Results in Homotopy Theory of Gauge Groups
Written by Mitsunobu Tsutaya.
Last update: February 13, 2022.
I don’t think that I could collect all the related works here and I think that this list might include
some incorrect descriptions. Indeed, I have read only a few papers in this list. If there are any
mistake here, please let me know.
1. Some selected results in chronological order• James, 1963: A study of the space of bundle maps.
• Gottlieb, 1972: BG ' Map(B, BG) f .
• Kono, 1991: The first result on the classification of the homotopy types of gauge groups.
• Crabb–Sutherland, 2000: Finiteness of homotopy equivalence types of gauge groups.
2. Homotopy types of G
General results on homotopy types.
• G(P) ' Ω Map(B, BG) f where f : B → BG is the classifying map of P (Gottlieb, 1972;
Atiyah–Bott, 1982).
• Finiteness of homotopy types of gauge groups for G a connected Lie group and B a finite
complex (Crabb–Sutherland, 2000).
• Fibrewise decomposition of the adjoint bundle induced from an automorphism on G (Kishimoto–Kono, 2010).
General homotopy decompositions over spheres.
• Theriault, 2010: p-local decomposition G for any simple Lie group G over S 4 such thatp > n` + 1 (p-locally trivial case) and some decompositions for p-locally non-trivial cases
when G = SU(n).
• Kishimoto–Kono–Tsutaya, 2013: p-local decomposition G ' B1 × · · · × B p−1 for any Liegroup G over S n such that πn−1 (G) ⊗ Q  Q.
• Kishimoto–Kono–Theriault, 2014: p-local refined decompositions of G for G = SU(n), Sp(n)of any rank over S 4 .
Consider the cases when π2n−1 (G)  Z. Let Pk be the principal G-bundle classified by k : S 2n →BG for a fixed generator  ∈ π2n (BG)  Z.
Complete classifications over S 4 .
• SU(2)-bundles over S 4 .
– Kono, 1991: G(Pk ) ' G(Pk0 ) if and only if (12, k) = (12, k0 ).
• SU(3)-bundles over S 4 .
– Hamanaka–Kono, 2006: G(Pk ) ' G(Pk0 ) if and only if (24, k) = (24, k0 ).
• SO(3)-bundles over S 4 (SO(3) = PU(2)).
– Kamiyama–Kishimoto–Kono–Tsukuda, 2007: G(Pk ) ' G(Pk0 ) if and only if (12, k) =(12, k0 ).
• PU(3)-bundles over S 4 .
– Hasui–Kishimoto–Kono–Sato, 2016: G(Pk ) ' G(Pk0 ) if and only if (24, k) = (24, k0 ).
• U(2)-bundles over S 4 .
– Cutler, 2018: G(Pk ) ' G(Pk0 ) if and only if (12, k) = (12, k0 ).
• U(3)-bundles over S 4 .
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2– Cutler, 2018: G(Pk ) ' G(Pk0 ) if and only if (24, k) = (24, k0 ).
Complete classifications over spheres of other dimensions.
• SU(3)-bundles over S 6 :– Hamanaka–Kaji–Kono, 2007: G(Pk ) ' G(Pk0 ) if and only if (120, k) = (120, k0 ).
• Sp(2)-bundles over S 8 .
– Hamanaka–Kaji–Kono, 2008: G(Pk ) ' G(Pk0 ) if and only if (140, k) = (140, k0 ).
Complete classifications over other manifolds.
• SU(2)-bundles over a sphere.
– Claudio–Spreafico, 2009: Classification of the homotopy types of G for B = S n suchthat 5 < n ≤ 25 except n = 21.
• SU(2)-bundles over a simply connected closed 4-manifold B:– Kono–Tsukuda, 1996: G(Pk ) ' G(Pk0 ) if and only if (12, d(B)k) = (12, d(B)k0 ), whered(B) is the parity of the intersection form of B.
• SU(3)-bundles over a simply connected closed spin 4-manifold B:– Theriault, 2012: G(Pk ) ' G(Pk0 ) if and only if (24, k) = (24, k0 ).
Complete local classifications.
• SU(5)-bundles over S 4 : G(Pk )(p) ' G(Pk0 )(p) for all prime p if and only if (120, k) = (120, k0 ).
– Hamanaka–Kono, 2006: The order of [Σ4 CP2 , G(Pk )] is determined by (120, k).
– Theriault, 2013: The order of h, idi is 120.
• Sp(2)-bundles over S 4 :– Sutherland, 1992: If G(Pk )(p) ' G(Pk0 )(p) for all prime p, then (10, k) = (10, k0 ).
– Choi–Hirato–Mimura, 2008: 40h, idi = 0 (the precise order was not determined).
– Theriault, 2010: G(Pk )(p) ' G(Pk0 )(p) for all prime p if and only if (40, k) = (40, k0 ) (theorder of h, idi is 40 and some observation on the homotopy sets).
• PSp(2)-bundles over S 4 .
– Hasui–Kishimoto–Kono–Sato, 2016: G(Pk ) ' G(Pk0 ) for all prime p if and only if(40, k) = (40, k0 ).
• U(5)-bundles over S 4 :– Cutler, 2018: G(Pk )(p) ' G(Pk0 )(p) for all prime p if and only if (120, k) = (120, k0 ).
• SU(3)-bundles over a simply connected closed non-spin 4-manifold B:– Theriault, 2012: G(Pk ) ' G(Pk0 ) for all prime p if and only if (12, k) = (12, k0 ).
• U(n)-bundles over an orientable surface:– Sutherland, 1992: G(Pk ) ' G(Pk0 ) for all prime p if and only if (n, k) = (n, k0 ).
– Theriault, 2011: Homotopy decomposition of G(P) for all prime p.
Rational homotopy types.
• Crabb–Sutherland, 2000: The rationalized gauge group is equivalent to that of the trivial
bundle when G is a connected Lie group.
• Wockel, 2007: Rational homotopy groups of gauge groups for G a Lie group and B a sphereor a surface.
• Félix–Oprea, 2009: Rational homotopy groups of gauge groups for G a Lie group.
Other results on the homotopy types of gauge groups.
• SU(n)-bundles over S 4 .

3– Sutherland, 1992: The order of π2n−4 (G(Pk )) and π2n−2 (G(Pk )) are computed for n > 2.
These results imply that if G(Pk )(p) ' G(Pk0 )(p) for all prime p, then (n(n2 − 1)/(n +1, 2), k) = (n(n2 − 1)/(n + 1, 2), k0 ).
– Hamanaka–Kono, 2006: The order of [Σ2n−6 CP2 , G(Pk )] is 21 (n − 2)!(n(n2 − 1), k)(n −1)!(n, k) (the orders of such groups for k and k0 coincide if and only if (n(n2 − 1), k) =(n(n2 − 1), k0 )).
– Kishimoto–Kono–Tsutaya, 2013: A partial converse of the above results for G = SU(n)of “low rank”.
• SU(n)-bundles over spheres.
– Kishimoto–Kono–Tsutaya, 2014: Some lassification of p-local homotopy types of G
for G = SU(n) of “low rank” and B = S 2d where d = 2, 3, . . . , n.
– Theriault, 2017: Some classification of p-local homotopy types of G for G = SU(n)of “low rank” and B = S 4 . This also bounds the order of the Samelson product S 3 ∧ΣCPn−1 → SU(n).
• SU(n)-bundles over B a simply connected closed spin 4-manifold.
– Sutherland, 1992: The order of π2n−4 (G(Pk )) and π2n−2 (G(Pk )) are computed for n > 2.
These results imply that if G(Pk )(p) ' G(Pk0 )(p) for all prime p, then (n(n2 − 1)/(n +1, 2), k) = (n(n2 − 1)/(n + 1, 2), k0 ).
• Sp(n)-bundles over B a simply connected closed spin 4-manifold.
– Sutherland, 1992: If G(Pk )(p) ' G(Pk0 )(p) for all prime p, then (n(2n + 1), k) = (n(2n +1), k0 ) when n is even and (4n(2n + 1), k) = (4n(2n + 1), k0 ) when n is odd.
• SU(3)-bundles over general spaces.
– Kono–Theriault, 2013: The order of hid, idi is 120.
• Sp(3)-bundles over S 4 .
– Cutler, 2018: G(Pk )(p) ' G(Pk0 )(p) for all prime p if and only if (N, k) = (N, k0 ) whereN = 84, 168 or 336.
• G2 -bundles over S 4 .
– Kishimoto–Theriault–Tsutaya, 2017: G(Pk )(p) ' G(Pk0 )(p) for all prime p if and only if(N, k) = (N, k0 ) where N = 84 or 168.
• Gauge groups over non-orientable surfaces.
– Theriault, 2013: Some homotopy decomposition and applications to the moduli spaceof flat connections.
• U(n)-bundles over CP2 .
– Cutler, 2018: Some classification of the homotopy types and the decompositions of G.
• p-local fibrewise triviality of adjoint bundles.
– Kono–Tsukuda, 2010: Under some assumptions for the base spaces. Explicit applications to SU(2)-bundles over 4-manifolds and to universal bundles.
3. Multiplicative properties of G
General results.
• Finiteness of H-types of gauge groups for G a Lie group and B a finite complex (Crabb–
Sutherland, 2000).
• Finiteness of An -types (n < ∞) of gauge groups for G a Lie group and B a finite complex(Tsutaya, 2012).
• Relation between the splitting as An -spaces and the homotopy commutativity (Kishimoto–Kono, 2010).

4Classification of An -types.
• Crabb–Sutherland 2000: H-types of G(P) for B = S 4 , G = SU(2).
• Tsutaya, 2012, 2012, 2015, 2018:– Some classifications of p-local An -types of G(P) for B = S 4 , G = SU(2).
– Complete classification of fibrewise A3 -types of the adjoint bundles for B = S 4 , G =SU(2).
Homotopy commutativities.
• Crabb–Sutherland, 1992: Homotopy commutativity of SU(2)-bundles over S 4 and someother cases.
• Crabb–Sutherland–Zhang, 1999: Homotopy nilpotency of G.
• Kishimoto–Kono–Theriault 2013: p-local homotopy commutativity of G over S 4 .
• Hasui–Kishimoto–Tsutaya 2019: p-local higher homotopy commutativity of G for P =EG|BnG and bundles over spheres.
4. (Co)homology of G
• Terzić, 2005: rational cohomology of G(P) for B a compact simply connected 4-manifold,G a semisimple compact simply connected Lie group.
• Choi, 2008: mod p homology (Pontryagin ring) of G(P) for B = S 4 , G = SU(n).
• Choi, 2008: mod p homology (Pontryagin ring) of G(P) for B = S 4 , G = G2 .
• Theriault, 2012, mod p (odd prime) homology (Pontryagin ring) of G(P) for B a simply
connected closed 4-manifold, G = SU(n), Sp(n) and Spin(n) except for some cases.
5. Homotopy types of BG ' Map(B, BG) f• Gottlieb, 1972; Atiyah–Bott, 1982: BG(P) ' Map(B, BG) f where f is the classifying mapof P.
• On Map(S 4 , B SU(2)).
– Tsukuda, 1996: If Map(S 4 , B SU(2))k ' Map(S 4 , B SU(2))k0 , then (p, k) = (p, k0 ) for
any prime p.
– Tsukuda, 2001: Map(S 4 , B SU(2))k ' Map(S 4 , B SU(2))k0 if and only if |k| = |k0 |.
– Tsutaya, 2012: Correction of 2-local computation in Tsukuda’s result.
• Map(B, B SU(2)) for B a simply connected closed 4-dimensional manifold.
– Tsukuda, 1996: Map(S 4 , B SU(2))k ' Map(S 4 , B SU(2))k0 implies (k, p) = (k0 , p) for
any prime p (Observations on the k-invariants).
– Kono–Tsukuda, 2000:∗ Map(B, B SU(2))k ' Map(B, B SU(2))k0 if and only if |k| = |k0 | when B admits anorientation-reversing homotopy equivalence.
∗ Map(B, B SU(2))k ' Map(B, B SU(2))k0 if and only if k = k0 when B does not
admit any orientation-reversing homotopy equivalence.
• Map(B, B U(n)) for B a closed orientable surface.
– Sutherland, 1992: Map(B, B U(n))k ' Map(B, B U(n))k+n for any k ∈ Z. If (n, k) =(n, k0 ), then (Map(B, B U(n))k )∧p ' (Map(B, B U(n))k0 )∧p for any prime p.
• Kishimoto–Tsutaya, 2016: There are infinitely many different homotopy types among thepath components of Map(S n , BG) if it has infinitely many path components.
Stable homotopy types.

5• Bauer–Crabb–Spreafico, 2001: Stable homotopy decompositions of BG for G = U(2), SO(3)and B = S 2 .
6. (Co)homology of BG ' Map(B, BG) f
Over 4-dimensional manifolds.
• Masbaum, 1991: Some computations on H ∗ (Map(B, B SU(2))) is done for various coefficients and B a 4-dimensional manifold.
• Tsukuda, 1997: On the cohomology of BG for G = SO(3) and B = S 2 .
• Choi, 1997: Mod 2 and rational homology groups of Map(S 4 , B Sp(n)) and Map∗ (S 4 , B Sp(n))
for all path components (Serre SS of the evaluation fibre sequence collapses).
• Choi–Yoon, 1998: Mod p homology groups of Map∗ (S 4 , B F4 ).
• Terzić, 2005: rational cohomology of BG(P) for B a compact simply connected 4-manifold,G a semisimple compact simply connected Lie group.
• Choi, 2006: Mod p homology group of Map(S 4 , B Sp(n)) and Map∗ (S 4 , B Sp(n)) for some
path components.
Other cases.
• Kaji, 2006: H i (BG; Z) (i ≤ 3) for G a simply connected compact Lie group and B a closed
connected 3-manifold.
7. Other results• Tsukuda, 1997: Study of maps BS 1 → BG(P).
• Tsukuda, 1998: Diffeomorphically isomorphism types of gauge groups.
• Piccinini–Spreafico, 2000: Conjugacy classes in gauge groups.
• Theriault, 2014: Multiplicative decomposition of G for G = U(∞), SU(∞) and B a simply
connected closed 4-manifold or a closed orientable surface.
• West, 2017: Homotopy types of BG∗ , G∗ and G for gauge transformations over real surfaces.
8. Open problems
On the homotopy types of G.
• Does the finiteness by Crabb–Sutherland (and by Tsutaya) also hold for non-connected compact Lie groups?
– What can we say about the rationalizations of gauge groups?
• Is there any case when G(Pk )(p) ' G(Pk0 )(p) for any prime p but G(Pk ) ; G(Pk0 ) integrally?
• Classify the homotopy types of G(Pk ) for principal G-bundles over S 2n` where {n1 ≤ · · · ≤ n` }is the type of a compact connected Lie group G.
– In this setting, the homotopy types are completely determined by the p-local homotopy
types.
– For simply connected simple Lie groups, the cases when G = SU(n) (n ≥ 4), Sp(n)(n ≥ 3), Spin(n) (n ≥ 7), G2 , F4 , E6 , E7 , E8 are still open.
On the homotopy types of BG.
• Does the infiniteness of BG by Kishimoto–Tsutaya also hold p-locally?
Other problems.
• Find any concrete applications of the study of homotopy types of the gauge groups or theirclassifying spaces to gauge theory.

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9. Unordered list
• Kono, 1991 [Kon91]– For G = SU(2) and B = S 4 , G(Pk ) ' G(Pk0 ) if and only if (12, k) = (12, k0 ) (k, k0 ∈π4 (B SU(2))  Z).
• Terzic, 2016 [Ter16]– The Pontryagin ring of the based loop space of gauge groups and their classifying spaces
over Q.
• Cutler, 2018 [Cut18]– For B = S 4 , G = U(n), SU(n) (n ≥ 3) and corresponding principal bundles Pk and P0k(k, k0 ∈ π4 (B U(n))  π4 (B SU(n))  Z), G(Pk )  G(P0k ) × S 1 as topological groups.
– For B = S 4 , G = U(2), SU(2) and corresponding principal bundles Pk and P0k (k, k0 ∈π4 (B U(2))  π4 (B SU(2))  Z), G(P2k )  G(P02k ) × S 1 as topological groups.
– For B = S 4 , G = U(2), PU(2) and corresponding principal bundles Pk and P0k (k, k0 ∈π4 (B U(2))  π4 (B PU(2))  Z), G(P2k+1 )  G(P02k+1 ) × S 1 as topological groups.
– For G = U(2) and B = S 4 , G(Pk ) ' G(Pk0 ) if and only if (12, k) = (12, k0 ) (k, k0 ∈π4 (B U(2))  Z).
– For G = U(3) and B = S 4 , G(Pk ) ' G(Pk0 ) if and only if (24, k) = (24, k0 ) (k, k0 ∈π4 (B U(3))  Z).
– For G = U(5) and B = S 4 , G(Pk )(p) ' G(Pk0 )(p) for any prime p if and only if (120, k) =(120, k0 ) (k, k0 ∈ π4 (B U(5))  Z).
– For G = U(2) and B = CP2 , some classification results are given.
• Choi, 2019 [Cho19]– For G = F4 , B = S 4 and p , 2, 13,
H∗ (G(Pk ); F p )  H∗ (F4 ; F p ) ⊗ H∗ (Ω4 F4 ; F p )as an algebra.
– For G = F4 , B = S 4 and p , 2, 13, the Pontryagin ring H∗ (G(Pk ); F p ) is also determinedwhen k . 0 mod p and p = 2, 13.
• Hasui–Kishimoto–So–Theriault, 2019 [HKST19]– For G exceptional p torsion-free except for (G, p) = (E7 , 5) and B = S 4 , G(Pk )(p) 'G(Pk0 )(p) if and only if min{v p (γ(G)), v p (k)} = min{v p (γ(G)), v p (k0 )}, whereγ(G2 ) = 3 · 7,γ(F4 ) = 52 · 13,γ(E6 ) = 52 · 7 · 13,γ(E7 ) = 7 · 11 · 19,γ(E8 ) = 72 · 112 · 13 · 19 · 31.

• Hasui–Kishimoto–Tsutaya, 2019 [HKT19]– For G simple, G(EnG)(p) is a Sugawara Ck -space if p > (n + k)n` .
• Kishimoto–Kono, 2019 [KK19]– For G = Sp(n) and B = S 4 , G(Pk ) ' G(Pk0 ) implies (4n(2n + 1), k) = (4n(2n + 1), k0 )(k, k0 ∈ π4 (B Sp(n))  Z).
– For G = Sp(n), B = S 4 and (p − 1)2 + 1 ≥ 2n, G(Pk )(p) ' G(Pk0 )(p) if and only ifmin{v p (4n(2n + 1)), v p (k)} = min{v p (4n(2n + 1)), v p (k0 )}.
• Membrillo-Solis, 2019 [MS19]– For simple G and B a S 3 -bundle over S 4 , some classification results are given.
– For G = SU(2) and B = S 7 , G(Pk ) ' G(Pk0 ) if and only if (3, k) = (3, k0 ) (k, k0 ∈π7 (B SU(2))  Z/12Z).

7– For G = SU(3) and B = S 7 , G(Pk )(3) ' G(Pk0 )(3) if and only if (3, k) = (3, k0 ) (k, k0 ∈π7 (B SU(3))  Z/6Z).
– For G = G2 and B = S 7 , G(Pk )(3) ' G(Pk0 )(3) if and only if (3, k) = (3, k0 ) (k, k0 ∈π7 (BG2 )  Z/3Z).
• Mohammadi–Asadi-Golmankhaneh, 2019 [MAG19]– For G = SU(4) and B = S 8 , G(Pk ) ' G(Pk0 ) implies (420, k) = (420, k0 ) (k, k0 ∈π8 (B SU(4))  Z); (3360, k) = (3360, k0 ) implies G(Pk )(p) ' G(Pk0 )(p) .
– Comment: as the authors pointed out, the classification does not depend on whether M
is spin or non-spin.
• So, 2019 [So19]– For G = SU(n) and B a simply-connected closed non-spin 4-manifold, G(Pk ) ' G(Pk0 )implies ( 12 n(n2 − 1), k) = ( 12 n(n2 − 1), k0 ) if n is odd or (n(n2 − 1), k) = (n(n2 − 1), k0 ) if n
is even (k, k0 ∈ [M, B SU(n)]  π4 (B SU(n))  Z).
– The order of the Samelson product S 4 ∧ SU(n) → SU(n) is divisible by 12 n(n2 − 1) if n
is odd or by n(n2 − 1) if n is even.
• So–Theriault, 2019 [ST19]– For G = Sp(2) and B a simply-connected closed 4-manifold, G(Pk )(p) ' G(Pk0 )(p) for
any prime if and only if (40, k) = (40, k0 ) (k, k0 ∈ [M, B Sp(2)]  π4 (B Sp(2))  Z).
– Comment: as the authors pointed out, the classification does not depend on whether M
is spin or non-spin.
• Theriault, 2019 [The19]– For G homotopy commutative and B = ΣX ∪ CA a homotopy cofiber of sum of Whitehead products between suspension spaces, the following principal fibration splits:
Map∗ (ΣA, BG) → Map∗ (B, BG) → Map∗ (ΣX, BG).
– Some applications to gauge groups of this fact are given. Note that BG∗ (P) ' Map(B, BG)αwith the classifying map α : B → BG of P.
• Mohammadi–Asadi-Golmankhaneh, 2020 [MAG20]– For G = SU(n) (n ≥ 3) and B = S 6 , G(P2k ) ' G(P2k0 ) implies ((n − 1)n(n + 1)(n + 2), k) =((n − 1)n(n + 1)(n + 2), k0 ) (k, k0 ∈ π6 (B SU(n))  Z).
• Huang, 2021 [Hua21]– For simple G and B a non-simply-connected closed 5-manifold, some homotopy decompositions are given.
• Kishimoto–Mebrillo-Solis–Theriault, 2021 [KMST21]– For G = (S 3 )n /{±1} and B = S 4 , G(Pk1 ,...,kn )(p) ' G(Pk10 ,...,kn0 )(p) for any prime p if and only
if {(12, k1 ), . . . , (12, kn )} = {(12, k10 ), . . . , (12, kn0 )} as multisets ((k1 , . . . , kn ), (k10 , . . . , kn0 ) ∈π4 (B(S 3 )n /{±1})  Zn ).
– Comment: Their result is the classification of localized homotopy types. But the methodof Kono [Kon91] could improve their result to the one for integral homotopy types.
• Mebrillo-Solis–Theriault, 2021 [MST21]– For G = U(n) (n < p) and B = P2 (p), G(Pk ) ' G(Pk0 ) (k.k0 ∈ [P2 (p), B U(n)]  Z/pZ).
– For G = U(p) and B = P2 (p), G(Pk ) ' G(Pk0 ) if and only if (p, k) = (p, k0 ) (k, k0 ∈[P2 (p), B U(n)]  Z/pZ).
– For G = U(p) (p = 3, 5) and B = L(p, q), G(Pk ) ' G(Pk0 ) if and only if (p, k) = (p, k0 )(k, k0 ∈ [L(p, q), B U(n)]  Z/pZ).
– It also contains partial results on other cases.
• Mohammadi, 2021 [Moh21]

8– For G = PSp(2) and B = S 4 , G(Pk ) ' G(Pk0 ) implies (140, k) = (140, k0 ) (k, k0 ∈π4 (B Sp(2))  Z); (140, k) = (140, k0 ) implies ΩG(Pk ) ' ΩG(Pk0 ).
– For G = PSp(3) and B = S 4 , G(Pk ) ' G(Pk0 ) implies (84, k) = (84, k0 ) (k, k0 ∈π4 (B Sp(3))  Z); (672, k) = (672, k0 ) implies ΩG(Pk )(p) ' ΩG(Pk0 )(p) for any primep.
• Rea, 2021 [Rea21]– For G = PU(5) and B = S 4 , G(Pk )(p) ' G(Pk0 )(p) for any prime p implies (120, k) =(120, k0 ) (k, k0 ∈ π4 (B PU(5))  Z).
– For G = PU(3) and B = S 6 , G(Pk ) ' G(Pk0 ) if and only if (120, k) = (120, k0 ) (k, k0 ∈π6 (B PU(3))  Z).
– It also shows the coincidence of the order of the Samelson products S 2i ∧SU(n) → SU(n)and S 2i ∧ PU(n) → PU(n).
• Takeda, 2021 [Tak21]– For G = U(n) and B = S 2 ,H ∗ (BG(Pk ); Z) = Z[c1 , . . . , cn , x1 , x2 , . . .]/(hn , hn+1 , . . .),where hi = kci +iX(−1) j s j (x1 , . . . , x j )ci− j .
j=1

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