Approximation of continuous gauge transformations

As indicated in Appendix A and Section 5.2, obtaining a good knowledge of the (low-dimensional) homotopy groups of an infinite-dimensional Lie group is an important task. The goal of this sec-tion is to make the homotopy groups of the gauge group more accessible by reducing their compu-tation to the continuous case, i.e., we shall prove thatπ_n(Gau(P)) is isomorphic toπ_n(Gau_c(P)).

Since continuous maps are much more flexible than smooth maps are, this will make the com-putation of the homotopy groups easier, as explained in Chapter 4.

This chapter was mainly inspired by [Ne02a, Section A.3] and [Hi76, Chapter 2].

We first provide the facts on the group of continuous gauge transformations that we shall need later on.

Remark 3.2.1. LetP = (K, π:P→X) be a continuous principal K-bundle. Then the same mapping as in the smooth case (cf. Remark 3.1.2) yields an isomorphism

Gau_c(P)∼=C(P, K)^K:={γ∈C(P, K) :γ(p·k) =k⁻¹·γ(p)·k for all p∈P, k∈K}, andC(P, K)^Kis a topological group as a closed subgroup ofC(P, K)_c. We equip Gau_c(P) with the topology defined by this isomorphism.

LetV = (Vi, σi)_i∈I be a closed continuous trivialising system of P. ThenQ

i∈IC(Vi, K)c is a topological group with

G_c,V(P) :=

(

(γi)i∈I ∈Y

i∈I

C(Vi, K) :γi(m) =kij(m)·γj(m)·kji(m) for all m∈Vi∩Vj

)

as a closed subgroup. Then G_c,V(P)3(γ_i)_i∈I 7→

p7→k_σi(p)⁻¹·γ_i(π(p))·k_σi(p) if p∈π⁻¹(V_i)

∈C(P, K)^K, defines an isomorphism of groups and a straightforward verification shows that this map also defines an isomorphism of topological groups. In exactly the same way one shows that

G_c,V(P) :=

(

(γ_i)_i∈I ∈Y

i∈I

C(V_i, K) :γ_i(m) =k_ij(m)·γ_j(m)·k_ji(m) for all m∈V_i∩V_j )

is also isomorphic toC(P, K)^K as a topological group.

If, in addition, X is compact and (Vi)i∈I also covers X, then there exists a finite subcover (Vi)i=1,...,n of X. Since eachC(Vi, K) is a Lie group [GN07a], the same argumentat as in the proof of Proposition 3.1.8 shows thatC(P, K)^K, with the subspace-topology fromC(P, K)c, can be turned into a Lie group.

We collect some concepts and facts from general topology that we shall use throughout this chapter.

Remark 3.2.2. IfX is a topological space, then a collection of subsets (Ui)i∈I of X is called locally finiteif eachx∈Xhas a neighbourhood that has non-empty intersection with only finitely manyU_i, andX is calledparacompact if each open cover has a locally finite refinement. IfX is the union of countably many compact subsets, then it is called σ-compact, and if each open cover has a countable subcover, it is calledLindel¨of.

Now let M be a finite-dimensional manifold with corners, which is in particular locally com-pact and locally connected. For these spaces, [Du66, Theorems XI.7.2+3] imply thatM is para-compact if and only if each component is σ-compact or, equivalently, Lindel¨of. Furthermore, [Du66, Theorem VIII.2.2] implies thatM is normal in each of these cases.

Remark 3.2.3. If Ui

i∈Iis a locally finite cover ofM by compact sets, then for fixedi∈I, the intersectionU_i∩U_jis non-empty for only finitely manyj∈I. Indeed, for everyx∈U_i, there is an open neighbourhoodU_x ofxsuch thatI_x:={j∈I:U_x∩U_j6=∅}is finite. SinceU_i is com-pact, it is covered by finitely many of these sets, say byU_x1, . . . , U_xn. ThenJ :=I_x1∪. . .∪I_xn is the finite set of indicesj∈J such thatU_i∩U_j is non-empty, proving the claim.

We now start the business of approximating continuous maps by smooth ones. In the case of functions with values in locally convex spaces, this is quite easy.

Proposition 3.2.4. IfM is a finite-dimensionalσ-compact manifold with corners, then for each locally convex space Y the space C^∞(M, Y)is dense in C(M, Y)_c. Iff ∈C(M, Y)has compact support and U is an open neighbourhood ofsupp(f), then each neighbourhood of f in C(M, Y)_c contains a smooth function whose support is contained inU.

Proof. The proof of [Ne02a, Theorem A.3.1] carries over without changes.

Corollary 3.2.5. If M is a finite-dimensional σ-compact manifold with corners and V is an open subset of the locally convex spaceY, thenC^∞(M, V)is dense inC(M, V)_c.

Proof. Since each open subset ofC(M, V)_c is also open inC(M, Y)_c, this follows immediately from the previous proposition.

We are now aiming for a similar statement for gauge transformations. In order to do so, we need to localise the smoothing process from Proposition 3.2.4. This means to organise an inductive smoothing process in a way that

• at each step, we smooth the function on a region, where it takes values in an open subset ofK, which is diffeomorphic to an open convex zero neighbourhood ofk

• when doing so, we should not vary the function in a region, where it is already smooth.

The following lemma provides the tool for this “localised” smoothing process.

Lemma 3.2.6. Let M be a finite-dimensional σ-compact manifold with corners, Y be a locally convex space,W ⊆Y be open and convex and letf :M →W be continuous. IfL⊆M is closed andU ⊆M is open such that f is smooth on a neighbourhood ofL\U, then each neighbourhood of f inC(M, Y)_c contains a continuous map g:M →W, which is smooth on a neighbourhood of Land which equalsf on M\U.

Proof. (cf. [Hi76, Theorem 2.5]) Let A⊆M be an open set containingL\U such thatf A is smooth. ThenL\A⊆U is closed inM so that there existsV ⊆U open with

L\A⊆V ⊆V ⊆U

Then {U, M\V} is an open cover of M, and there exists a smooth partition of unity {f₁, f₂} subordinated to this cover. Then

Gf:C(M, W)c→C(M, Y)c, Gf(γ)(x) =f1(x)γ(x) +f2(x)f(x) is continuous sinceγ7→f1γ andf1γ7→f1γ+f2f are continuous.

If γis smooth on A∪V then so is Gf(γ), becausef1 andf2 are smooth, f is smooth on A and f2

_V ≡0. Note that L⊆A∪(L\A)⊆A∪V, so thatA∪V is an open neighbourhood of L. Furthermore we have Gf(γ) =γ on V and Gf(γ) =f onM\U. Since Gf(f) =f, there is for each open neighbourhoodO off an open neighbourhoodO⁰ off such thatGf(O⁰)⊆O. By the preceding Corollary there is a smooth functionh∈O⁰ such thatg:=G_f(h) has the desired properties.

We first aim for a generalisation of the previous lemma to functions with values in a locally convex Lie groupK. Note that we used a convexity argument in the proof of the previous lemma, showing that the local convexity ofKwill be crucial for the generalisation to work.

Lemma 3.2.7. Let M be a finite-dimensional σ-compact manifold with corners, K be a Lie group, W ⊆K be diffeomorphic to an open convex subset of k and f :M →W be continuous.

If L⊆M is closed andU ⊆M is open such thatf is smooth on a neighbourhood ofL\U, then each neighbourhood of f inC(M, W)_c contains a map which is smooth on a neighbourhood ofL and which equalsf on M\U.

Proof. Letϕ:W →ϕ(W)⊆kbe the postulated diffeomorphism. If bK1, V1c ∩. . .∩ bKn, Vnc is an open neighbourhood of f ∈C(M, K)c, where we may assume that Vi ⊆W, then bK1, ϕ(V1)c ∩. . .∩ bKn, ϕ(Vn)c is an open neighbourhood of ϕ◦f in C(M, ϕ(W))c. We ap-ply Lemma 3.2.6 to this open neighbourhood to obtain a maph. Thenϕ⁻¹◦hhas the desired properties.

Proposition 3.2.8. Let M be a connected paracompact finite-dimensional manifold with cor-ners,K be a Lie group andf ∈C(M, K). IfL⊆M is closed andU ⊆M is open such thatf is smooth on a neighbourhood ofL\U, then each open neighbourhood O off inC(M, K)_c contains a map g, which is smooth on a neighbourhood of Land equalsf onM\U.

Proof. We recall the properties of the topology on M from Remark 3.2.2. If f is smooth on the open neighbourhood A of L\U, then there exists an open set A⁰⊆M such that L\U ⊆A⁰⊆A⁰⊆A. We choose an open cover (Wj)_j∈J of f(M), where each Wj is an open subset of K diffeomorphic to an open zero neighbourhood of k and set Vj:=f⁻¹(Wj). Since eachx∈M has an open neighbourhoodVx,j with Vx,j compact andVx,j ⊆Vj for some j∈J, we may redefine the cover (Vj)_j∈J such thatVj is compact andf(Vj)⊆Wj for allj ∈J.

SinceM is paracompact, we may assume that the cover (Vj)_j∈Jis locally finite, and sinceM is normal, there exists a cover (V_i⁰)i∈Isuch that for eachi∈Ithere exists aj∈Jsuch thatV_i⁰⊆Vj. SinceM is also Lindel¨of, we may assume that the latter is countable, i.e.,I=N⁺:={1,2, . . .}.

Hence M is also covered by countably many of the V_j and we may thus assume V_i⁰⊆V_i and f(V_i)⊆W_i for eachi∈N⁺ Furthermore we set V₀:=∅ andV₀⁰ :=∅. Observe that both covers are locally finite by their construction. Define

Li:=L∩V_i⁰\(V₀⁰∪. . .∪V_i−1⁰ )

which is closed and contained in Vi. Since L\A⁰ ⊆U we then haveLi\A⁰ ⊆Vi∩U and there exist open subsetsUi⊆Vi∩U such thatLi\A⁰⊆Ui⊆Ui ⊆Vi∩U. We claim that there exist functionsgi ∈O,i∈N⁰, satisfying

gi =gi−1 on M\Ui for i >0, g_i(V_j)⊆W_j for all i, j∈N0and

gi is smooth on a neighbourhood of L0∪. . .∪Li∪A⁰.

For i= 0 we have nothing to show, hence we assume that the gi are defined for i < a. We consider the set

Q:={γ∈C(Va, Wa) :γ=ga−1onVa\Ua}, which is a closed subspace ofC(Va, Wa)c. Then the map

F :Q→C(M, Wa), F(γ)(x) =

γ(x) if x∈U_a g_a−1(x) if x∈M\Ua

is continuous sinceUa is closed. Note that, by induction,g_a−1(Va)⊆Wa, whence g_a−1|_V

a ∈Q.

SinceFis continuous andF(g_a−1|_V

a) =g_a−1, there exists an open setO⁰ ⊆C(Va, Wa) containing g_a−1|_V

a such thatF(O⁰∩Q)⊆O.

Since (Vj)j∈N0 is locally finite andVj is compact, the set{j∈N⁰:Ua∩Vj6=∅}is finite and hence

O⁰⁰=O⁰∩ \

j∈N0

bUa∩V_j, W_jc

is an open neighbourhood of g_a−1|_V

a in C(V_a, W_a)_c by induction. We now apply Lemma 3.2.7 with to the manifold with corners Va, the closed set L⁰_a:= (L∩V_a⁰)∪(A⁰∩Va)⊆Va, the open setUa⊆Va, g_a−1|_V

a∈Q⊆C(Va, Wa) and the open neighbourhood O⁰⁰of g_a−1|_V

a. Due to the construction we haveLa\Ua⊆A⁰∩Va andL∩V_a⁰⊆L0∪. . .∪La. Hence we have

L⁰_a\Ua⊆(L0∪. . .∪L_a−1∪(La\Ua))∪(A⁰∩Va\Ua)⊆L1∪. . .∪L_a−1∪(A⁰∩Va) so that g_a−1|_V

a is smooth on a neighbourhood ofL⁰_a\Ua. We thus obtain a maph∈O⁰⁰which is smooth on a neighbourhood ofL⁰_a and which coincides with ga−1|_V

a onVa\Ua⊇Va\Ua, hence is contained inO⁰⁰∩Q, and we set g_a:=F(h). Since h(U_a∩V_j)⊆W_j and g_a−1(V_j)⊆W_j, we haveF(h)(Vj)⊆Wj. FurthermoreF(h) inherits the smoothness properties fromg_a−1onM\Ua, fromhonVaand sinceLa ⊆L∩V_a⁰, it has the desired smoothness properties onM. This finishes the construction of thegi.

We now constructg. First we setm(x) := max{i:x∈Vi}andn(x) := max{i:x∈Vi}. Then obviouslyn(x)≤m(x) and eachx∈M has a neighbourhood on whichg_n(x), . . . , g_m(x) coincide sinceUi⊆Viandgi=g_i−1onM\Ui. Henceg(x) :=g_n(x)(x) defines a continuous function onM. Ifx∈L, thenx∈L0∪. . .∪L_n(x) and thusg is smooth on a neighbourhood ofx. Ifx∈M\U, thenx /∈U1∪. . .∪U_n(x) and thusg(x) =f(x).

To make the following technical proofs more readable, we first introduce some notation.

Remark 3.2.9. In the remaining section, multiple lower indices on subsets ofM always indicate intersections, namelyU_1···r:=U₁∩. . .∩U_r.

The following technical Lemma will make the smoothing process work.

Lemma 3.2.10. Let M be a manifold with corners that is covered locally finitely by countably many compact sets Ui

i∈N. Moreover, letkij:Uij →Kbe continuous functions into a Lie group K so that kij =k⁻¹_ji holds for all i, j∈N. Then for any convex centred chart ϕ:W →ϕ(W) of K, each sequence of open unit neighbourhoods (W_j⁰)_j∈N with W_j⁰ ⊆W and each α∈N, there areϕ-convex open unit neighbourhoodsW_ij^α⊆W inK for indicesi < j andW_j^α⊆W_j⁰ forj∈N that satisfy

kji(x)·(W_ij^α)⁻¹·W_i^α·kij(x)⊆W_j^α for all x∈Uijαandi < j, (3.3) kji(x)·(W_ij^α)⁻¹·W_in^α·kij(x)⊆W_jn^α for all x∈Uijnα andi < j < n (3.4) Proof. Initially, we setW_i^α:=W_i⁰for alli, respectivelyW_ij^α:=W for alli < j, disregarding the conditions (3.3) and (3.4). These sets are shrinked later so that they satisfy (3.3) and (3.4).

Our first goal is to satisfy (3.3). We note that the condition in (3.3) becomes trivial ifUjα is empty, because this impliesUijα=∅. So we need to consider at most finitely many conditions on (3.3) corresponding to the finitely many j∈N such that Ujα6=∅, and we deal with those inductively in decreasing order ofj, starting with the maximal such index.

For fixed j and alli < j with Uijα6=∅, we describe below how to make the ϕ-convex unit neighbourhoodsW_ij^αandW_i^α on the left hand side smaller so that the corresponding conditions (3.3) are satisfied. Making W_ij^α and W_i^α smaller does not compromise any conditions on W_ij^α0

and W_j^α0 for j⁰ > j that we guaranteed before, because these sets can only appear on the left hand side of such conditions.

To satisfy condition (3.3) for giveni < j andW_j^α, we note that the function ϕ_ij :U_ijα×K×K→K, (x, k, k⁰)7→k_ji(x)·k⁻¹·k⁰·k_ij(x)

is continuous and maps all the points (x, e, e) for x∈U_ijα to e. Hence we may choose open neighbourhoods U_x of x and ϕ-convex open unit neighbourhoods W_x⊆W_ij^α and W_x⁰ ⊆W_i^α

such that ϕ_ij(U_x×W_x×W_x⁰)⊆W_j^α. Since U_ijα is compact, it is covered by finitely many Ux, say by (Ux)_x∈F for a finite set F ⊆Uijα. Then we replaceW_ij^α and W_i^α by their subsets T

x∈FWxand T

x∈FW_x⁰, respectively, which areϕ-convex open unit neighbourhoods inK such thatϕij(Uijα×W_ij^α×W_i^α)⊆W_j^α, in other words, (3.3) is satisfied

Our second goal is to make the setsW_ij^αalso satisfy (3.4), which is non-trivial for the finitely many triples (i, j, n)∈N³withi < j < nthat satisfyU_ijnα6=∅. We can argue as above, except for a slightly more complicated order of processing the setsW_jn^α on the right hand side. Namely, we define the following total order

(i, j)<(i⁰, j⁰) :⇔ j < j⁰ or (j=j⁰ andi < i⁰) (3.5) on pairs of real numbers, in particular on pairs of indices (i, j) inN×N withi < j. Note that this guarantees (i, j)<(j, n) and (i, n)<(j, n) whenever i, j, n are as in condition (3.4). We process the pairs (j, n) withUijnα6=∅for someiin descending order, starting with the maximal such pair. At each step, we fixW_jn^α and makeW_ij^αandW_in^α smaller for all relevanti < j so that (3.4) is satisfied. This does not violate any conditions (3.3) or (3.4) that we guaranteed earlier in the process, because W_ij^α and W_in^α can only appear on the left hand side of such conditions.

For the choice of the smaller unit neighbourhoods, we use the continuous function ϕ_ijn:U_ijnα×K×K→K, (x, k, k⁰)7→k_ji(x)·k⁻¹·k⁰·k_ij(x)

and the compactness ofUijnα and argue as before. We thus accomplish our second goal.

We are now ready to prove the generalisation of Proposition 3.2.4. This proposition is the first hint that the spacesC(P, K)^K andC^∞(P, K)^K are topologically closely related.

Proposition 3.2.11. If P is a smooth principal K-bundle over the connected, paracompact finite-dimensional manifold with cornersM, thenGau_c(P) is dense inGau(P).

Proof. Let (Uj)j∈Jbe a trivialising open cover ofM. Proposition 3.3.3 yields locally finite open covers U_i^[λ]

i∈N ofM for everyλ∈ {0,∞} ∪ 1 +¹₃N

such that the closuresU^[λ]_i are compact manifolds with corners and

U^[∞]_i ⊆U_i^[j+1]⊆U^[j+1]_i ⊆U_i^[j+2/3]⊆U^[j+2/3]_i ⊆U_i^[j+1/3]⊆U^[j+1/3]_i ⊆U_i^[j]⊆U_i^[0]⊆U^[0]_i ⊆Ui

holds for all i, j∈N, where Ui denotes a suitable set of the cover (Uj)_j∈J for every i∈N. Furthermore , let

kij:U^[0]_ij →K

be a the transition functions of a fixed cocycle arising from the trivialising cover. By Remark 3.2.1, we may identify Gauc(P) with

G^[∞](P) :={(γi)_i∈N∈Y

i∈N

C(U^[∞]_i , K) :γi(x) =kij(x)·γj(x)·kji(x) for all x∈U^[∞]_ij } or with

G^[0](P) :={(γi)_i∈N∈Y

i∈N

C(U^[0]_i , K) :γi(x) =kij(x)·γj(x)·kji(x) for all x∈U^[0]_ij}, and eachγ= (γ_i)_i∈N∈G^[∞](P) is given by the restriction of some uniquely determined element ofG^[0](P).

Let ϕ:W →ϕ(W)⊆kbe a convex centred chart of K. Then a basic open neighbourhood of (γ_i)_∈N in G^[∞](P) is given by

{(γ_i⁰)_i∈N∈G^[∞](P) : (γ⁰_i·γ_i⁻¹)(U^[∞]_i )⊆W_i for all i≤m} (3.6)

for open unit neighbourhoodsW_i⊆W. Then

kji(x)·kij(x) =e∈Wj for all x∈U^[0]_ij and i < j≤m

and a compactness argument as in Lemma 3.2.10 yields open unit neighbourhoodsW_i⁰⊆Kwith kji(x)·W_i⁰·kij(x)⊆Wj for all x∈U^[0]_ij and i < j≤m (3.7) Fori≥m, we setW_i⁰ =W. We shall inductively construct smooth mapseγi:U^[0]_i →Ksuch that

(a) eγj=kji·eγi·kij pointwise onU^[j]_ij for alli < j ∈N, (b) eγi·γ_i⁻¹

U^[i]_iα

⊆W_i^αfor alli, α∈N and (c) (eγi·γ_i⁻¹)(U^[∞]_i )⊆Wi for alli≤m

are satisfied at each step, where theW_i^αareϕ-convex unit neighbourhoods provided by Lemma 3.2.10 that we apply to the countable compact cover U^[0]_i

i∈N, to the transition functions k_ij, and to (W_i⁰)_i∈N. Then (eγi|

U^[∞]_i )_i∈N is an element of G^[∞](P), contained in the basic open neighbourhood (3.6) and thus establishes the assertion.

To construct the smooth functionγe1:U^[0]₁ →K, we apply Proposition 3.2.8 to the continuous mapf :=γ₁ onM :=A:=U :=U^[0]₁ and to the open neighbourhood

O1:=

bU^[1]₁ , W1c ∩ \

α∈N

j

U^[1]_1α, W₁^αk

·γ1

of γ₁, which is indeed open, since only finitely many U^[1]_1α are non-empty. By construction, eγ₁ satisfies (b) and (c). To construct the smooth function eγj :U^[0]_j →K inductively for j >1, we need three steps:

• In order to satisfy (b) in the end, we define a map

eγ⁰_j: [

i<j

U^[j−1]_ij →K, eγ⁰_j(x) :=kji(x)·eγi(x)·kij(x) forx∈U^[j−1]_ij .

Ifxis an element of bothU^[j−1]_ij andU^[j−1]_i0j fori⁰ < i < j, condition (a) forj−1 and the cocycle condition assert that the so-defined values foreγ_j⁰(x) agree.

• This definition ofeγ⁰_j, along with properties (a), (b) and (3.3) assert that ϕ_j(x) :=eγ_j⁰(x)·γj(x)⁻¹=k_ji(x)·eγ_i(x)·kij(x)·γj(x)⁻¹=k_ji(x)·eγ_i(x)·γ_i(x)⁻¹

| {z }

∈W_i^α

·kij(x)∈W_j^α holds for all x∈U^[j−1]_ijα , i < j and αin N. Furthermore, (3.7) ensures that if j≤m, we have

ϕ_j(x) =k_ji(x)·eγ_i(x)·γ_i(x)⁻¹

| {z }

∈W_i^α⊆W_i⁰

·kij(x)∈W_j

for x∈U^[j−1]_ij and all i < j. So we may apply Lemma 3.3.1 to A:=S

i<jU^[j−1]_ij and B :=S

i<jU^[j−2/3]_ij to fade outϕj to a continuous map Φj onM :=U^[0]_j . Then Φjcoincides withϕ_i onB, mapsU^[j]_jα intoW_j^αand ifj≤malsoU^[j−1]_j intoW_j.

• Accordingly, Φ_j·γ_j is an element of the open neighbourhood Oj:= j

U^[j−1]_j , Wj

k∩ \

α∈N

j

U^[j]_jα, W_j^αk

·γj

of γ_j and is smooth on S

i<jU_ij^[j−2/3]. If we apply Proposition 3.2.8 to M :=A:=U^[0]_j , U :=M \S

i<jU^[j−1/3]_ij , Oj, and to f := Φj·γj, then we obtain a smooth map eγj:U^[0]_j →K.

The mapeγjsatisfies (a), because so doesγe_j⁰, with which it coincides onS

i<jU^[j]_ij. Moreover, (b) and (c) are satisfied due to the choice ofOj. This concludes the construction.

In combination with the fact that C^∞(P, K)^K is dense inC(P, K)^K, the following fact will provide the isomorphismπn(C^∞(P, K)^K)∼=πn(C(P, K)^K), which we are aiming for.

Lemma 3.2.12. Let P be a smooth principal K-bundle over the compact base M, having the property SUB with respect to the smooth closed trivialising system V = (Vi, σi)i=1,...,n and let ϕ:W →W⁰ be the corresponding convex centred chart of K (cf. Definition 3.1.7). If (γi)i=1,...,n∈G_V(P) represents an element of C^∞(P, K)^K (cf. Remark 3.1.6), which is close to identity, in the sense that γi(Vi)⊆W, then (γi)i=1,...,n is homotopic to the constant map (x7→e)i=1,...,n.

Proof. Since the map ϕ_∗:U :=G_V(P)∩

n

Y

i=1

C^∞(V_i, W)→g(P), (γ_i⁰)_i=1,...,n7→(ϕ◦γ_i⁰)_i=1,...,n, is a chart ofG_V(P) (cf. Proposition 3.1.8) andϕ∗(U)⊆g_V(P) is convex, the map

[0,1]3t7→ϕ⁻¹_∗ t·ϕ_∗((γi)i=1,...,n)

∈G_V(P) defines the desired homotopy.

We finally obtain the main theorem of this section.

Theorem 3.2.13 (Weak homotopy equivalence for Gau(P)). Let P be a smooth princi-pal K-bundle over the compact manifold M (possibly with corners). IfP has the property SUB, then the natural inclusion ι: Gau(P),→Gauc(P) of smooth into continuous gauge transforma-tions is a weak homotopy equivalence, i.e., the induced mappings πn(Gau(P))→πn(Gauc(P)) are isomorphisms of groups forn∈N⁰.

Proof. We identify Gau(P) with C^∞(P, K)^K and Gau_c(P) with C(P, K)^K. To see that πn(ι) is surjective, consider the continuous principal K-bundle pr^∗(P) obtained form P by pulling it back along the projection pr :Sⁿ×M →M. Then pr^∗(P) is isomorphic to (K,id×π,Sⁿ×P,Sⁿ×M), where K acts triv-ially on the first factor of Sⁿ×P. We have with respect to this action C(pr^∗(P), K)^K ∼=C(Sⁿ×P, K)^K and C^∞(pr^∗(P))^K ∼=C^∞(Sⁿ×P, K)^K. The isomor-phism C(Sⁿ, G0)∼=C_∗(Sⁿ, G0)oG0=C_∗(Sⁿ, G)oG0, where C_∗(Sⁿ, G) denotes the space of base-point-preserving maps from Sⁿ to G, yields πn(G) =π0(C_∗(Sⁿ, G)) =π0(C(Sⁿ, G0)) for any topological groupG. We thus get a map

π_n(C^∞(P, K)^K) =π₀(C_∗(Sⁿ, C^∞(P, K)^K)) =

π0(C(Sⁿ, C^∞(P, K)^K₀))→^η π0(C(Sⁿ, C(P, K)^K₀)),

whereη is induced by the inclusionC^∞(P, K)^K ,→C(P, K)^K.

If f ∈C(Sⁿ×P, K) represents an element [F]∈π₀(C(Sⁿ, C(P, K)^K₀)) (recall C(P, K)^K∼=G_c,V(P)⊆Qn

i=1C(V_i, K) andC(Sⁿ, C(V_i, K))∼=C(Sⁿ×V_i, K)), then there exists fe∈C^∞(Sⁿ×P, K)^K which is contained in the same connected component of C(Sⁿ×P, K)^K asf (cf. Proposition 3.2.11). Since feis in particular smooth in the second argument, it follows thatferepresents an element Fe∈C(Sⁿ, C^∞(P, K)^K). Since the connected components and the arc components of C(Sⁿ×P, K)^K coincide (since it is a Lie group, cf. Remark 3.2.1), there exists a path

τ: [0,1]→C(Sⁿ×P, K)^K₀

such that t7→τ(t)·f is a path connecting f and fe. Since Sⁿ is connected it follows that C(Sⁿ×P, K)^K₀∼=C(Sⁿ, C(P, K)^K)0⊆C(Sⁿ, C(P, K)^K₀). Thus τ represents a path in C(Sⁿ, C(P, K)^K₀ )) connecting F and Fe whence [F] = [F]e ∈π0(C(Sⁿ, C(P, K)^K₀)). Thatπn(ι) is injective follows with Lemma 3.2.12 as in [Ne02a, Theorem A.3.7].

This theorem makes the homotopy groups of gauge groups accessible in terms of constructions for continuous mappings. This will be done in Chapter 4.

Im Dokument Infinite-Dimensional Lie Theory for Gauge Groups (Seite 34-42)