\) as its multiplicative order in a group contained in \([{\mathbb {C}}{\mathbb {P}}^2\wedge G, BG]\).

Apply \([-\wedge G, BG]\) to cofibration (4) to obtain an exact sequence of sets

$$\begin{aligned} {[}\Sigma ^3G, BG]\overset{(\Sigma \eta )^*}{\longrightarrow }[\Sigma ^4G, BG]\overset{q^*}{\longrightarrow }[{\mathbb {C}}{\mathbb {P}}^2\wedge G, BG]. \end{aligned}$$

All terms except \([{\mathbb {C}}{\mathbb {P}}^2\wedge G, BG]\) are groups and \((\Sigma \eta )^*\) is a group homomorphism since \(\Sigma \eta \) is a suspension. We want to refine this exact sequence so that the last term is replaced by a group. Observe that \({\mathbb {C}}{\mathbb {P}}^2\) is the cofiber of \(\eta \) and so there is a coaction \(\psi :{\mathbb {C}}{\mathbb {P}}^2\rightarrow {\mathbb {C}}{\mathbb {P}}^2\vee S^4\). We show that the coaction gives a group structure on \(Im(q^*)\).

Lemma 2.1

Let Y be a space and let \(A\overset{f}{\rightarrow }B\overset{g}{\rightarrow }C\overset{h}{\rightarrow }\Sigma A\) be a cofibration sequence. If \(\Sigma A\) is homotopy cocommutative, then \(Im(h^*)\) is an abelian group and

$$\begin{aligned} {[}\Sigma B, Y]\overset{(\Sigma f)^*}{\longrightarrow }[\Sigma A, Y]\overset{h^*}{\longrightarrow }Im(h^*)\longrightarrow 0 \end{aligned}$$

is an exact sequence of groups and group homomorphisms.

Proof

Apply \([-,Y]\) to the cofibration to get an exact sequence of sets

$$\begin{aligned} {[}\Sigma B, Y]\overset{(\Sigma f)^*}{\longrightarrow }[\Sigma A, Y]\overset{h^*}{\longrightarrow }[C, Y]. \end{aligned}$$
(6)

Note that \([\Sigma B, Y]\) and \([\Sigma A, Y]\) are groups, and \((\Sigma f)^*\) is a group homomorphism. We will replace [CY] by \(Im(h^*)\) and define a group structure on it such that \(h^*:[\Sigma A, Y]\rightarrow Im(h^*)\) is a group homomorphism.

For any \(\alpha \) and \(\beta \) in \([\Sigma A, Y]\), we define a binary operator \(\boxtimes \) on \(Im(h^*)\) by

$$\begin{aligned} h^*\alpha \boxtimes h^*\beta =h^*(\alpha +\beta ). \end{aligned}$$

To check this is well-defined we need to show \(h^*(\alpha +\beta )\simeq h^*(\alpha '+\beta )\simeq h^*(\alpha +\beta ')\) for any \(\alpha ,\alpha ',\beta ,\beta '\) satisfying \(h^*\alpha \simeq h^*\alpha '\) and \(h^*\beta \simeq h^*\beta '\).

First we show \(h^*(\alpha +\beta )\simeq h^*(\alpha '+\beta )\). By definition, we have

$$\begin{aligned} h^*(\alpha +\beta )=(\alpha +\beta )\circ h=\triangledown \circ (\alpha \vee \beta )\circ \sigma \circ h, \end{aligned}$$

where \(\sigma :\Sigma A\rightarrow \Sigma A\vee \Sigma A\) is the comultiplication and \(\triangledown :Y\vee Y\rightarrow Y\) is the folding map. Since C is a cofiber, there is a coaction \(\psi :C\rightarrow C\vee \Sigma A\) such that \(\sigma \circ h\simeq (h\vee {\mathbb {1}})\circ \psi \).

Then we obtain a string of equivalences

$$\begin{aligned} h^*(\alpha +\beta )= & {} \triangledown \circ (\alpha \vee \beta )\circ \sigma \circ h\\\simeq & {} \triangledown \circ (\alpha \vee \beta )\circ (h\vee {\mathbb {1}})\circ \psi \\\simeq & {} \triangledown \circ (\alpha '\vee \beta )\circ (h\vee {\mathbb {1}})\circ \psi \\\simeq & {} \triangledown \circ (\alpha '\vee \beta )\circ \sigma \circ h\\= & {} h^*(\alpha '+\beta ) \end{aligned}$$

The third line is due to the assumption \(h^*\alpha \simeq h^*\alpha '\). Therefore we have \(h^*(\alpha +\beta )\simeq h^*(\alpha '+\beta )\). Since \(\Sigma A\) is cocommutative, \([\Sigma A, Y]\) is abelian and \(h^*(\alpha +\beta )\simeq h^*(\beta +\alpha )\). Then we have

$$\begin{aligned} h^*(\alpha +\beta )\simeq h^*(\beta +\alpha )\simeq h^*(\beta '+\alpha )\simeq h^*(\alpha +\beta '). \end{aligned}$$

This implies \(\boxtimes \) is well-defined.

Due to the associativity of \(+\) in \([\Sigma A, Y]\), \(\boxtimes \) is associative since

$$\begin{aligned} (h^*\alpha \boxtimes h^*\beta )\boxtimes h^*\gamma= & {} h^*(\alpha +\beta )\boxtimes h^*\gamma \\= & {} h^*((\alpha +\beta )+\gamma )\\= & {} h^*(\alpha +(\beta +\gamma ))\\= & {} h^*\alpha \boxtimes h^*(\beta +\gamma )\\= & {} h^*\alpha \boxtimes (h^*\beta \boxtimes h^*\gamma ). \end{aligned}$$

Clearly the trivial map \(*:C\rightarrow Y\) is the identity of \(\boxtimes \) and \(h^*(-\alpha )\) is the inverse of \(h^*\alpha \). Therefore \(\boxtimes \) is indeed a group multiplication.

By definition of \(\boxtimes \), \(h^*:[\Sigma A, Y]\rightarrow Im(h^*)\) is a group homomorphism, and hence an epimorphism. Since \([\Sigma A, Y]\) is abelian, so is \(Im(h^*)\). We replace [CY] by \(Im(h^*)\) in (6) to obtain a sequence of groups and group homomorphisms

$$\begin{aligned} {[}\Sigma B, Y]\overset{(\Sigma f)^*}{\longrightarrow }[\Sigma A, Y]\overset{h^*}{\longrightarrow }Im(h^*)\longrightarrow 0. \end{aligned}$$

The exactness of (6) implies \(ker(h^*)=Im(\Sigma f)^*\), so the sequence is exact. \(\square \)

Applying Lemma 2.1 to cofibration \(\Sigma ^3G\rightarrow \Sigma ^2G\rightarrow {\mathbb {C}}{\mathbb {P}}^2\wedge G\) and the space \(Y=BG\), we obtain an exact sequence of abelian groups

$$\begin{aligned}{}[\Sigma ^3G, BG]\overset{(\Sigma \eta )^*}{\longrightarrow }[\Sigma ^4G, BG]\overset{q^*}{\longrightarrow }Im(q^*)\longrightarrow 0. \end{aligned}$$
(7)

In the middle square of (5) \(\partial '_k\simeq q^*\partial _k\), so \(\partial '_k\) is in \(Im(q^*)\). For any number m, \(q^*(m\partial _k)=mq^*\partial _k\), so the “order” of \(\partial '_k\) defined in [18] coincides with the multiplicative order of \(\partial '_k\) in \(Im(q^*)\). The exact sequence (7) allows us to compare the orders of \(\partial _1\) and \(\partial '_1\).

Lemma 2.2

Let m be the order of \(\partial _1\) and let \(m'\) be the order of \(\partial '_1\). Then m is \(m'\) or \(2m'\).

Proof

By exactness of (7), there is some \(f\in [\Sigma ^3G, BG]\) such that \((\Sigma \eta )^*f\simeq m'\partial _1\). Since \(\Sigma \eta \) has order 2, \(2m'\partial _1\) is null homotopic. It follows that \(2m'\) is a multiple of m. Since m is greater than or equal to \(m'\), m is either \(m'\) or \(2m'\). \(\square \)

When \(G=SU(2)\), the order m of \(\partial _1\) is 12 and the order \(m'\) of \(\partial '_1\) is 6 [12]. When \(G=SU(3)\), \(m=24\) and \(m'=12\) [18]. When \(G=Sp(2)\), \(m=40\) and \(m'=20\) [15]. It is natural to ask whether \(m=2m'\) for all G.

In the \(S^4\) case, Theorem 1.1 gives a sufficient condition for \({\mathcal {G}}_k(S^4)\simeq {\mathcal {G}}_l(S^4)\) when localized rationally or at any prime. In the \({\mathbb {C}}{\mathbb {P}}^2\) case, Theriault showed a similar counting statement, in which the sufficient condition depends on the order of \(\partial _1\) instead of \(\partial '_1\).

Theorem 2.3

(Theriault, [18]) Let m be the order of \(\partial _1\). If \((m,k)=(m,l)\), then \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\) is homotopy equivalent to \({\mathcal {G}}_l({\mathbb {C}}{\mathbb {P}}^2)\) when localized rationally or at any prime.

Lemma 2.2 can be used to improve the sufficient condition of Theorem 2.3.

Theorem 2.4

Let \(m'\) be the order of \(\partial '_1\). If \((m',k)=(m',l)\), then \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\) is homotopy equivalent to \({\mathcal {G}}_l({\mathbb {C}}{\mathbb {P}}^2)\) when localized rationally or at any prime.

Proof

By Lemma 2.2, m is either \(m'\) or \(2m'\). If \(m=m'\), then the statement is same as Theorem 2.3. Assume \(m=2m'\). Localize at an odd prime p. Let \(p^r\) be the p-component of m, that is \(m=p^r\cdot q\) where q is coprime to p. Observe that \(m\circ \partial _1\simeq (p^r\cdot q)\circ \partial _1\simeq p^r\circ \partial _1\) since the power map \(q:\Omega ^3_0G\rightarrow \Omega ^3_0G\) is a homotopy equivalence. Therefore \(p^r\) is the order of \(\partial _1\) after localization. The hypothesis \((m', k)=(m',l)\) implies \((p^r, k)=(p^r, l)\), so a homotopy equivalence \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\simeq {\mathcal {G}}_l({\mathbb {C}}{\mathbb {P}}^2)\) follows by Theorem 2.3. A similar argument works for rational localization. Now it remains to consider the case where \(m=2m'\) when localized at 2.

Assume \(m=2^n\) and \(m'=2^{n-1}\). For any k, \((2^{n-1},k)=2^i\) where i an integer such that \(0\le i\le n-1\). If \(i\le n-2\), then \(k=2^it\) for some odd number t and \((2^{n-1},k)=2^i\). The sufficient condition \((2^{n-1},k)=(2^{n-1},l)\) is equivalent to \((2^n,k)=(2^n,l)\). Again the homotopy equivalence \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\simeq {\mathcal {G}}_l({\mathbb {C}}{\mathbb {P}}^2)\) follows by Theorem 2.3. If \(i=n-1\), then \((2^n,k)\) is either \(2^n\) or \(2^{n-1}\). We claim that \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\) has the same homotopy type for both \((2^n,k)=2^n\) or \((2^n,k)=2^{n-1}\).

Consider fibration (3)

$$\begin{aligned} {\text {Map}}^*_0({\mathbb {C}}{\mathbb {P}}^2, G)\longrightarrow {\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\longrightarrow G\overset{\partial '_k}{\longrightarrow }{\text {Map}}^*_0({\mathbb {C}}{\mathbb {P}}^2,BG). \end{aligned}$$

If \((2^n,k)=2^n\), then \(k=2^nt\) for some number t. By linearity of Samelson products, \(\partial _k\simeq k\partial _1\). Since \(\partial '_k\simeq q^*k\partial _1\simeq q^*2^nt\partial _1\) and \(\partial _1\) has order \(2^n\), \(\partial '_k\) is null homotopic and we have

$$\begin{aligned} {\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\simeq G\times {\text {Map}}^*_0({\mathbb {C}}{\mathbb {P}}^2,G). \end{aligned}$$

If \((2^n,k)=2^{n-1}\), then \(k=2^{n-1}t\) for some odd number t. Writing \(t=2s+1\) gives \(k=2^ns+2^{n-1}\). Since \(\partial '_k\simeq q^*k\partial _1\simeq q^*(2^ns+2^{n-1})\partial _1\simeq q^*2^{n-1}\partial _1\) and \(\partial '_1\) has order \(2^{n-1}\), \(\partial '_k\) is null homotopic and we have

$$\begin{aligned} {\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\simeq G\times {\text {Map}}^*_0({\mathbb {C}}{\mathbb {P}}^2,G). \end{aligned}$$

The same is true for \({\mathcal {G}}_l({\mathbb {C}}{\mathbb {P}}^2)\) and hence \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\simeq {\mathcal {G}}_l({\mathbb {C}}{\mathbb {P}}^2)\). \(\square \)

3 Plan for the proofs of Theorems 1.5 and 1.6

From this section onward, we will focus on SU(n)-gauge groups over \({\mathbb {C}}{\mathbb {P}}^2\). There is a fibration

$$\begin{aligned} SU(n)\longrightarrow SU(\infty )\overset{p}{\longrightarrow }W_n, \end{aligned}$$
(8)

where \(p:SU(\infty )\rightarrow W_n\) is the projection and \(W_n\) is the symmetric space \(SU(\infty )/SU(n)\). Then we have

$$\begin{aligned} {\tilde{H}}^{*}(SU(\infty ))= & {} \Lambda (x_3,\ldots ,x_{2n-1},\ldots ),\\ {\tilde{H}}^{*}(SU(n))= & {} \Lambda (x_3,\ldots ,x_{2n-1}),\\ {\tilde{H}}^{*}(BSU(n))= & {} {\mathbb {Z}}[c_2,\ldots ,c_n],\\ {\tilde{H}}^{*}(W_n)= & {} \Lambda (\bar{x}_{2n+1},\bar{x}_{2n+3},\ldots ), \end{aligned}$$

where \(x_{2n+1}\) has degree \(2n+1\), \(c_i\) is the \(i{\text {th}}\) universal Chern class and \(x_{2i+1}=\sigma (c_{i+1})\) is the image of \(c_{i+1}\) under the cohomology suspension \(\sigma \), and \(p^*(\bar{x}_{2i+1})=x_{2i+1}\). Furthermore, \(H^{2n}(\Omega W_n)\cong {\mathbb {Z}}\) and \(H^{2n+2}(\Omega W_n)\cong {\mathbb {Z}}\) are generated by \(a_{2n}\) and \(a_{2n+2}\), where \(a_{2i}\) is the transgression of \(x_{2i+1}\).

The \((2n+4)\)-skeleton of \(W_n\) is \(\Sigma ^{2n-1}{\mathbb {C}}{\mathbb {P}}^2\) for n odd, and is \(S^{2n+3}\vee S^{2n+1}\) for n even, so its homotopy groups are as follows:

(9)

The canonical map \(\epsilon :\Sigma {\mathbb {C}}{\mathbb {P}}^{n-1}\rightarrow SU(n)\) induces the inclusion \(\epsilon _*:H_*(\Sigma {\mathbb {C}}{\mathbb {P}}^{n-1})\rightarrow H_*(SU(n))\) of the generating set. Let C be the quotient \({\mathbb {C}}{\mathbb {P}}^{n-1}/{\mathbb {C}}{\mathbb {P}}^{n-3}\) and let \(\bar{q}:\Sigma {\mathbb {C}}{\mathbb {P}}^{n-1}\rightarrow \Sigma C\) be the quotient map. Then there is a diagram

where \((\partial '_k)_*\) sends f to \(\partial '_k\circ f\) and the rows are induced by fibration (3). In particular, in the second row the map \(\epsilon :\Sigma {\mathbb {C}}{\mathbb {P}}^{n-1}\rightarrow SU(n)\) is sent to \((\partial '_k)_*(\epsilon )=\partial '_k\circ \epsilon \). In Sect. 4, we use unstable K-theory to calculate the order of \(\partial '_1\circ \epsilon \), giving a lower bound on the order of \(\partial '_1\). Furthermore, in [5] Hamanaka and Kono considered an exact sequence similar to the first row to give a necessary condition for \({\mathcal {G}}_k(S^4)\simeq {\mathcal {G}}_l(S^4)\). In Sect. 5 we follow the same approach and use the first row to give a necessary condition for \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\simeq {\mathcal {G}}_l({\mathbb {C}}{\mathbb {P}}^2)\).

We remark that it is difficult to use only one of the two rows to prove both Theorems 1.5 and 1.6. On the one hand, \(\partial '_1\circ \epsilon \) factors through a map \(\bar{\partial }:\Sigma C\rightarrow {\text {Map}}^*({\mathbb {C}}{\mathbb {P}}^2, BSU(n))\). There is no obvious method to show that \(\bar{\partial }\) and \(\partial '_1\circ \epsilon \) have the same orders except direct calculation. Therefore we cannot compare the orders of \(\bar{\partial }\) and \(\partial '_1\) to prove Theorem 1.5 without calculating the order of \(\partial '_1\circ \epsilon \). On the other hand, applying the method used in Sect. 5 to the second row gives a much weaker conclusion than Theorem 1.6. This is because \([\Sigma C,B{\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)]\) is a much smaller group than \([\Sigma {\mathbb {C}}{\mathbb {P}}^{n-1},B{\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)]\) and much information is lost by the map \(\bar{q}^*\).

4 A lower bound on the order of \(\partial '_1\)

The restriction of \(\partial _1\) to \(\Sigma {\mathbb {C}}{\mathbb {P}}^{n-1}\) is \(\partial _1\circ \epsilon \), which is the triple adjoint of the composition

$$\begin{aligned} \langle {\imath ,\epsilon }\rangle :S^3\wedge \Sigma {\mathbb {C}}{\mathbb {P}}^{n-1}\overset{\imath \wedge \epsilon }{\longrightarrow }SU(n)\wedge SU(n)\overset{\langle {{\mathbb {1}},{\mathbb {1}}}\rangle }{\longrightarrow }SU(n). \end{aligned}$$

Since \(SU(n)\simeq \Omega BSU(n)\), we can further take its adjoint and get

$$\begin{aligned} \rho :\Sigma S^3\wedge \Sigma {\mathbb {C}}{\mathbb {P}}^{n-1}\overset{\Sigma \imath \wedge \epsilon }{\longrightarrow }\Sigma SU(n)\wedge SU(n)\overset{[ev,ev]}{\longrightarrow }BSU(n), \end{aligned}$$

where [evev] is the Whitehead product of the evaluation map

$$\begin{aligned} ev:\Sigma SU(n)\simeq \Sigma \Omega BSU(n)\rightarrow BSU(n) \end{aligned}$$

with itself. Similarly, the restriction \(\partial '_1\circ \epsilon \) is adjoint to the composition

$$\begin{aligned} \rho ':{\mathbb {C}}{\mathbb {P}}^2\wedge \Sigma {\mathbb {C}}{\mathbb {P}}^{n-1}\overset{q\wedge {\mathbb {1}}}{\longrightarrow }S^4\wedge \Sigma {\mathbb {C}}{\mathbb {P}}^{n-1}\overset{\Sigma \imath \wedge \epsilon }{\longrightarrow }\Sigma SU(n)\wedge SU(n)\overset{[ev,ev]}{\longrightarrow }BSU(n). \end{aligned}$$

Since we will frequently refer to the facts established in [4, 5], it is easier to follow their setting and consider its adjoint

$$\begin{aligned} \gamma =\tau (\rho '\circ T):{\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1}\rightarrow SU(n), \end{aligned}$$

where \(T:\Sigma {\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1}\rightarrow {\mathbb {C}}{\mathbb {P}}^2\wedge \Sigma {\mathbb {C}}{\mathbb {P}}^{n-1}\) is the swapping map and \(\tau :[\Sigma {\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1},BSU(n)]\rightarrow [{\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1},SU(n)]\) is the adjunction. By adjunction, the orders of \(\partial '_1\circ \epsilon ,\rho '\) and \(\gamma \) are the same. We will calculate the order of \(\gamma \) using unstable K-theory to prove Theorem 1.5.

Apply \([{\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1},-]\) to fibration (8) to obtain the exact sequence

$$\begin{aligned} {\tilde{K}}^0({\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1})\overset{p_*}{\longrightarrow }[{\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1},\Omega W_n]\longrightarrow [{\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1},SU(n)]\longrightarrow 0. \end{aligned}$$

Since \({\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1}\) is a CW-complex with even dimensional cells, the last item \([{\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1},SU(\infty )]\cong {\tilde{K}}^1({\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1})\) is zero. First we identify the term \([{\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1},\Omega W_n]\).

Lemma 4.1

We have the following:

  • \([\Sigma ^{2n-4}{\mathbb {C}}{\mathbb {P}}^2,\Omega W_n]\cong {\mathbb {Z}}\);

  • \([\Sigma ^{2n-3}{\mathbb {C}}{\mathbb {P}}^2,\Omega W_n]=0\) for n odd;

  • \([\Sigma ^{2n-2}{\mathbb {C}}{\mathbb {P}}^2,\Omega W_n]\cong {\mathbb {Z}}\oplus {\mathbb {Z}}\).

Proof

First, apply \([\Sigma ^{2n-4}-, \Omega W_n]\) to cofibration (4) to obtain the exact sequence

$$\begin{aligned} \pi _{2n}(W_n)\longrightarrow \pi _{2n+1}(W_n)\longrightarrow [\Sigma ^{2n-4}{\mathbb {C}}{\mathbb {P}}^2,\Omega W_n]\longrightarrow \pi _{2n-1}(W_n). \end{aligned}$$

We refer to Table (9) freely for the homotopy groups of \(W_n\). Since \(\pi _{2n-1}(W_n)\) and \(\pi _{2n}(W_n)\) are zero, \([\Sigma ^{2n-4}{\mathbb {C}}{\mathbb {P}}^{n-1},\Omega W_n]\) is isomorphic to \(\pi _{2n+1}(W_n)\cong {\mathbb {Z}}\).

Second, apply \([\Sigma ^{2n-3}-, \Omega W_n]\) to (4) to obtain

$$\begin{aligned} \pi _{2n+2}(W_n)\longrightarrow [\Sigma ^{2n-3}{\mathbb {C}}{\mathbb {P}}^2,\Omega W_n]\longrightarrow \pi _{2n}(W_n). \end{aligned}$$

Since \(\pi _{2n}(W_n)\) and \(\pi _{2n+2}(W_n)\) are zero for n odd, so is \([\Sigma ^{2n-3}{\mathbb {C}}{\mathbb {P}}^2,\Omega W_n]\).

Third, apply \([\Sigma ^{2n-2}-, \Omega W_n]\) to (4) to obtain

$$\begin{aligned}&\pi _{2n+2}(W_n)\overset{\eta _1}{\longrightarrow }\pi _{2n+3}(W_n)\longrightarrow [\Sigma ^{2n-2}{\mathbb {C}}{\mathbb {P}}^2,\Omega W_n]\\&\quad \overset{j}{\longrightarrow }\pi _{2n+1}(W_n)\overset{\eta _2}{\longrightarrow }\pi _{2n+2}(W_n), \end{aligned}$$

where \(\eta _1\) and \(\eta _2\) are induced by Hopf maps \(\Sigma ^{2n}\eta :S^{2n+3}\rightarrow S^{2n+2}\) and \(\Sigma ^{2n-1}\eta :S^{2n+2}\rightarrow S^{2n+1}\), and j is induced by the inclusion \(S^{2n+1}\hookrightarrow \Sigma ^{2n-2}{\mathbb {C}}{\mathbb {P}}^2\) of the bottom cell. When n is odd, \(\pi _{2n+2}(W_n)\) is zero and \(\pi _{2n+1}(W_n)\) and \(\pi _{2n+3}(W_n)\) are \({\mathbb {Z}}\), so \([\Sigma ^{2n-2}{\mathbb {C}}{\mathbb {P}}^{n-1},\Omega W_n]\) is \({\mathbb {Z}}\oplus {\mathbb {Z}}\). When n is even, the \((2n+4)\)-skeleton of \(W_n\) is \(S^{2n+1}\vee S^{2n+3}\). The inclusions

$$\begin{aligned} \begin{array}{c c c} i_1:S^{2n+1}\rightarrow S^{2n+1}\vee S^{2n+3}&\quad \text {and}&\quad i_2:S^{2n+3}\rightarrow S^{2n+1}\vee S^{2n+3} \end{array} \end{aligned}$$

generate \(\pi _{2n+1}(W_n)\) and the \({\mathbb {Z}}\)-summand of \(\pi _{2n+3}(W_n)\), and the compositions

$$\begin{aligned} \begin{array}{c c c} j_1:S^{2n+2}\overset{\Sigma ^{2n-1}\eta }{\longrightarrow }S^{2n+1}\overset{i_1}{\longrightarrow }W_n&\text {and}&j_2:S^{2n+3}\overset{\Sigma ^{2n}\eta }{\longrightarrow }S^{2n+2}\overset{\Sigma ^{2n-1}\eta }{\longrightarrow }S^{2n+1}\overset{i_1}{\longrightarrow }W_n \end{array} \end{aligned}$$

generate \(\pi _{2n+2}(W_n)\) and the \({\mathbb {Z}}/2{\mathbb {Z}}\)-summand of \(\pi _{2n+3}(W_n)\) respectively. Since \(\eta _1\) sends \(j_1\) to \(j_2\), the cokernel of \(\eta _1\) is \({\mathbb {Z}}\). Similarly, \(\eta _2\) sends \(i_1\) to \(j_1\), so \(\eta _2:{\mathbb {Z}}\rightarrow {\mathbb {Z}}/2{\mathbb {Z}}\) is surjective. This implies the preimage of j is a \({\mathbb {Z}}\)-summand. Therefore \([\Sigma ^{2n-2}{\mathbb {C}}{\mathbb {P}}^2,\Omega W_n]\cong {\mathbb {Z}}\oplus {\mathbb {Z}}\). \(\square \)

Let C be the quotient \({\mathbb {C}}{\mathbb {P}}^{n-1}/{\mathbb {C}}{\mathbb {P}}^{n-3}\). Since \(\Omega W_n\) is \((2n-1)\)-connected, \([{\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1},\Omega W_n]\) is isomorphic to \([{\mathbb {C}}{\mathbb {P}}^2\wedge C,\Omega W_n]\) which is easier to determine.

Lemma 4.2

The group \([{\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1},\Omega W_n]\cong [{\mathbb {C}}{\mathbb {P}}^2\wedge C,\Omega W_n]\) is isomorphic to \({\mathbb {Z}}^{\oplus 3}\).

Proof

When n is even, C is \(S^{2n-2}\vee S^{2n-4}\). By Lemma 4.1, \([{\mathbb {C}}{\mathbb {P}}^2\wedge C, \Omega W_n]\) is \([\Sigma ^{2n-2}{\mathbb {C}}{\mathbb {P}}^2, \Omega W_n]\oplus [\Sigma ^{2n-4}{\mathbb {C}}{\mathbb {P}}^2, \Omega W_n]\cong {\mathbb {Z}}^{\oplus 3}\).

When n is odd, C is \(\Sigma ^{2n-6}{\mathbb {C}}{\mathbb {P}}^2\). Apply \([\Sigma ^{2n-6}{\mathbb {C}}{\mathbb {P}}^2\wedge -, \Omega W_n]\) to cofibration (4) to obtain the exact sequence

$$\begin{aligned}&[\Sigma ^{2n-3}{\mathbb {C}}{\mathbb {P}}^2,\Omega W_n]\longrightarrow [\Sigma ^{2n-2}{\mathbb {C}}{\mathbb {P}}^2,\Omega W_n]\longrightarrow [\Sigma ^{2n-6}{\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^2,\Omega W_n]\\&\quad \longrightarrow [\Sigma ^{2n-4}{\mathbb {C}}{\mathbb {P}}^2,\Omega W_n]\longrightarrow [\Sigma ^{2n-3}{\mathbb {C}}{\mathbb {P}}^2,\Omega W_n] \end{aligned}$$

By Lemma 4.1, the first and the last terms \([\Sigma ^{2n-3}{\mathbb {C}}{\mathbb {P}}^2, \Omega W_n]\) are zero, while the second term \([\Sigma ^{2n-2}{\mathbb {C}}{\mathbb {P}}^2, \Omega W_n]\) is \({\mathbb {Z}}\oplus {\mathbb {Z}}\) and the fourth \([\Sigma ^{2n-4}{\mathbb {C}}{\mathbb {P}}^2, \Omega W_n]\) is \({\mathbb {Z}}\). Therefore \([{\mathbb {C}}{\mathbb {P}}^2\wedge C,\Omega W_n]\) is \({\mathbb {Z}}^{\oplus 3}\). \(\square \)

Define \(a:[{\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1}, \Omega W_n]\rightarrow H^{2n}({\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1})\oplus H^{2n+2}({\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1})\) to be a map sending \(f\in [{\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1}, \Omega W_n]\) to \(a(f)=f^*(a_{2n})\oplus f^*(a_{2n+2})\). The cohomology class \(\bar{x}_{2n+1}\) represents a map \(\bar{x}_{2n+1}:W_n\rightarrow K({\mathbb {Z}},2n+1)\) and \(a_{2n}=\sigma (\bar{x}_{2n+1})\) represents its loop \(\Omega \bar{x}_{2n+1}:\Omega W_n\rightarrow \Omega K({\mathbb {Z}},2n+1)\). Similarly \(a_{2n+2}=\sigma (\bar{x}_{2n+3})\) represents a loop map. This implies a is a group homomorphism. Furthermore, \(a_{2n}\) and \(a_{2n+2}\) induce isomorphisms between \(H^i(\Omega W_n)\) and \(H^i(K(2n,{\mathbb {Z}})\times K(2n+2,{\mathbb {Z}}))\) for \(i=2n\) and \(2n+2\). Since \([{\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1},\Omega W_n]\) is a free \({\mathbb {Z}}\)-module by Lemma 4.2a is a monomorphism. Consider the diagram

(10)

In the left square, \(\Phi \) is defined to be \(a\circ p^*\). In the right square, \(\psi \) is the quotient map and b is defined as follows. Any \(f\in [{\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1},SU(n)]\) has a preimage \(\tilde{f}\) and b(f) is defined to be \(\psi (a(\tilde{f}))\). An easy diagram chase shows that b is well-defined and injective. Since b is injective, the order of \(\gamma \in [{\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1}, SU(n)]\) equals the order of \(b(\gamma )\in Coker(\Phi )\). In [4], Hamanaka and Kono gave an explicit formula for \(\Phi \).

Theorem 4.3

(Hamanaka and Kono [4]) Let Y be a CW-complex. For any \(f\in {\tilde{K}}^0(Y)\) we have

$$\begin{aligned} \Phi (f)=n!ch_{2n}(f)\oplus (n+1)!ch_{2n+2}(f), \end{aligned}$$

where \(ch_{2i}(f)\) is the \(2i{\text {th}}\) part of ch(f).

Let u and v be the generators of \(H^2({\mathbb {C}}{\mathbb {P}}^2)\) and \(H^2({\mathbb {C}}{\mathbb {P}}^{n-1})\). For \(1\le i\le n-1\), denote \(L_i\) and \(L'_i\) as the generators of \({\tilde{K}}^0({\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1})\) with Chern characters \(ch(L_i)=u^2(e^v-1)^i\) and \(ch(L'_i)=(u+\frac{1}{2}u^2)\cdot (e^v-1)^i\). By Theorem 4.3 we have

$$\begin{aligned} \Phi (L_i)= & {} n(n-1)A_iu^2v^{n-2}+n(n+1)B_iu^2v^{n-1},\\ \Phi (L'_i)= & {} \frac{n(n-1)}{2}A_iu^2v^{n-2}+nB_iuv^{n-1}+\frac{n(n+1)}{2}B_iu^2v^{n-1}, \end{aligned}$$

where

$$\begin{aligned} \begin{array}{c c c} A_i=\sum \nolimits ^i_{j=1}(-1)^{i+j}\left( {\begin{array}{c}i\\ j\end{array}}\right) j^{n-2} \quad &{}\text {and} &{}\quad B_i=\sum \nolimits ^i_{j=1}(-1)^{i+j}\left( {\begin{array}{c}i\\ j\end{array}}\right) j^{n-1}. \end{array} \end{aligned}$$

Write an element \(xu^2v^{n-2}+yuv^{n-1}+zu^2v^{n-1}\in H^{2n}({\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1})\oplus H^{2n+2}({\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1})\) as (xyz). Then the coordinates of \(\Phi (L_i)\) and \(\Phi (L'_i)\) are \((n(n-1)A_i, 0, n(n+1)B_i)\) and \((\frac{n(n-1)}{2}A_i, nB_i, \frac{n(n+1)}{2}B_i)\) respectively.

Lemma 4.4

For \(n\ge 3\), \(Im(\Phi )\) is spanned by \((\frac{n(n-1)}{2},n,\frac{n(n+1)}{2})\), \((n(n-1),0,0)\) and (0, 2n, 0).

Proof

By definition, \(Im(\Phi )=span\{\Phi (L_i),\Phi (L'_i)\}^{n-1}_{i=1}\). For \(i=1\), \(A_1=B_1=1\). Then

$$\begin{aligned} \Phi (L_1)= & {} (n(n-1), 0, n(n+1))\\= & {} 2\left( \frac{1}{2}n(n-1), n, \frac{1}{2}n(n+1)\right) -(0,2n,0)\\= & {} 2\Phi (L'_1)-(0,2n,0) \end{aligned}$$

Equivalently \((0,2n,0)=2\Phi (L'_1)-\Phi (L_1)\), so \(span\{\Phi (L_1),\Phi (L'_1)\}=span\{\Phi (L'_1),(0,2n,0)\}\). For other i’s,

$$\begin{aligned} \Phi (L_i)= & {} (n(n-1)A_i, 0, n(n+1)B_i)\\= & {} 2\left( \frac{1}{2}n(n-1)A_i, nB_i, \frac{1}{2}n(n+1)B_i\right) -(0,2nB_i,0)\\= & {} 2\Phi (L'_i)-B_i(0,2n,0) \end{aligned}$$

is a linear combination of \(\Phi (L'_i)\) and (0, 2n, 0), so

$$\begin{aligned} Im(\Phi )=span\{\Phi (L'_1),\ldots ,\Phi (L'_{n-1}),(0,2n,0)\}. \end{aligned}$$

We claim that \(span\{\Phi (L'_i)\}^{n-1}_{i=1}=span\{\Phi (L'_1), (n(n-1),0,0)\}\). Observe that

$$\begin{aligned} \Phi (L'_i)= & {} \left( \frac{n(n-1)}{2}A_i,nB_i,\frac{n(n+1)}{2}B_i\right) \\= & {} \left( \frac{n(n-1)}{2}B_i,nB_i,\frac{n(n+1)}{2}B_i\right) +\left( \frac{n(n-1)}{2}(A_i-B_i),0,0\right) \\= & {} B_i\Phi (L'_1)+\frac{A_i-B_i}{2}\cdot (n(n-1),0,0). \end{aligned}$$

The difference

$$\begin{aligned} A_i-B_i= & {} \sum ^i_{j=1}(-1)^{i+j}\left( {\begin{array}{c}i\\ j\end{array}}\right) j^{n-2}-\sum ^i_{j=1}(-1)^{i+j}\left( {\begin{array}{c}i\\ j\end{array}}\right) j^{n-1}\\= & {} \sum ^i_{j=1}(-1)^{i+j+1}\left( {\begin{array}{c}i\\ j\end{array}}\right) (j^{n-1}-j^{n-2})\\= & {} \sum ^i_{j=1}(-1)^{i+j+1}\left( {\begin{array}{c}i\\ j\end{array}}\right) (j-1)j^{n-2} \end{aligned}$$

is even since each term \((j-1)j^{n-2}\) is even and \(n\ge 3\). Therefore \(\frac{A_i-B_i}{2}\) is an integer and \(\Phi (L'_i)\) is a linear combination of \(\Phi (L'_1)\) and \((n(n-1),0,0)\).

Furthermore,

$$\begin{aligned} \Phi (L'_2)= & {} B_2\Phi (L'_1)+(A_2-B_2)\left( \frac{n(n-1)}{2},0,0\right) \\= & {} B_2\Phi (L'_1)-2^{n-3}(n(n-1),0,0) \end{aligned}$$

and

$$\begin{aligned} \Phi (L'_3)= & {} B_3\Phi (L'_1)+(A_3-B_3)\left( \frac{n(n-1)}{2},0,0\right) \\= & {} B_3\Phi (L'_1)-(3^{n-2}-3\cdot 2^{n-3})(n(n-1),0,0). \end{aligned}$$

For \(n=3\), \(B_2=2\) and \(\Phi (L'_2)=2\Phi (L'_1)-(n(n-1),0,0)\), so we have

$$\begin{aligned} span\{\Phi (L'_i)\}^{n-1}_{i=1}=span\{\Phi (L'_1),\Phi (L'_2)\}=span\{\Phi (L'_1),(n(n-1),0,0)\}. \end{aligned}$$

For \(n\ge 4\), since \(2^{n-3}\) and \(3^{n-2}-3\cdot 2^{n-3}\) are coprime to each other, there exist integers s and t such that \(2^{n-3}s+(3^{n-2}-3\cdot 2^{n-3})t=1\) and

$$\begin{aligned} (n(n-1),0,0)=(sB_2+tB_3)\Phi (L'_1)-s\Phi (L'_2)-t\Phi (L'_3). \end{aligned}$$

Therefore \((n(n-1),0,0)\) is a linear combination of \(\Phi (L'_1),\Phi (L'_2)\) and \(\Phi (L'_3)\). This implies \(span\{\Phi (L'_1),(n(n-1),0,0)\}=span\{\Phi (L'_i)\}^{n-1}_{i=1}\).

Combine all these together to obtain

$$\begin{aligned} Im(\Phi )= & {} span\{\Phi (L_i),\Phi (L'_i)\}^{n-1}_{i=1}\\= & {} span\{\Phi (L'_1),(n(n-1),0,0),(0,2n, 0)\}\\= & {} span\left\{ \left( \frac{n(n-1)}{2},n,\frac{n(n+1)}{2}\right) ,(n(n-1),0,0),(0,2n,0)\right\} . \end{aligned}$$

\(\square \)

Back to diagram (10). The map \(\gamma \) has a lift \(\tilde{\gamma }:{\mathbb {C}}{\mathbb {P}}^2\wedge {\mathbb {C}}{\mathbb {P}}^{n-1}\rightarrow \Omega W_n\). By exactness, the order of \(\gamma \) equals the minimum number m such that \(m\tilde{\gamma }\) is contained in \(Im(p_*)\). Since a and b are injective, the order of \(\gamma \) equals the minimum number \(m'\) such that \(m'a(\tilde{\gamma })\) is contained in \(Im(\Phi )\).

Lemma 4.5

Let \(\alpha :\Sigma X\rightarrow SU(n)\) be a map for some space X. If \(\alpha ':{\mathbb {C}}{\mathbb {P}}^2\wedge X\rightarrow SU(n)\) is the adjoint of the composition

$$\begin{aligned} {\mathbb {C}}{\mathbb {P}}^2\wedge \Sigma X\overset{q\wedge {\mathbb {1}}}{\longrightarrow }\Sigma S^3\wedge \Sigma X\overset{\Sigma \imath \wedge \alpha }{\longrightarrow }\Sigma SU(n)\wedge SU(n)\overset{[ev,ev]}{\longrightarrow }BSU(n), \end{aligned}$$

then there is a lift \(\tilde{\alpha }\) of \(\alpha '\) such that \(\tilde{\alpha }^*(a_{2i})=u^2\otimes \Sigma ^{-1}\alpha ^*(x_{2i-3})\), where \(\Sigma \) is the cohomology suspension isomorphism.

Proof

In [4, 5], Hamanaka and Kono constructed a lift \(\Gamma :\Sigma SU(n)\wedge SU(n)\rightarrow W_n\) of [evev] such that \(\Gamma ^*(\bar{x}_{2i+1})=\sum _{j+k=i-1}\Sigma x_{2j+1}\otimes x_{2k+1}\). Let \(\tilde{\Gamma }\) be the composition

$$\begin{aligned} \tilde{\Gamma }:{\mathbb {C}}{\mathbb {P}}^2\wedge \Sigma X\overset{q\wedge {\mathbb {1}}}{\longrightarrow }\Sigma S^3\wedge \Sigma X\overset{\Sigma \imath \wedge \alpha }{\longrightarrow }\Sigma SU(n)\wedge SU(n)\overset{\Gamma }{\longrightarrow }W_n. \end{aligned}$$

Then we have

$$\begin{aligned} \tilde{\Gamma }^*(\bar{x}_{2i+1})= & {} (q\wedge {\mathbb {1}})^*(\Sigma \imath \wedge \alpha )^*\Gamma ^*(\bar{x}_{2i+1})\\= & {} (q\wedge {\mathbb {1}})^*(\Sigma \imath \wedge \alpha )^*\left( \sum _{j+k=i-1}\Sigma x_{2j+1}\otimes x_{2k+1}\right) \\= & {} (q\wedge {\mathbb {1}})^*(\Sigma u_3\otimes \alpha ^*(x_{2i-3}))\\= & {} u^2\otimes \alpha ^*(x_{2i-3}), \end{aligned}$$

where \(u_3\) is the generator of \(H^3(S^3)\).

Let \(T:\Sigma {\mathbb {C}}{\mathbb {P}}^2\wedge X\rightarrow {\mathbb {C}}{\mathbb {P}}^2\wedge \Sigma X\) be the swapping map and let \(\tau :[\Sigma {\mathbb {C}}{\mathbb {P}}^2\wedge X,W_n]\rightarrow [{\mathbb {C}}{\mathbb {P}}^2\wedge X,\Omega W_n]\) be the adjunction. Take \(\tilde{\alpha }:{\mathbb {C}}{\mathbb {P}}^2\wedge X\rightarrow \Omega W_n\) to be the adjoint of \(\tilde{\Gamma }\), that is \(\tilde{\alpha }=\tau (\tilde{\Gamma }\circ T)\). Then \(\tilde{\alpha }\) is a lift of \(\alpha '\). Since

$$\begin{aligned} (\tilde{\Gamma }\circ T)^*(\bar{x}_{2i+1})=T^*\circ \tilde{\Gamma }^*(\bar{x}_{2i+1})=T^*(u^2\otimes \alpha ^*(x_{2i-3}))=\Sigma u^2\otimes \Sigma ^{-1}\alpha ^*(x_{2i-3}), \end{aligned}$$

we have \(\tilde{\alpha }^*(a_{2i})=u^2\otimes \Sigma ^{-1}\alpha ^*(x_{2i-3})\). \(\square \)

Lemma 4.6

In diagram (10), \(\gamma \) has a lift \(\tilde{\gamma }\) such that \(a(\tilde{\gamma })=u^2v^{n-2}\oplus u^2v^{n-1}\).

Proof

Recall that \(\gamma \) is the adjoint of the composition

$$\begin{aligned} \rho ':{\mathbb {C}}{\mathbb {P}}^2\wedge \Sigma {\mathbb {C}}{\mathbb {P}}^{n-1}\overset{q\wedge {\mathbb {1}}}{\longrightarrow }\Sigma S^3\wedge \Sigma {\mathbb {C}}{\mathbb {P}}^{n-1}\overset{\Sigma \imath \wedge \epsilon }{\longrightarrow }\Sigma SU(n)\wedge SU(n)\overset{[ev,ev]}{\longrightarrow }BSU(n). \end{aligned}$$

Now we use Lemma 4.5 and take \(\alpha \) to be \(\epsilon :\Sigma {\mathbb {C}}{\mathbb {P}}^{n-1}\rightarrow SU(n)\). Then \(\gamma \) has a lift \(\tilde{\gamma }\) such that \(\tilde{\gamma }^*(a_{2i})=u^2\otimes \Sigma ^{-1}\epsilon ^*(x_{2i-3})=u^2\otimes v^{i-2}\). This implies

$$\begin{aligned} a(\tilde{\gamma })=\tilde{\gamma }^*(a_{2n})\oplus \tilde{\gamma }^*(a_{2n+2})=u^2v^{n-2}\oplus u^2v^{n-1}. \end{aligned}$$

\(\square \)

Now we can calculate the order of \(\partial '_1\circ \epsilon \), which gives a lower bound on the order of \(\partial '_1\).

Theorem 4.7

When \(n\ge 3\), the order of \(\partial '_1\circ \epsilon \) is \(\frac{1}{2}n(n^2-1)\) for n odd and \(n(n^2-1)\) for n even.

Proof

Since \(\partial '_1\circ \epsilon \) is adjoint to \(\gamma \) , it suffices to calculate the order of \(\gamma \). By Lemma 4.4, \(Im(\Phi )\) is spanned by \((\frac{1}{2}n(n-1),n,\frac{1}{2}n(n+1)),(n(n-1),0,0)\) and (0, 2n, 0). By Lemma 4.6, \(a(\tilde{\gamma })\) has coordinates (1, 0, 1). Let m be a number such that \(ma(\tilde{\gamma })\) is contained in \(Im(\Phi )\). Then

$$\begin{aligned} m(1, 0, 1)=s\left( \frac{1}{2}n(n-1),n,\frac{1}{2}n(n+1)\right) +t(n(n-1),0,0)+r(0,2n,0) \end{aligned}$$

for some integers st and r. Solve this to get

$$\begin{aligned} \begin{array}{c c c} m=\frac{1}{2}tn(n^2-1),&s=-2r,&s=t(n-1). \end{array} \end{aligned}$$

Since \(s=-2r\) is even, the smallest positive value of t satisfying \(s=t(n-1)\) is 1 for n odd and 2 for n even. Therefore m is \(\frac{1}{2}n(n^2-1)\) for n odd and \(n(n^2-1)\) for n even. \(\square \)

For SU(n)-gauge groups over \(S^4\), the order m of \(\partial _1\) has the form \(m=n(n^2-1)\) for \(n=3\) and 5 [5, 19]. If p is an odd prime and \(n<(p-1)^2+1\), then m and \(n(n^2-1)\) have the same p-components [9, 20]. These facts suggest it may be the case that \(m=n(n^2-1)\) for any \(n>2\). In fact, one can follow the method Hamanaka and Kono used in [5] and calculate the order of \(\partial \circ \epsilon \) to obtain a lower bound \(n(n^2-1)\) for n odd. However, it does not work for the n even case since \([S^4\wedge {\mathbb {C}}{\mathbb {P}}^{n-1},\Omega W_n]\) is not a free \({\mathbb {Z}}\)-module. An interesting corollary of Theorem 4.7 is to give a lower bound on the order of \(\partial _1\) for n even.

Corollary 4.8

When n is even and greater than 2, the order of \(\partial _1\) is at least \(n(n^2-1)\).

Proof

The order of \(\partial '_1\circ \epsilon \) is a lower bound on the order of \(\partial '_1\), which is either the same as or half of the order of \(\partial _1\) by Lemma 2.2. The corollary follows from Theorem 4.7. \(\square \)

5 A necessary condition for \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\simeq {\mathcal {G}}_l({\mathbb {C}}{\mathbb {P}}^2)\)

In this section we follow the approach in [5] to prove Theorem 1.6. The techniques used are similar to that in Sect. 4, except we are working with the quotient \(\Sigma C=\Sigma {\mathbb {C}}{\mathbb {P}}^{n-1}/\Sigma {\mathbb {C}}{\mathbb {P}}^{n-1}\) instead of \(\Sigma {\mathbb {C}}{\mathbb {P}}^{n-1}\). When n is odd, C is \(\Sigma ^{2n-6}{\mathbb {C}}{\mathbb {P}}^2\), and when n is even, C is \(S^{2n-2}\vee S^{2n-4}\). Apply \([\Sigma C,-]\) to fibration (3) to obtain the exact sequence

$$\begin{aligned}&[\Sigma C, SU(n)]\overset{(\partial '_k)_*}{\longrightarrow }[\Sigma C, {\text {Map}}^*_0({\mathbb {C}}{\mathbb {P}}^2,BSU(n))]\\&\quad \longrightarrow [\Sigma C, B{\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)]\longrightarrow [\Sigma C, BSU(n)], \end{aligned}$$

where \((\partial '_k)_*\) sends \(f\in [\Sigma C, SU(n)]\) to \(\partial '_k\circ f\in [\Sigma C, {\text {Map}}^*_0({\mathbb {C}}{\mathbb {P}}^2, BSU(n))]\). Since \(BSU(n)\rightarrow BSU(\infty )\) is a 2n-equivalence and \(\Sigma C\) has dimension \(2n-1\), \([\Sigma C, BSU(n)]\) is \({\tilde{K}}^0(\Sigma C)\) which is zero. Similarly, \([\Sigma C, SU(n)]\cong [\Sigma ^2C, BSU(n)]\) is \({\tilde{K}}^0(\Sigma ^2C)\cong {\mathbb {Z}}\oplus {\mathbb {Z}}\). Furthermore, by adjunction we have \([\Sigma C, {\text {Map}}^*_0({\mathbb {C}}{\mathbb {P}}^2,BSU(n))]\cong [\Sigma C\wedge {\mathbb {C}}{\mathbb {P}}^2, BSU(n)]\). The exact sequence becomes

$$\begin{aligned} {\tilde{K}}^0(\Sigma ^2C)\overset{(\partial '_k)_*}{\longrightarrow }[\Sigma C\wedge {\mathbb {C}}{\mathbb {P}}^2, BSU(n)]\longrightarrow [\Sigma C, B{\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)]\longrightarrow 0. \end{aligned}$$
(11)

This implies \([\Sigma C, B{\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)]\cong [C, {\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)]\) is \(Coker(\partial '_k)_*\). Also, apply \([{\mathbb {C}}{\mathbb {P}}^2\wedge C, -]\) to fibration (8) to obtain the exact sequence

$$\begin{aligned}&[{\mathbb {C}}{\mathbb {P}}^2\wedge C, \Omega SU(\infty )]\overset{p_*}{\longrightarrow }[{\mathbb {C}}{\mathbb {P}}^2\wedge C, \Omega W_n]\nonumber \\&\quad \longrightarrow [{\mathbb {C}}{\mathbb {P}}^2\wedge C, SU(n)]\longrightarrow [{\mathbb {C}}{\mathbb {P}}^2\wedge C, SU(\infty )]. \end{aligned}$$
(12)

Observe that \([{\mathbb {C}}{\mathbb {P}}^2\wedge C, \Omega SU(\infty )]\cong {\tilde{K}}^0({\mathbb {C}}{\mathbb {P}}^2\wedge C)\) is \({\mathbb {Z}}^{\oplus 4}\) and \([{\mathbb {C}}{\mathbb {P}}^2\wedge C, SU(\infty )]\cong {\tilde{K}}^1({\mathbb {C}}{\mathbb {P}}^2\wedge C)\) is zero. Combine exact sequences (11) and (12) to obtain the diagram

where \(a(f)=f^*(a_{2n})\oplus f^*(a_{2n+2})\) for any \(f\in [{\mathbb {C}}{\mathbb {P}}^2\wedge C, \Omega W_n]\), and \(\Phi \) is defined to be \(a\circ p_*\). By Lemma 4.2\([{\mathbb {C}}{\mathbb {P}}^2\wedge C,\Omega W_n]\) is free. Following the same argument in Sect. 4 implies the injectivity of a.

Our strategy to prove Theorem 1.6 is as follows. If \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\) is homotopy equivalent to \({\mathcal {G}}_l({\mathbb {C}}{\mathbb {P}}^2)\), then \([C, {\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)]\cong [C, {\mathcal {G}}_l({\mathbb {C}}{\mathbb {P}}^2)]\) and exactness in (12) implies that \(Im(\partial '_k)_*\) and \(Im(\partial '_l)_*\) have the same order in \([{\mathbb {C}}{\mathbb {P}}^2\wedge C, SU(n)]\), resulting in a necessary condition for a homotopy equivalence \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\simeq {\mathcal {G}}_l({\mathbb {C}}{\mathbb {P}}^2)\). To calculate the order of \(Im(\partial '_k)_*\), we will find a preimage \(\tilde{\partial }_k\) of \(Im(\partial '_k)_*\) in \([{\mathbb {C}}{\mathbb {P}}^2\wedge C,\Omega W_n]\). Since a is injective, we can embed \(\tilde{\partial }_k\) into \(H^{2n}({\mathbb {C}}{\mathbb {P}}^2\wedge C)\oplus H^{2n+2}({\mathbb {C}}{\mathbb {P}}^2\wedge C)\) and work out the order of \(Im(\partial '_k)_*\) there.

Let \(u, v_{2n-4}\) and \(v_{2n-2}\) be generators of \(H^2({\mathbb {C}}{\mathbb {P}}^2)\), \(H^{2n-4}(C)\) and \(H^{2n-2}(C)\). Then we write an element \(xu^2v_{2n-4}+yuv_{2n-2}+zu^2v_{2n-2}\in H^{2n}({\mathbb {C}}{\mathbb {P}}^2\wedge C)\oplus H^{2n+2}({\mathbb {C}}{\mathbb {P}}^2\wedge C)\) as (xyz). First we need to find the submodule Im(a).

Lemma 5.1

For n odd, Im(a) is \(\{(x, y, z)|x+y\equiv z\pmod {2}\}\), and for n even, Im(a) is \(\{(x, y, z)|y\equiv 0\pmod {2}\}\).

Proof

When n is odd, C is \(\Sigma ^{2n-6}{\mathbb {C}}{\mathbb {P}}^2\) and the \((2n+3)\)-skeleton of \(\Omega W_n\) is \(\Sigma ^{2n-2}{\mathbb {C}}{\mathbb {P}}^2\). To say \((x, y, z)\in Im(a)\) means there exists \(f\in [{\mathbb {C}}{\mathbb {P}}^2\wedge C,\Omega W_n]\) such that

$$\begin{aligned} \begin{array}{c c c} f^*(a_{2n})=xu^2v_{2n-4}+yuv_{2n-2}&\text {and}&f^*(a_{2n+2})=zu^2v_{2n-2}. \end{array} \end{aligned}$$
(13)

Reducing to homology with \({\mathbb {Z}}/2{\mathbb {Z}}\)-coefficients, we have

$$\begin{aligned} \begin{array}{c c c} Sq^2(u)=u^2,&Sq^2(v_{2n-4})=v_{2n-2},&Sq^2(a_{2n})=a_{2n+2}. \end{array} \end{aligned}$$

Apply \(Sq^2\) to (13) to get \(x+y\equiv z\pmod {2}\). Therefore Im(a) is contained in \(\{(x, y, z)|x+y\equiv z\pmod {2}\}\). To show that they are equal, we need to show that (1, 0, 1), (0, 1, 1) and (0, 0, 2) are in Im(a). Consider maps

$$\begin{aligned} \begin{array}{l} f_1:{\mathbb {C}}{\mathbb {P}}^2\wedge C\overset{q_1}{\longrightarrow }S^4\wedge C\simeq \Sigma ^{2n-2}{\mathbb {C}}{\mathbb {P}}^2\hookrightarrow \Omega W_n\\ f_2:{\mathbb {C}}{\mathbb {P}}^2\wedge C\overset{q_2}{\longrightarrow }{\mathbb {C}}{\mathbb {P}}^2\wedge S^{2n-2}\hookrightarrow \Omega W_n\\ f_3:{\mathbb {C}}{\mathbb {P}}^2\wedge C\overset{q_3}{\longrightarrow }S^{2n+2}\overset{\theta }{\longrightarrow }\Omega W_n \end{array} \end{aligned}$$

where \(q_1,q_2\) and \(q_3\) are quotient maps and \(\theta \) is the generator of \(\pi _{2n+3}(W_n)\). Their images are

$$\begin{aligned} \begin{array}{c c c} a(f_1)=(1, 0, 1)&a(f_2)=(0, 1, 1)&a(f_3)=(0, 0, 2) \end{array} \end{aligned}$$

respectively, so \(Im(a)=\{(x, y, z)|x+y\equiv z\pmod {2}\}\).

When n is even, C is \(S^{2n-2}\vee S^{2n-4}\) and the \((2n+3)\)-skeleton of \(\Omega W_n\) is \(S^{2n+2}\vee S^{2n}\). Reducing to homology with \({\mathbb {Z}}/2{\mathbb {Z}}\)-coefficients, \(Sq^2(v_{2n-4})=0\) and \(Sq^2(a_{2n})=0\). Apply \(Sq^2\) to (13) to get \(y\equiv 0\pmod {2}\). Therefore Im(a) is contained in \(\{(x, y, z)|y\equiv 0\pmod {2}\}\). To show that they are equal, we need to show that (1, 0, 0), (0, 2, 0) and (0, 0, 1) are in Im(a). The maps

$$\begin{aligned} \begin{array}{l} f'_1:{\mathbb {C}}{\mathbb {P}}^2\wedge C\overset{q'_1}{\longrightarrow }S^4\wedge (S^{2n-2}\vee S^{2n-4})\overset{p_1}{\longrightarrow }S^4\wedge S^{2n-4}\hookrightarrow \Omega W_n\\ f'_2:{\mathbb {C}}{\mathbb {P}}^2\wedge C\overset{q'_2}{\longrightarrow }S^4\wedge (S^{2n-2}\vee S^{2n-4})\overset{p_2}{\longrightarrow }S^4\wedge S^{2n-2}\hookrightarrow \Omega W_n \end{array} \end{aligned}$$

where \(q'_1\) and \(q'_2\) are quotient maps and \(p_1\) and \(p_2\) are pinch maps, have images \(a(f'_1)=(1, 0, 0)\) and \(a(f'_2)=(0, 0, 1)\). To find (0, 2, 0), apply \([-\wedge S^{2n-2},\Omega W_n]\) to cofibration (4) to obtain the exact sequence

$$\begin{aligned} \pi _{2n+3}(W_n)\longrightarrow [{\mathbb {C}}{\mathbb {P}}^2\wedge S^{2n-2},\Omega W_n]\overset{i^*}{\longrightarrow }\pi _{2n+1}(W_n)\overset{\eta ^*}{\longrightarrow }\pi _{2n+2}(W_n) \end{aligned}$$

where \(i^*\) is induced by the inclusion \(i:S^2\hookrightarrow {\mathbb {C}}{\mathbb {P}}^2\) and \(\eta ^*\) is induced by Hopf map \(\eta \). The third term \(\pi _{2n+1}(W_n)\cong {\mathbb {Z}}\) is generated by \(i':S^{2n+1}\rightarrow W_n\), the inclusion of the bottom cell, and the fourth term \(\pi _{2n+2}(W_n)\cong {\mathbb {Z}}/2{\mathbb {Z}}\) is generated by \(i'\circ \eta \), so \(\eta ^*:{\mathbb {Z}}\rightarrow {\mathbb {Z}}/2{\mathbb {Z}}\) is a surjection. By exactness \([{\mathbb {C}}{\mathbb {P}}^2\wedge S^{2n-2},\Omega W_n]\) has a \({\mathbb {Z}}\)-summand with the property that \(i^*\) sends its generator g to \(2i'\). Therefore the composition

$$\begin{aligned} f'_3:{\mathbb {C}}{\mathbb {P}}^2\wedge (S^{2n-2}\vee S^{2n-4})\overset{pinch}{\longrightarrow }{\mathbb {C}}{\mathbb {P}}^2\wedge S^{2n-2}\overset{g}{\longrightarrow }\Omega W_n \end{aligned}$$

has image (0, 2, 0). It follows that \(Im(a)=\{(x, y, z)|y\equiv 0\pmod {2}\}\). \(\square \)

Now we split into the n odd and n even cases to calculate the order of \(Im(\partial '_k)_*\).

5.1 The order of \(Im(\partial '_k)_*\) for n odd

When n is odd, C is \(\Sigma ^{2n-6}{\mathbb {C}}{\mathbb {P}}^2\). First we find \(Im(\Phi )\) in Im(a). For \(1\le i\le 4\), let \(L_i\) be the generators of \({\tilde{K}}^0({\mathbb {C}}{\mathbb {P}}^2\wedge C)\cong {\mathbb {Z}}^{\oplus 4}\) with Chern characters

$$\begin{aligned} \begin{array}{l l} ch(L_1)=\left( u+\frac{1}{2}u^2\right) \cdot \left( v_{2n-4}+\frac{1}{2}v_{2n-2}\right) &{}ch(L_2)=\left( u+\frac{1}{2}u^2\right) v_{2n-2}\\ ch(L_3)=u^2\left( v_{2n-4}+\frac{1}{2}v_{2n-2}\right) &{}ch(L_4)=u^2v_{2n-2}. \end{array} \end{aligned}$$

By Theorem 4.3, we have

$$\begin{aligned} \Phi (L_1)= & {} \frac{n!}{2}u^2v_{2n-4}+\frac{n!}{2}uv_{2n-2}+\frac{(n+1)!}{4}u^2v_{2n-2}\\ \Phi (L_2)= & {} n!uv_{2n-2}+\frac{(n+1)!}{2}u^2v_{2n-2}\\ \Phi (L_3)= & {} n!u^2v_{2n-4}+\frac{(n+1)!}{2}u^2v_{2n-2}\\ \Phi (L_4)= & {} (n+1)!u^2v_{2n-2}. \end{aligned}$$

By Lemma 5.1, Im(a) is spanned by (1, 0, 1), (0, 1, 1) and (0, 0, 2). Under this basis, the coordinates of the \(\Phi (L_i)\)’s are

$$\begin{aligned} \begin{array}{l@{\quad }l} \Phi (L_1)=\left( \frac{n!}{2},\frac{n!}{2},\frac{(n-3)\cdot n!}{8}\right) , &{}\Phi (L_2)=\left( 0,n!,\frac{(n-1)\cdot n!}{4}\right) ,\\ \Phi (L_3)=\left( n!,0,\frac{(n-1)\cdot n!}{4}\right) , &{}\Phi (L_4)=\left( 0,0,\frac{(n+1)!}{2}\right) . \end{array} \end{aligned}$$

We represent their coordinates by the matrix

$$\begin{aligned} M_{\Phi }=L \begin{pmatrix} \frac{n(n-1)}{2} &{}\quad \frac{n(n-1)}{2} &{}\quad \frac{n(n-1)(n-3)}{8}\\ 0 &{}\quad n(n-1) &{}\quad \frac{n(n-1)^2}{4}\\ n(n-1) &{}\quad 0 &{}\quad \frac{n(n-1)^2}{4}\\ 0 &{}\quad 0 &{}\quad \frac{n(n^2-1)}{2} \end{pmatrix}, \end{aligned}$$

where \(L=(n-2)!\). Then \(Im(\Phi )\) is spanned by the row vectors of \(M_{\Phi }\).

Next, we find a preimage of \(Im(\partial '_k)_*\) in \([{\mathbb {C}}{\mathbb {P}}^2\wedge C,\Omega W_n]\). In exact sequence (11) \({\tilde{K}}^0(\Sigma ^2C)\) is \({\mathbb {Z}}\oplus {\mathbb {Z}}\). Let \(\alpha _1\) and \(\alpha _2\) be its generators with Chern classes

$$\begin{aligned} \begin{array}{l l} c_{n-1}(\alpha _1)=(n-2)!\Sigma ^2v_{2n-4} &{}c_n(\alpha _1)=\frac{(n-1)!}{2}\Sigma ^2v_{2n-2}\\ c_{n-1}(\alpha _2)=0 &{}c_n(\alpha _2)=(n-1)!\Sigma ^2v_{2n-2}. \end{array} \end{aligned}$$

Lemma 5.2

For \(i=1,2\), \((\partial '_k)_*(\alpha _i)\) has a lift \(\tilde{\alpha }_{i,k}:{\mathbb {C}}{\mathbb {P}}^2\wedge C\rightarrow \Omega W_n\) such that

$$\begin{aligned} a(\tilde{\alpha }_{i,k})=ku^2\otimes \Sigma ^{-2}c_{n-1}(\alpha _i)\oplus ku^2\otimes \Sigma ^{-2}c_{n}(\alpha _i). \end{aligned}$$

Proof

For dimension and connectivity reasons, \(\alpha _i:\Sigma ^2C\rightarrow BSU(\infty )\) lifts through \(BSU(n)\rightarrow BSU(\infty )\). Label the lift \(\Sigma ^2C\rightarrow BSU(n)\) by \(\alpha _i\) as well. Let \(\alpha '_i:\Sigma C\rightarrow SU(n)\) be the adjoint of \(\alpha _i\). Then \((\partial '_k)_*(\alpha _i)\) is the adjoint of the composition

$$\begin{aligned} {\mathbb {C}}{\mathbb {P}}^2\wedge \Sigma C\overset{q\wedge {\mathbb {1}}}{\longrightarrow }\Sigma S^3\wedge \Sigma C\overset{\Sigma k\imath \wedge \alpha '_i}{\longrightarrow }\Sigma SU(n)\wedge SU(n)\overset{[ev,ev]}{\longrightarrow }BSU(n). \end{aligned}$$

By Lemma 4.5, \((\partial '_k)_*(\alpha _i)\) has a lift \(\tilde{\alpha }_{i,k}\) such that \(\tilde{\alpha }_{i,k}^*(a_{2j})=ku^2\otimes \Sigma ^{-1}(\alpha ')^*(x_{2j-3})\). Since \(\sigma (c_{j-1})=x_{2j-3}\), we have \(\tilde{\alpha }_{i,k}^*(a_{2j})=ku^2\otimes \Sigma ^{-2}c_{j-1}(\alpha _i)\) and

$$\begin{aligned} a(\tilde{\alpha }_{i,k})=ku^2\otimes \Sigma ^{-2}c_{n-1}(\alpha _i)\oplus ku^2\otimes \Sigma ^{-2}c_{n}(\alpha _i). \end{aligned}$$

\(\square \)

By Lemma 5.2, \((\partial '_k)_*(\alpha _1)\) and \((\partial '_k)_*(\alpha _2)\) have lifts

$$\begin{aligned} \begin{array}{c c c} \tilde{\alpha }_{1,k}=(n-2)!ku^2v_{2n-4}+\frac{(n-1)!}{2}ku^2v_{2n-2}&\quad \text {and}&\quad \tilde{\alpha }_{2,k}=(n-1)!ku^2v_{2n-2}. \end{array} \end{aligned}$$

We represent their coordinates by the matrix

$$\begin{aligned} M_{\partial }=kL \begin{pmatrix} 1 &{}\quad 0 &{}\quad \frac{n-3}{4}\\ 0 &{}\quad 0 &{}\quad \frac{n-1}{2} \end{pmatrix}. \end{aligned}$$

Let \(\tilde{\partial }_k=span\{\tilde{\alpha }_{1,k}, \tilde{\alpha }_{2,k}\}\) be the preimage of \(Im(\partial '_k)_*\) in \([{\mathbb {C}}{\mathbb {P}}^2\wedge C,\Omega W_n]\). Then \(\tilde{\partial }_k\) is spanned by the row vectors of \(M_{\partial }\).

Lemma 5.3

When n is odd, the order of \(Im(\partial '_k)_*\) is

$$\begin{aligned} |Im(\partial '_k)_*|=\frac{\frac{1}{2}n(n^2-1)}{(\frac{1}{2}n(n^2-1), k)}\cdot \frac{n}{(n,k)}. \end{aligned}$$

Proof

Suppose \(n=4m+3\) for some integer m. Then

$$\begin{aligned} M_{\Phi }=(4m+3)L \begin{pmatrix} 2m+1 &{}\quad 2m+1 &{}\quad 2m^2+m\\ 0 &{}\quad 4m+2 &{}\quad 4m^2+4m+1\\ 4m+2 &{}\quad 0 &{}\quad 4m^2+4m+1\\ 0 &{}\quad 0 &{}\quad 8m^2+12m+4 \end{pmatrix} \end{aligned}$$

and

$$\begin{aligned} M_{\partial }=kL \begin{pmatrix} 1 &{}\quad 0 &{}\quad m\\ 0 &{}\quad 0 &{}\quad 2m+1 \end{pmatrix}. \end{aligned}$$

Transform \(M_{\Phi }\) into Smith normal form

$$\begin{aligned} A\cdot M_{\Phi }\cdot B=(4m+3)L \begin{pmatrix} (2m+1) &{} &{}\\ &{}\quad (2m+1) &{}\\ &{} &{}\quad (2m+1)(4m+4)\\ &{} &{}\quad 0 \end{pmatrix}, \end{aligned}$$

where

$$\begin{aligned} \begin{array}{c c c} A= \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ -2 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 4m+2 &{}\quad 1 &{}\quad -(2m+1) &{}\quad 0\\ 4 &{}\quad -2 &{}\quad -2 &{}\quad 1 \end{pmatrix} &{}\text {and} &{}B= \begin{pmatrix} 1 &{}\quad -m &{}-(2m+1)\\ 0 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 1 &{}\quad 2 \end{pmatrix}. \end{array} \end{aligned}$$

The matrix B represents a basis change in Im(a) and A represents a basis change in \(Im(\Phi )\). Therefore \([{\mathbb {C}}{\mathbb {P}}^2\wedge C, SU(n)]\) is isomorphic to

$$\begin{aligned} \frac{{\mathbb {Z}}}{\frac{1}{2}(4m+3)!{\mathbb {Z}}}\oplus \frac{{\mathbb {Z}}}{\frac{1}{2}(4m+3)!{\mathbb {Z}}}\oplus \frac{{\mathbb {Z}}}{\frac{1}{2}(4m+4)!{\mathbb {Z}}}. \end{aligned}$$

We need to find the representation of \(\tilde{\partial }_k\) under the new basis represented by B. The new coordinates of \(\tilde{\alpha }_{1,k}\) and \(\tilde{\alpha }_{2,k}\) are the row vectors of the matrix

$$\begin{aligned} M_{\partial } \cdot \begin{pmatrix} 1 &{}\quad -m &{}\quad -(2m+1)\\ 0 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 1 &{}\quad 2 \end{pmatrix} = \begin{pmatrix} kL &{}\quad 0 &{}\quad -kL\\ 0 &{}\quad (2m+1)kL &{}\quad (4m+2)kL \end{pmatrix}. \end{aligned}$$

Apply row operations to get

$$\begin{aligned} \begin{pmatrix} 1 &{}\quad 0\\ 4m+2 &{}\quad 1 \end{pmatrix} \cdot \begin{pmatrix} kL &{}\quad 0 &{}\quad -kL\\ 0 &{}\quad (2m+1)kL &{}\quad (4m+2)kL \end{pmatrix} = \begin{pmatrix} kL &{}\quad 0 &{}\quad -kL\\ (4m+2)kL &{}\quad (2m+1)kL &{}\quad 0 \end{pmatrix}. \end{aligned}$$

Let \(\mu =(kL,0,-kL)\) and \(\nu =((4m+2)kL, (2m+1)kL,0)\). Then

$$\begin{aligned} \tilde{\partial }_k=\{x\mu +y\nu \in [{\mathbb {C}}{\mathbb {P}}^2\wedge C,\Omega W_n]|x, y\in {\mathbb {Z}}\}. \end{aligned}$$

If \(x\mu +y\nu \) and \(x'\mu +y'\nu \) are the same modulo \(Im(\Phi )\), then we have

$$\begin{aligned}\left\{ \begin{array}{llll} &{}xkL+(4m+2)ykL \equiv x'kL+(4m+2)y'kL &{}\pmod {(2m+1)(4m+3)L}\\ &{}(2m+1)ykL \equiv (2m+1)y'kL &{}\pmod {(2m+1)(4m+3)L}\\ &{}xkL \equiv x'kL &{}\pmod {(2m+1)(4m+3)(4m+4)L} \end{array}\right. \end{aligned}$$

These conditions are equivalent to

$$\begin{aligned}\left\{ \begin{array}{r c l l} xk &{}\equiv &{}x'k &{}\pmod {(2m+2)(4m+3)(4m+2)}\\ yk &{}\equiv &{}y'k &{}\pmod {(4m+3)} \end{array}\right. \end{aligned}$$

This implies that there are \(\displaystyle {\frac{(2m+2)(4m+3)(4m+2)}{((2m+2)(4m+3)(4m+2), k)}}\) distinct values of x and \(\displaystyle {\frac{4m+3}{(4m+3,k)}}\) distinct values of y, so we have

$$\begin{aligned} |Im(\partial '_k)_*|=\frac{(2m+2)(4m+3)(4m+2)}{((2m+2)(4m+3)(4m+2), k)}\cdot \frac{4m+3}{(4m+3,k)}. \end{aligned}$$

When \(n=4m+1\), we can repeat the calculation above to obtain

$$\begin{aligned} |Im(\partial '_k)_*|=\frac{2m(4m+2)(4m+1)}{(2m(4m+2)(4m+1), k)}\cdot \frac{4m+1}{(4m+1,k)}. \end{aligned}$$

\(\square \)

5.2 The order of \(Im(\partial '_k)_*\) for n even

When n is even, C is \(S^{2n-2}\vee S^{2n-4}\). For \(1\le i\le 4\), let \(L_i\) be the generators of \({\tilde{K}}^0({\mathbb {C}}{\mathbb {P}}^2\wedge C)\cong {\mathbb {Z}}^{\oplus 4}\) with Chern characters

$$\begin{aligned} \begin{array}{l l} ch(L_1)=\left( u+\frac{1}{2}u^2\right) v_{2n-4} &{}ch(L_2)=u^2v_{2n-4}\\ ch(L_3)=\left( u+\frac{1}{2}u^2\right) v_{2n-2} &{}ch(L_4)=u^2v_{2n-2}. \end{array} \end{aligned}$$

By Theorem 4.3, we have

$$\begin{aligned} \Phi (L_1)= & {} \frac{n!}{2}u^2v_{2n-4}\\ \Phi (L_2)= & {} n!u^2v_{2n-4}\\ \Phi (L_3)= & {} n!uv_{2n-2}+\frac{(n+1)!}{2}u^2v_{2n-2}\\ \Phi (L_4)= & {} (n+1)!u^2v_{2n-2}. \end{aligned}$$

By Lemma 5.1, Im(a) is spanned by (1, 0, 0), (0, 2, 0) and (0, 0, 1). Under this basis, the coordinates of the \(\Phi (L_i)\)’s are

$$\begin{aligned} \begin{array}{l l} \Phi (L_1)=(\frac{n!}{2},0,0), &{}\quad \Phi (L_2)=(n!,0,0),\\ \Phi (L_3)=\left( 0,\frac{n!}{2},\frac{(n+1)!}{2}\right) , &{} \quad \Phi (L_4)=(0,0,(n+1)!). \end{array} \end{aligned}$$

We represent the coordinates of \(\Phi (L_i)\)’s by the matrix

$$\begin{aligned} M_{\Phi }=\frac{n(n-1)}{2}L \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0\\ 2 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad n+1\\ 0 &{}\quad 0 &{}\quad 2n+2 \end{pmatrix} \end{aligned}$$

Then \(Im(\Phi )\) is spanned by the row vectors of \(M_{\Phi }\).

In exact sequence (11) \({\tilde{K}}^0(\Sigma ^2C)\) is \({\mathbb {Z}}\oplus {\mathbb {Z}}\). Let \(\alpha _1\) and \(\alpha _2\) be its generators with Chern classes

$$\begin{aligned} \begin{array}{l l} c_{n-1}(\alpha _1)=(n-2)!\Sigma ^2v_{2n-4} &{}c_n(\alpha _1)=0\\ c_{n-1}(\alpha _2)=0 &{}c_n(\alpha _2)=(n-1)!\Sigma ^2v_{2n-2}. \end{array} \end{aligned}$$

By Lemma 5.2, \((\partial '_k)_*(\alpha _1)\) and \((\partial '_k)_*(\alpha _2)\) have lifts

$$\begin{aligned} \begin{array}{c c c} \tilde{\alpha }_{1, k}=(n-2)!ku^2v_{2n-4}&\text {and}&\tilde{\alpha }_{2, k}=(n-1)!ku^2v_{2n-2}. \end{array} \end{aligned}$$

We represent their coordinates by a matrix

$$\begin{aligned} M_{\partial }=kL \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad n-1 \end{pmatrix}. \end{aligned}$$

Then the preimage \(\tilde{\partial }_k=span\{\tilde{\alpha }_{1, k}, \tilde{\alpha }_{2, k}\}\) of \(Im(\partial '_k)_*\) is spanned by the row vectors of \(M_{\partial }\). We calculate as in the proof of Lemma 5.3 to obtain the following lemma.

Lemma 5.4

When n is even, the order of \(Im(\partial '_k)_*\) is

$$\begin{aligned} |Im(\partial '_k)_*|=\frac{\frac{1}{2}n(n-1)}{\left( \frac{1}{2}n(n-1), k\right) }\cdot \frac{n(n+1)}{(n(n+1),k)}. \end{aligned}$$

5.3 Proof of Theorem 1.6

Before comparing the orders of \(Im(\partial '_k)_*\) and \(Im(\partial '_k)_*\), we prove a preliminary lemma.

Lemma 5.5

Let n be an even number and let p be a prime. Denote the p-component of t by \(\nu _p(t)\). If there are integers k and l such that

$$\begin{aligned} \nu _p\left( \frac{1}{2}n,k\right) \cdot \nu _p(n,k)=\nu _p\left( \frac{1}{2}n,l\right) \cdot \nu _p(n,l), \end{aligned}$$

then \(\nu _p(n, k)=\nu _p(n,l)\).

Proof

Suppose p is odd. If p does not divide n, then \(\nu _p(n,k)=\nu _p(n,l)=1\), so the lemma holds. If p divides n, then \(\nu _p(\frac{1}{2}n,k)=\nu _p(n,k)\). The hypothesis becomes \(\nu _p(n,k)^2=\nu _p(n,l)^2\), implying that \(\nu _p(n, k)=\nu _p(n,l)\).

Suppose \(p=2\). Let \(\nu _2(n)=2^r\), \(\nu _2(k)=2^t\) and \(\nu _2(l)=2^s\). Then the hypothesis implies

$$\begin{aligned} min(r-1,t)+min(r,t)=min(r-1,s)+min(r,s). \end{aligned}$$
(14)

To show \(\nu _2(n,k)=\nu _2(n,l)\), we need to show \(min(r,t)=min(r,s)\). Consider the following cases: (1) \(t,s\ge r\), (2) \(t,s\le r-1\), (3) \(t\le r-1,s\ge r\) and (4) \(s\le r-1,t\ge r\).

Case (1) obviously gives \(min(r,t)=min(r,s)\). In case (2), when \(t,s\le r-1\), equation (14) implies \(2t=2s\). Therefore \(t=s\) and \(min(r,t)=min(r,s)\).

It remains to show cases (3) and (4). For case (3) with \(t\le r-1,s\ge r\), equation (14) implies

$$\begin{aligned} 2t=min(r-1, s)+r. \end{aligned}$$

Since \(s\ge r\), \(min(r-1,s)=r-1\) and the right hand side is \(2r-1\) which is odd. However, the left hand side is even, leading to a contradiction. This implies that this case does not satisfy the hypothesis. Case (4) is similar. Therefore \(\nu _2(n, k)=\nu _2(n, l)\) and the asserted statement follows. \(\square \)

Proof of Theorem 1.6

In exact sequence (11), \([C,{\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)]\) is \(Coker(\partial '_k)_*\). By hypothesis, \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\) is homotopy equivalent to \({\mathcal {G}}_l({\mathbb {C}}{\mathbb {P}}^2)\), so \(|Im(\partial '_k)_*|=|Im(\partial '_k)_*|\). The n odd and n even cases are proved similarly, but the even case is harder.

When n is even, by Lemma 5.4 the order of \(Im(\partial '_k)_*\) is

$$\begin{aligned} |Im(\partial '_k)_*|=\frac{\frac{1}{2}n(n-1)}{\left( \frac{1}{2}n(n-1),k\right) }\cdot \frac{n(n+1)}{(n(n+1),k)}, \end{aligned}$$

so we have

$$\begin{aligned} \left( \frac{1}{2}n(n-1),k\right) \cdot (n(n+1),k)=\left( \frac{1}{2}n(n-1),l\right) \cdot (n(n+1),l). \end{aligned}$$
(15)

We need to show that

$$\begin{aligned} \nu _p(n(n^2-1),k)=\nu _p(n(n^2-1),l) \end{aligned}$$
(16)

for all primes p. Suppose p does not divide \(\frac{1}{2}n(n^2-1)\). Equation (16) holds since both sides are 1. Suppose p divides \(\frac{1}{2}n(n^2-1)\). Since \(n-1\), n and \(n+1\) are coprime, p divides only one of them. If p divides \(n-1\), then \(\nu _p(\frac{1}{2}n,k)=\nu _p(n,k)=\nu _p(n+1,k)=1\). Equation (15) implies \(\nu _p(n-1,k)=\nu _p(n-1,l)\). Since

$$\begin{aligned} \nu _p(n(n^2-1),k)=\nu _p(n-1,k)\cdot \nu _p(n,k)\cdot \nu _p(n+1,k), \end{aligned}$$

this implies equation (16) holds. If p divides \(n+1\), then equation (16) follows from a similar argument. If p divides n, then equation (15) implies \(\nu _p(\frac{1}{2}n,k)\cdot \nu _p(n,k)=\nu _p(\frac{1}{2}n,l)\cdot \nu _p(n,l)\). By Lemma 5.5\(\nu _p(n, k)=\nu _p(n,l)\), so equation (16) holds.

When n is odd, by Lemma 5.3 the order of \(Im(\partial '_k)_*\) is

$$\begin{aligned} |Im(\partial '_k)_*|=\frac{\frac{1}{2}n(n^2-1)}{\left( \frac{1}{2}n(n^2-1),k\right) }\cdot \frac{n}{(n,k)}, \end{aligned}$$

so we have

$$\begin{aligned} \left( \frac{1}{2}n(n^2-1),k\right) \cdot (n,k)=\left( \frac{1}{2}n(n^2-1),l\right) \cdot (n,l). \end{aligned}$$

We can argue as above to show that for all primes p,

$$\begin{aligned} \nu _p\left( \frac{1}{2}n(n^2-1),k\right) =\nu _p\left( \frac{1}{2}n(n^2-1),l\right) . \end{aligned}$$

\(\square \)