This paper studies the automorphism groups of Cuntz-Toeplitz algebras, focusing on their homotopy groups, providing new insights into their topological structure and symmetries.
Contribution
It determines the homotopy groups of automorphism groups of Cuntz-Toeplitz algebras, a novel topological analysis of these operator algebra automorphisms.
Findings
01
Computed the homotopy groups of automorphism groups
02
Identified topological invariants of automorphisms
03
Enhanced understanding of Cuntz-Toeplitz algebra symmetries
Abstract
The Cuntz-Toeplitz algebra En+1 for n≥1 is the universal C*-algebra generated by n+1 isometries with mutually orthogonal ranges. In this paper, we investigate the automorphism groups of the Cuntz-Toeplitz algebras and determine their homotopy groups.
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Full text
The homotopy groups of the automorphism groups of Cuntz-Toeplitz algebras
The Cuntz-Toeplitz algebra En+1 for n≥1 is the universal C*-algebra generated by n+1 isometries with mutually orthogonal ranges. In this paper, we determine the homotopy groups of the automorphism group of En+1.
1 Introduction
The Cuntz-Toeplitz algebra En+1 for n≥1 is the universal C*-algebra generated by n+1 isometries with mutually orthogonal ranges. In this paper, we investigate the automorphism groups of the Cuntz-Toeplitz algebras and determine their homotopy groups.
The homotopy groups of the automorphism groups are necessary to classify the continuous fields of C*-algebras. However, there are only few classes of C*-algebras whose homotopy groups of the automorphism groups are determined.
To the best knowledge of the author’s, the homotopy groups are known only for Kirchberg algebras [6, 12, 4], strongly self-absorbing C*-algebras [9], and simple AF-algebras [22, 30]. The rough strategy of computation of the homotopy groups in the previous work is as follows.
First, we show the weak homotopy equivalence between the automorphism group and the endomorphism semi-group.
Then we compute the homotopy groups of the endomorphism semi-group from the K-theoretic or KK-theoretic data of the C*-algebra.
We illustrate the strategy in the case of Kirchberg algebras where we have a powerful tool, Kirchberg-Phillips’ classification theorem.
Regarding a continuous map ρ:X→End(A⊗K) as an element in Hom(A⊗K,C(X)⊗A⊗K), we can associate a KK-class KK(ρ)∈KK(A,C(X)⊗A) to ρ.
Therefore the homotopical data of End(A⊗K) are recovered from the KK-theoretic data, and we can directly compute the general homotopy sets by the map [X,AutA⊗K]→KK(A,C(X)⊗A) (see [6, Proposition 5.8, Theorem 4.6]).
In general, there is no such powerful tool and the homotopy groups are computed for only exceptional classes of C*-algebras.
The strongly self-absorbing C*-algebras are such examples. Dadarlat and Pennig show in [9, Theorem 2.3] that the automorphism group AutD is contractible for every unital strongly self-absorbing C*-algebra D.
Using a certain fibration, they determine the homotopy groups of Aut(D⊗K).
In this paper, we use similar fibrations in the case mentioned above.
Let {Ti}i=1n+1 be the canonical generators of En+1 and let End0En+1 be the path component of idEn+1 of the semi-group of unital endomorphisms of En+1.
We denote by e the minimal projection 1−∑i=1n+1TiTi∗.
Then the map End0En+1∋ρ↦∑i=1n+1ρ(Ti)Ti∗∈UEn+1(1−e) is a homeomorphism where UEn+1 is the unitary group of En+1.
Our proof of the main result is based on the the fact that the map UEn+1→UEn+1(1−e) defined by the right multiplication by 1−e gives a fibration with a fibre S1.
Theorem 1.1**.**
The homotopy groups of AutEn+1 are as follows :
[TABLE]
To prove Theorem 1.1, we show that the inclusion map AutEn+1→End0En+1 is a weak homotopy equivalence (Theorem 3.14).
Corollary 1.2**.**
Let X be a compact CW complex.
The following sequence is an exact sequence of pointed sets and the first 4 terms give an exact sequence of groups :
[TABLE]
The group AuteEn+1 is the subgroup of all automorphisms that fix the minimal projection e∈En+1.
The original motivation of this work is to investigate the structure of continuous fields of the Cuntz algebras beyond Dadarlat’s work [10] using the Cuntz-Toeplitz extensions, and we will hopefully come back to this subject in the near future.
We discuss the group structure of the homotopy sets [X,AutEn+1] and [X,AutOn+1] in [16].
We organize this paper as follows.
In section 2, we give some preliminaries to compute the homotopy groups.
We introduce several fibration sequences with the help of [9, Lemma 2.8, 2.16 and Corollary 2.9].
As a consequence, the homotopy groups of the connected component of the endomorphism semi-group, denoted by End0En+1, are obtained.
In section 3, we show the weak homotopy equivalence of End0En+1 and AutEn+1. The main ingredient of the proof is Pimsner–Popa–Voiculescu’s non-commutative Weyl–von Neumann type theorem.
2 Preliminaries
2.1 Notation and the basic facts of the theory of C*-algebras
Let A be a unital C*-algebra and let UA be the group of unitary elements in A.
We denote by U0A the path component of 1A of UA.
For a non-unital C*-algebra B, we denote its unitization by B∼.
The K-groups of A are denoted by Ki(A), i=0,1.
We denote by [p]0 the class of the projection p in K0(A), and denote by [u]1 the class of the unitary u in K1(A).
Let SA be the suspension of A, the set of A-valued functions on [0,1] that vanish at [math] and 1.
For the K-theory, we refer to [1, 17].
For a topological space Y and two elements y0 and y1, we denote y0∼hy1 in Y if there is a continuous path from y0 to y1.
Two unitaries u,v∈UA are homotopy equivalent if u∼hv in UA.
There is a natural map UA/∼h→K1(A) from the set of homotopy classes of unitaries to the K1-group.
We say A is K1-injective if the map is injective. For the non-unital C*-algebra B, it is K1-injective if the natural map UB∼/∼h→K1(B) is injective.
For example, the algebra A⊗K is K1-injective by the definition of the K1-group.
We denote by K the algebra of compact operators of infinite dimensional separable Hilbert space H.
For A⊗K, we denote by M(A⊗K) the multiplier algebra of A⊗K, and denote by Q(A⊗K) its quotient by A⊗K.
We denote the quotient map by π.
We remark that Q(A⊗K) is K1-injective,
(see [21, Section 1.13]).
We identify M(K) with B(H) where B(H) is the algebra of the bounded operators on H.
For A=C(X), we denote by Cs∗b(X,B(H)) the set of B(H)-valued bounded continuous functions on X with respect to the strong* operator topology (abbreviated to SOT*).
This is a realization of the multiplier algebra M(C(X)⊗K)
(see [29, Proposition 2.57]).
We refer to [13, Theorem 1] for the K-theory of the multiplier algebra, and a generalization of Kuiper’s theorem.
Theorem 2.1**.**
Let A be a unital C-algebra. Then UM(A⊗K) is contractible with respect to the norm topology, and we have
Ki(M(A⊗K))=0,i=1,2.*
Let A, B and C be C*-algebras. An extension C of A by B⊗K is an exact sequence
[TABLE]
and the Busby invariant of the extension is the induced map τ:A→Q(B⊗K). We refer to [1] for the definition of the Busby invariant.
The extension is called trivial if the above exact sequence splits.
The extension is called essential if τ is injective, and called unital if τ is unital.
We refer to [1] for the basic facts of the theory of extensions of C*-algebras. There are two equivalence relations of unital extensions, the strong unitary equivalence and the weak unitary equivalence.
Definition 2.2**.**
Let A and B be C-algebras.
Two Busby invariants τi:A→Q(B⊗K), i=1,2 are said to be strongly unitarily equivalent if there exists a unitary U∈UM(B⊗K) satisfying τ1=Adπ(U)∘τ2. They are said to be weakly unitarily equivalent if there exists a unitary u∈UQ(B⊗K) with τ1=Adu∘τ2.
We denote the strong unitary equivalence by ∼s.u.e and denote the weak unitary equivalence by ∼w.u.e.
We denote τ1∼sτ2 if there exists two trivial extensions ρ1 and ρ2 satisfying τ1⊕ρ1∼s.u.eτ2⊕ρ2.
We denote by Ext(A,B⊗K) the set of the equivalence classes of the Busby invariants with respect to the equivalence relation ∼s.*
We note that the weak unitary equivalence, ∼w.u.e induces the equivalence ∼s(see [1, Proposition 5.6.4]).
In this paper, we deal with the extensions of the Cuntz algebras by C(X)⊗K.
One has a universal coefficient theorem of Ext-groups.
Let A and B be separable C-algebras, with A in the bootstrap class.
Then there is an unnaturally splitting short exact sequence*
[TABLE]
If ⨁i=0,1Hom(Ki(A),Ki+1(B))=0, we have an isomorphism
Ext(A,B⊗K)→⨁i=0,1ExtZ1(Ki(A),Ki(B))
that sends a class of Busby invariant [τ] of an extension 0→B⊗K→Cτ→A→0 to the class of group extension of the commutative groups [Ki(B)→Ki(Cτ)→Ki(A)]∈ExtZ1(Ki(A),Ki(B)) for i=0,1.
Let En+1 be the universal C*-algebra generated by n+1 isometries with mutually orthogonal ranges and let {Ti}i=1n+1 be the canonical generator of En+1.
It is called the Cuntz-Toeplitz algebra.
The closed two-sided ideal generated by the minimal projection e:=1−∑i=1n+1TiTi∗ is isomorphic to the compact operators K, which is known to be the only closed non-trivial two-sided ideal.
Consider the full Fock space F(Cn+1) and the left creations {Ti}i=1n+1 (see [26, Section 1]).
Then one has K⊂C∗({Ti}i=1n+1)=En+1⊂B(F(Cn+1)).
In this paper, we frequently identify K∼ with K+C1En+1⊂B(F(Cn+1)).
Let π:En+1→On+1 be the quotient map by the ideal K, and let Si:=π(Ti).
The quotient algebra On+1 is the universal simple C*-algebra generated by n+1 isometries with the relation :Sj∗Si=δij,1=∑i=1n+1SiSi∗.
We denote by O∞ the universal C*-algebra generated by the countably infinite isometories with mutually orthogonal ranges.
The algebras On+1 and O∞ are called the Cuntz algebras, whose K-groups are the following :
The Cuntz algebras are the Kirchberg algebras, and they tensorially absorb O∞, On+1⊗O∞≅On+1.
The algebra that tensorially absorbs O∞ has K1-injectivity by the lemma below.
Let A be a unital C-algebra. Then the natural map UA⊗O∞/∼h→K1(A⊗O∞) is bijective.
In particular, every unital C*-algebra that tensorially absorbs O∞ is K1-injective.*
Definition 2.5**.**
We denote by τ0 the Busby invariant of the extension
[TABLE]
The inclusion map C(X)⊗K↪C(X)⊗En+1 induces the Busby invariant τ=idC(X)⊗τ0:C(X)⊗On+1↪Q(C(X)⊗K) of the unital essential extension
[TABLE]
Since K and En+1 are KK-equivalent to C (see [26, Theorem 4.4]) the above exact sequence induces the following 6-term exact sequence :
[TABLE]
For a pointed topological space (X,x0), we denote by ΣX its reduced suspension with the base point x0.
For pointed topological spaces (X,x0),(Y,y0), we denote the set of the continuous maps from X to Y by Map(X,Y) and denote the set of base point preserving continuous maps by Map0(X,Y). We denote the homotopy set Map(X,Y)/∼h by [X,Y] and denote Map0(X,Y)/∼h by [X,Y]0. We remark that if Y is an H-space, the natural map [X,Y]0→[X,Y] is bijective.
Lemma 2.6**.**
Let (X,x0) be a based compact Hausdorff space. Then the natural map
[TABLE]
is a surjective isomorphism.
Proof.
Since K1(On+1)=0, we have
K1(C0(X,x0)⊗On+1)=K1(C(X)⊗On+1).
By Lemma 2.4, the natural map
[TABLE]
is an isomorphism.
Since UOn+1 is an H-space, we have
Let A be a C-algebra and let I be a two-sided closed ideal of A.
If A/I and I are K1-injective and the natural map US(A/I)∼→K1(S(A/I)) is surjective, then A is K1-injective.*
We refer to [28] for the surjectivity, the properly infinite full projections and the properly infiniteness of the C*-algebras.
Let A be a unital C-algebra, and let p and q be properly infinite full projections. Then there exists a partial isometry v with p=vv∗,q=v∗v, if and only if [p]0=[q]0 in K0(A).*
We show that the algebra C(X)⊗En+1 is K1-injective.
Proposition 2.10**.**
Let X be a compact Hausdorff space. Then, the map
[TABLE]
is an isomorphism.
Proof.
Surjectivity follows from the fact that C(X)⊗En+1 is properly infinite and Lemma 2.8. We identify SC0(X,x0)⊗On+1 with C0(ΣX,x0)⊗On+1. Since C(X)⊗On+1 is K1-injective by Lemma 2.4 and C(X)⊗K is K1-injective, it is sufficient to prove the surjectivity of the natural map U(SC(X)⊗On+1)∼→K1(SC(X)⊗On+1).
For the space Y:=[0,1]×X/({0}×X⊔{1}×X), we have SC(X)=C0(Y,y0). So we have the conclusion by Lemma 2.7.
∎
Let EndEn+1 be the semi-group of unital ∗-endomorphisms of En+1.
We topologize EndEn+1 by the point-wise norm topology, and let End0En+1 be the path component of idEn+1 in EndEn+1.
We denote by EndeEn+1 (resp. AuteEn+1) the subset of EndEn+1 (resp. AutEn+1) consisting of all elements fixing the minimal projection e.
Every automorphism of En+1 preserves the ideal of compact operators and induces an automorphism of On+1.
For α in AutEn+1, we denote by α~ the induced automorphism of On+1. This gives a group homomorphism AutEn+1→AutOn+1.
Lemma 2.11**.**
The set End0En+1 equals to a subset {ρ∈EndEn+1∣ρ(e)isaminimalprojectionofK} of EndEn+1, and the map
[TABLE]
is a homeomorphism.
Proof.
First, we show End0En+1⊂{ρ∈EndEn+1∣ρ(e):minimalprojection} because the converse is trivial. Consequently, the map End0En+1∋ρ↦uρ∈UEn+1(1−e) is well-defined.
If ρ(e) is a minimal projection, there exists a partial isometry v with vv∗=ρ(e),v∗v=e. Then the unitary v+uρ is in the path component of 1En+1 by the K1-injectivity of En+1. We take a norm continuous path of unitaries {ut}t∈[0,1] in UEn+1 from v+uρ to 1En+1, and we have the continuous path ρt:Ti↦utTi from ρ to idEn+1.
Second, we show the map End0En+1∋ρ↦uρ∈UEn+1(1−e) is a homeomorphism.
For every w∈UEn+1(1−e), we have the map ρw:Ti↦wTi by the universality of En+1.
The map UEn+1(1−e)∋w↦ρw∈End0En+1 is continuous because {Ti}i=1n+1 is the generator of En+1.
This gives the inverse of the map End0En+1∋ρ↦uρ∈UEn+1(1−e).
∎
2.2 Section algebras and the theory of extensions of C*-algebras
We use the following elementary fact.
Lemma 2.12**.**
Let A be a unital C-algebra, and let X be a compact metrizable space. Let P1 and P2 be principal AutA bundles over X. Let A1 and A2 be the section algebras of the associated bundles of P1 and P2 with fibre A respectively.
Then P1 and P2 are isomorphic if and only if there exists a C(X)-linear isomorphism φ:A1→A2.*
Let π:M(C(X)⊗K)→Q(C(X)⊗K) be the quotient map by the ideal C(X)⊗K.
We need the following technical theorem of the theory of extensions of C*-algebras.
Let X be a finite CW complex.
Let A be a separable simple unital C-algebra, and let μ:A→M(C(X)⊗K) and σ:A→Q(C(X)⊗K) be unital ∗-homomorphisms. Then σ⊕π∘μ and σ are strongly unitarily equivalent.*
The theorem above is a special case of [27, Theorem 2.10]. Since A is simple, the assumptions for it are satisfied.
Lemma 2.14**.**
Let X be a finite CW complex.
Let A be a separable simple unital C-algebra.
Suppose that A has a unital essential trivial extension π∘μ where μ:A↪M(C(X)⊗K) is a unital embedding.
Then two unital essential extensions τ1 and τ2 are weakly unitarily equivalent if and only if [τ1]=[τ2]inExt(A,C(X)⊗K).*
Proof.
We show that [τ1]=[τ2] implies τ1∼w.u.eτ2 as the other implication is always the case.
By definition, there exists a trivial extension π∘ρi such that τ1⊕π∘ρ1∼s.u.eτ2⊕π∘ρ2.
Adding π∘μ to the both side, we may assume that ρi(1A) is a properly infinite full projection in M(C(X)⊗K).
Since K0(M(C(X)⊗K))=0 and ρi(1A) is properly infinite full, there exists an isometry Vi with ViVi∗=ρi(1A).
Now we have τ1⊕π∘(AdV1∗∘ρ1)∼w.u.eτ2⊕π∘(AdV2∗∘ρ2).
It follows from Theorem 2.13 that τi∼s.u.eτi⊕π∘(AdVi∗∘ρi), and we have the conclusion.
∎
We have the following theorem of Paschke and Valette.
Let A and B be unital separable C-algebras, and assume that A is nuclear. Let μ:A→M(K) be a unital embedding with μ(A)∩K={0}. For the unital ∗-homomorphism τ:=π(1B⊗μ), we have an isomorphism*
[TABLE]
which sends the class of a unitary u∈τ(A)′∩Q(B⊗K) to the class of extension
[TABLE]
The following theorem holds from the argument of [24, Section 1, Theorem 1.5].
Let τ1 and τ2 be unital extensions of On+1 by K. Then τ1∼s.u.eτ2 if and only if τ1∼hτ2.
Proposition 2.17**.**
Let X be a compact Hausdorff space with Tor(K0(C(X)),Zn)=0, and let σ:On+1→Q(C(X)⊗K) be an arbitrary unital extension.
Then every element of K1(σ(On+1)′∩Q(C(X)⊗K)) is a n-torsion element, and the set U(σ(On+1)′∩Q(C(X)⊗K)) is contained in the path component of 1 of UQ(C(X)⊗K).
Proof.
By Theorem 2.3, all elements of Ext(SOn+1,C(X)⊗K) are n-torsion elements.
We define τ:=π∘(1C(X)⊗μ) where μ:On+1→M(K) is a unital embedding.
Since On+1 is simple, we have μ(On+1)∩K={0}. By Theorem 2.15, we have
[TABLE]
So we have [σ⊕n]=n[σ]=[τ]=0, and Lemma 2.14 gives a unitary w∈UQ(C(X)⊗K) with σ⊕n=Adw∘τ. We have an isomorphism
[TABLE]
We also have (σ(On+1)′∩Q(C(X)⊗K))⊗Mn≅(σ⊕n(On+1)′∩Q(C(X)⊗K)).
So we have
[TABLE]
Since K0(C(X))=K1(Q(C(X)⊗K)) has no n-torsion, we have [w]1=0∈K1(Q(C(X)⊗K)) for every w∈U(σ(On+1)′∩Q(C(X)⊗K)).
So we have the conclusion by K1-injectivity of Q(C(X)⊗K) (see [21, Section 1.13]).
∎
As an application of Lemma 2.14, we show in Proposition 2.22 that the group AutEn+1 is path connected.
The straightforward computation yields the lemma below.
Lemma 2.18**.**
We have the following isomorphisms of K-groups and Ext-groups :
Let τ0:On+1→Q(K) be the Busby invariant in Definition 2.5.
If a unitary u in UM(K) commutes with En+1 up to compact operators (i.e. [u,d]∈K for every d in En+1), there exists a self adjoint element h in Q(K) such that e2πih=π(u) and [h,a]=0 for every a in On+1.
Corollary 2.20**.**
The group N:={u∈UM(K)∣[u,En+1]⊂K} is path connected.
Lemma 2.21**.**
Let α be an automorphism of En+1, and Uα:=∑iα(ei1)ve1i be an implementing unitary of α↾K where v is a partial isometry with vv∗=α(e),v∗v=e.
Then we have α=AdUα↾En+1
Proof.
We show AdUα↾En+1=α.
Let F⊂K be the set of all finite rank projections.
Since α is an automorphism, the image α(K)=K contains a net {α(p)}p∈F that weakly converges to 1.
For every d∈En+1, we have α(p)α(d)=α(pd)=AdUα(pd)=α(p)AdUα(d), and α=AdUα↾En+1 holds.
∎
Proposition 2.22**.**
The group AutEn+1 is path connected.
Proof.
Let α be an automorphism of En+1 and let α~ be an induced automorphism of On+1. Since AutOn+1 is path connected, we take a path ht with h0=α~,h1=idOn+1.
We take two unital essential extensions
[TABLE]
[TABLE]
where τ0 is the Busby invariant in Definition 2.5, and we regard h:C[0,1]⊗On+1→C[0,1]⊗On+1 as a C[0,1]-linear isomorphism.
Since τ1∼hτ2, we have [τ1]=[τ2] in Ext(C[0,1]⊗On+1,C[0,1]⊗K).
We have [τ1(1C[0,1]⊗idOn+1)]=[τ2(1C[0,1]⊗idOn+1)] in Ext(On+1,C[0,1]⊗K).
By Lemma 2.14, there exists a unitary v∈UQ(C[0,1]⊗K) with τ2(1C[0,1]⊗idOn+1)=Adv∘τ1(1C[0,1]⊗idOn+1).
Since C[0,1] is in the center of Q(C[0,1]⊗K), we have τ2=Adv∘τ1.
We show [v]1=0 in K1(Q(C[0,1]⊗K)). By the construction of τ2 and v, the unitary v1 is in τ0(On+1)′∩Q(K). By Proposition 2.17, we have [v1]1=0 in K1(Q(K)). Since the map ev1∗:K1(Q(C[0,1]⊗K))→K1(Q(K)) is an isomorphism from Lemma 2.18, we have [v]1=0.
We take a unitary lift V∈UM(C[0,1]⊗K) of v.
It follows that AdV is a C[0,1]-linear isomorphism of C[0,1]⊗En+1.
Therefore the map [0,1]∋t↦AdVt∈AutEn+1 is continuous. Let Uα:=∑iα(ei1)ve1i be an implementing unitary of α restricted to K where v is a partial isometry satisfying vv∗=α(e11),v∗v=e11, and {eij} is a system of matrix units.
By Lemma 2.21, we have AdUα↾En+1=α.
We have Adπ(V0)=h0=α~=Adπ(Uα), and it follows that V0∗Uα commutes with En+1 up to compact operators.
By Corollary 2.20, the automorphism AdV0∗Uα is in the path component of idEn+1 in AutEn+1. Similary there is a continuous path from idEn+1 to AdV1 in AutEn+1. Therefore we have
[TABLE]
∎
2.3 Implementing unitaries of AutC(X)(C(X)⊗En+1)
Let AutC(X)(C(X)⊗En+1) be the group of C(X)-linear automorphisms of C(X)⊗En+1. We remark that the homotopy set [X,AutEn+1] is identified with the set of homotopy equivalence classes of the elements of AutC(X)(C(X)⊗En+1).
Let G be a compact topological group. We denote by BG its classifying space, and denote by EG the universal principal G-bundle over BG.
One realization of those spaces is as follows.
For a contractible space X equipped with a free G action, the quotient map X→X/G gives the universal bundle.
We refer to [14] for the basic facts about the classifying spaces.
Let H1 be the set vectors of norm 1 in a separable Hilbert space with the norm topology.
We identify H1 with the set {f∈L2[0,1]∣∣∣f∣∣2=1}.
There is a map ht:H1×[0,1]→H1 that sends (f,t) to (1[0,t]f+1[t,1])/∣∣1[0,t]f+1[t,1]∣∣2 where 1[a,b] is the characteristic function of [a,b].
This gives the deformation retraction to the set {1[0,1]}, and the space H1 is contractible, (see [29]).
The group S1 freely acts on H1 by the scalar multiplication.
Therefore we can adopt H1 as a model of ES1.
We identifies BS1 with the set consisting of all minimal projections, and the map ES1=H1∋ξ↦ξ⊗ξ∗∈BS1 gives the universal bundle where we denote by ξ⊗η∗ the operator H∋x↦⟨x,η⟩ξ∈H for ξ,η∈H.
The space BS1 is the Eilenberg-Maclane space K(Z,2) and we identifies the homotopy set [X,BS1] with H2(X) via the Chern classes of the line bundles.
Proposition 2.23**.**
Let X be a compact Hausdorff space, and let α:X→AutEn+1 be a continuous map. Let η be the map AutEn+1∋α↦α(e)∈BS1.
If the image of [α] by the map [X,AutEn+1]η∗[X,BS1] is zero, then there exists a unitary U in UM(C(X)⊗K) such that AdUx=αx.
Proof.
Let ξ0 be a norm 1 eigenvector corresponding to the minimal projection e.
By assumption, there exists a norm continuous section ξ:X→H1 with ξx⊗ξx∗=αx(e).
Using a system of matrix units {1C(X)⊗eij} with e=e11:=ξ0⊗ξ0∗,
we have a unitary Ux:=∑iαx(ei1)ξx⊗ξ0∗e1i.
Since ξx is norm continuous, U:X∋x↦Ux∈M(K) is SOT-continuous.
In particular, U:X→UM(K) is SOT*-continuous and U∈UM(C(X)⊗K).
Lemma 2.21 shows AdUx↾En+1=αx for every x∈X.
∎
Lemma 2.24**.**
Let X be a compact Hausdorff space and let α be an element of Map(X,AutOn+1). Then the map α:C(X)⊗En+1→C(X)⊗En+1 induces the identity map of the K-groups, Ki(α)=idKi(C(X)⊗En+1), i=1,2.
Proof.
Since α is C(X)-linear, we have the commutative diagram below :
[TABLE]
We have the conclusion from the KK-equivalence of C and En+1.
∎
Let r:AutEn+1→AutK be the restriction map. Then we have a commutative diagram below
[TABLE]
We remark that the map η:AutK→BS1 gives the homotopy equivalence (see [9, Lemma 2.8]).
Lemma 2.25**.**
The map η∗:[Sk,AutEn+1]→[Sk,BS1] is the zero map for k≥1.
Hence the map r∗:[Sk,AutEn+1]→[Sk,AutK] is also zero.
Proof.
If k=2, we have [Sk,BS1]=H2(Sk)=0.
We show the statement in the case of k=2. For every α in Map(S2,AutEn+1), the map K0(α):K0(C(S2)⊗En+1)→K0(C(S2)⊗En+1) is the identity by Lemma 2.24. Since the map K0(C(S2)⊗K)−nK0(C(S2)⊗En+1) is injective, [e]0=[α(e)]0 in K0(C(S2)⊗K). The group K~0C(S2) is generated by the Bott element, the class of the tautological line bundle (see [1, Section 9.2.10]), and the tautological line bundle is also a generator of H2(S2). Therefore we have η∗([α])=0 in H2(S2).
∎
2.4 Some fibration sequences
In this section, we introduce several fibrations to compute the homotopy groups of AutEn+1.
We refer to [5, Chap. 6] for the definition and the basic facts about fibrations.
Definition 2.26**.**
Let X,Y and Z be the topological spaces, and let π:X→Y be the continuous map.
The map π has the homotopy lifting property (abbreviated to HLP) for Z, if for every commuting diagram
[TABLE]
*there exists a continuous map g~:[0,1]×Z→X such that g~(0,z)=g(z) for every z in Z and π∘g~=f.
The map π:X→Y is a Serre fibration, if π has HLP for every n-disc, Dn.*
We remark that a Serre fibration has HLP for every CW complex. A fibration gives a long exact sequence of homotopy sets. We denote by ΩX or Ωx0X the loop space of the pointed set (X,x0).
Theorem 2.27**.**
Let (Z,z0) be a pointed CW complex.
Let π:(X,x0)→(Y,y0) be a Serre fibration with the fibre F:=π−1(y0). Then, there is a long exact sequence of groups (i≥1), and exact sequence of pointed sets (i≥0)
[TABLE]
In particular, we have the long exact sequence of the homotopy groups in the case of Z={z0}.
Let (X,x0),(Y,y0) be the pointed topological spaces. Then the natural map [ΣX,Y]0→[X,ΩY]0 is a bijection.
By the theorem of Hurewicz, every principal G-bundle is a fibration. Therefore we use the long exact sequence to compute the homotopy groups of the topological group G. We refer to the argument in [9, Lemma 2.8, 2.16, Corollary 2.9] for the proof of the following 4 lemmas.
Lemma 2.29**.**
Let p:UEn+1→UEn+1(1−e) be the multiplication by 1−e. Then, the map p is a principal S1-bundle that has the S1 action by the right multiplication of (1−e)+ze, z∈S1.
Lemma 2.30**.**
Let H1 be the set of vectors of norm 1 of the Hilbert space H, and ξ0∈H1 be a vector corresponding the minimal projection e. The map q:UEn+1→H1 that sends a unitary u to uξ0 is a fibration with the fiber U(1−e)En+1(1−e).
Remark 2.31**.**
Since H1 is contractible, it is follows from the long exact sequence of the homotopy groups induced by the fibration of Lemma 2.30 that the map
[TABLE]
is the weak homotopy equivalence. Hence the map EndeEn+1∋ρ↦e+∑iρ(Ti)Ti∗∈UEn+1 is the weak homotopy equivalence.
Lemma 2.32**.**
Let η:End0En+1→BS1 be the map that sends α to α(e) and let AuteEn+1 be the stabilizer subgroup of the minimal projection e. Then there is a principal AuteEn+1-bundle
[TABLE]
Remark 2.33**.**
Since the map η∗:[Sk,AutEn+1]→[Sk,BS1] is the zero map by Lemma 2.25, it follows from Lemma 2.32 that for every α in Map(Sk,AutEn+1), there exists α′ in Map(Sk,AuteEn+1) that is homotopic to α in Map(Sk,AutEn+1).
Lemma 2.34**.**
The following sequence gives a fibration :
[TABLE]
Remark 2.35**.**
In section 3, we show that the map AutEn+1→End0En+1 is a weak homotopy equivalence.
Hence the groups AuteEn+1 and EndeEn+1 are weakly homotopy equivalent from the long exact sequences and 5-lemma.
Then the map AuteEn+1∋α↦e+∑iα(Ti)Ti∗∈UEn+1 is the weak homotopy equivalence by the Remark 2.31.
By the fibration in Lemma 2.29, we know the homotopy groups of End0En+1.
Theorem 2.36**.**
The homotopy groups of End0En+1 are as follows:
[TABLE]
[TABLE]
Proof.
By Lemma 2.11, it is sufficient to compute the homotopy groups of UEn+1(1−e). By the fibration sequence
[TABLE]
we have the long exact sequence of the homotopy groups
[TABLE]
The map S1→UEn+1 sends a complex number z to a unitary 1−e+ze.
We have [Sk,UEn+1]0=[Sk,UEn+1]=K1(Sk) by K1-injectivity of C(Sk)⊗En+1. The map
[TABLE]
is the multiplication by −n, and so we have the conclusion.
∎
We remark that a generator of π1(End0En+1)=Zn is the canonical gauge action of S1 that is λz:Ti↦zTi for every z∈S1.
3 The main result
3.1 The homotopy groups of AutEn+1
In this section, by using the theory of extensions, we show that the inclusion map AutEn+1→End0En+1 is a weak homotopy equivalence.
First, we show [S2m,AutEn+1] is trivial, for m≥1.
Second, we show the surjectivity of the map [S2m−1,AutEn+1]→[S2m−1,End0En+1] for m≥1.
Finally, we show the injectivity of the map.
Let X be a compact Hausdorff space.
It is well-known in homotopy theory that every principal AutEn+1 bundle P over X comes from the classifying map X→BAutEn+1 [14, Section 4, Proposition 10.6].
So we identifies the equivalence class of a principal bundle [P] with the homotopy equivalence class of its classifying map and denote [P]∈[X,BAutEn+1].
For a bundle P, the section algebra of the associated bundle P×AutEn+1En+1 is a locally trivial continuous C(X)-algebra Γ(X,P×AutEn+1En+1).
Let k be a natural number.
For every α∈Map(Sk,AutEn+1), there is a principal AutEn+1-bundle Pα representing the class [α] in [Sk,AutEn+1]≅[Sk+1,BAutEn+1].
We construct a continuous field of En+1 over Sk+1 corresponding to Pα as follows.
We denote the interior of the closed k+1-disc by D∘k+1.
We view Sk+1 as a non-reduced suspension of Sk, that is, D∘k+1∪Sk∪D∘k+1, and view α a clutching function on Sk of two trivial bundles over D∘k+1∪Sk and Sk∪D∘k+1.
By the following lemma, we have [Sk,AutEn+1]=[Sk+1,BAutEn+1].
Let G be a path connected group. Let X be a topological space, and let SX be its non-reduced suspension. Then the map
[TABLE]
is bijective.
Definition 3.2**.**
We identify the section algebra of Pα×AutEn+1En+1 with the following algebra :
[TABLE]
and denote by Cα the essential ideal
[TABLE]
Let Aα be the quotient algebra of Bα by Cα :
[TABLE]
The algebra Aα is isomorphic to the section algebra Γ(Sk+1,Pα~×AutOn+1On+1), where α~ is the induced map in Map(Sk,On+1).
We remark that C(Sk+1) is identified with the algebra
[TABLE]
which is the center of Bα.
Since the map [Sk,AutEn+1]→[Sk,AutK] is zero map by Lemma 2.25, the associated bundle Pα×AutEn+1K is trivial. We fix a trivialization and obtain θα:Cα→C(Sk+1)⊗K. Thus we get a unital essential extension τθα
[TABLE]
where the isomorphism θα:Cα→C(Sk+1)⊗K depends on the trivialization of the bundle Pα×AutEn+1K.
Lemma 3.3**.**
Let m≥1 be a natural number. Then we have [S2m,AutEn+1]=0.
Proof.
Since [S2m,AutOn+1]=0 by [12, Theorem 7.4], there is a trivialization φα:C(S2m+1)⊗On+1→Aα that is C(S2m+1)-linear isomorphism for every α∈Map(S2m,AutOn+1). Consider two extensions of On+1 by C(S2m+1)⊗K :
[TABLE]
[TABLE]
where the map τ0 is the Busby invariant in Definition 2.5.
It follows from the construction that [evpt∘σα]=[evpt∘σ] in Ext(On+1,K). By Lemma 2.18, we have [σα]=[σ] in Ext(On+1,C(S2m+1)⊗K), and Lemma 2.14 yields that there exists a unitary w in UQ(C(S2m+1)⊗K) satisfying Adw∘σ=σα. There is another unitary U in UM(K) with Adπ(U)∘evpt∘σ=evpt∘σα by Theorem 2.16.
By Proposition 2.17, we have [evpt(w)]1=[evpt(w)π(U∗)]1=0 in K1Q(K), and Lemma 2.18 yields that [w]1=0. Therefore we have a unitary W that is a lift of w, and the map AdW:C(S2m+1)⊗En+1→θα(Bα) is a C(S2m+1)-linear isomorphism. From Lemma 2.12, the bundle Pα is isomorphic to the trivial bundle, and we have [α]=0 in [S2m,AutEn+1] by Lemma 3.1 and Proposition 2.22.
∎
Lemma 3.4**.**
Let m≥1 be a natural number. Then the map
[TABLE]
is surjective.
Proof.
Let τ be the map idC(S2m−1)⊗τ0 where τ0 is the Busby invariant in Definition 2.5.
We show that for every γ∈Map(S2m−1,AutOn+1) there exists a lift Γ∈Map(S2m−1,AutOn+1) with Γ~x=γx for every x∈S2m−1.
We recall the notation that Γ~x is an induced automorphism of On+1 from Γx.
For every γ in Map(S2m−1,AutOn+1), we regard γ as an element of AutC(S2m−1)(C(S2m−1)⊗On+1), and there are two extensions of On+1
[TABLE]
[TABLE]
For every x∈S2m−1, the map γx is homotopic to idOn+1 in AutOn+1 because AutOn+1 is path connected by [12, Theorem 1.1].
Hence we have evx∘σγ∼s.u.eevx∘σ by Theorem 2.16 because evx∘σγ∼hevx∘σ.
From Lemma 2.18, we have [σγ]=[σ] in Ext(On+1,C(S2m−1)⊗K), and σγ∼w.u.eσ by Lemma 2.14.
We have two unitaries v∈Q(C(S2m−1)⊗K) and V∈M(K) satisfying σγ=Adv∘σ and evpt∘σγ=Adπ(V)∘evpt∘σ.
So we have [evpt(v)]1=[π(V)∗evpt(v)]1=0 by Proposition 2.17, and Lemma 2.18 yields [v]1=0∈K1Q(C(S2m−1)⊗K).
Therefore we have σγ∼s.u.eσ, and there is a unitary Uγ∈UM(C(S2m−1)⊗K) with
[TABLE]
We have the following commutative diagram
[TABLE]
The map Γ:S2m−1∋x↦Ad(Uγ)x∈AutEn+1 is continuous and it is a lift of the map γ.
∎
For α′ in Map(S2m−1,AuteEn+1) with m≥1, we take the map θα′ as follows.
Let Uα′ be a unitary in UM(C(S2m−1)⊗(1−e)K(1−e)) of the form Uα′=∑i=1α′(1C(S2m−1)⊗ei1)(1C(S2m−1)⊗e1i) where {eij} is a system of matrix units with e11=e. By Theorem 2.1, there is a norm continuous path from 1−e to Uα′ in UM(C(Sk)⊗(1−e)K(1−e)). Adding the projection 1C(S2m−1)⊗e to the path, we have a norm continuous path U∈C[0,1]⊗M(C(S2m−1)⊗K) satisfying
[TABLE]
where we write 1C(S2m−1)⊗e simply by e.
We define two C(S2m)-algebras M and Mα′ :
[TABLE]
[TABLE]
The algebras M and Mα′ are C(S2m)-linearly isomorphic to M(C(S2m)⊗K) and we identify M with M(C(S2m)⊗K).
Definition 3.5**.**
We define a map θα′:Mα′→M by the C(S2m)-linear isomorphism
[TABLE]
The algebras Bα′ and Cα′ defined in Definition 3.2 are subalgebras of Mα′.
We denote by l the constant map S2m−1→{idEn+1} and denote by l~ the induced map S2m−1→{idOn+1}.
If α′ is homotopic to l in Map(S2m−1,EndeEn+1), then [α′~]=0 in [S2m−1,AutOn+1] because of [S2m−1,EndOn+1]=[S2m−1,AutOn+1] by [12, Proposition 6.1], and there is a trivialization φα′:C(S2m)⊗On+1→Aα′ .
We can explicitely construct φα′ from the homotopy between α′~−1 and l~.
Definition 3.6**.**
Let α′ be an element of Map(S2m−1,AuteEn+1) homotopic to l in Map(S2m−1,EndeEn+1), and let ht:[0,1]×S2m−1→AutOn+1 be a path from l~=h0 to α′~−1=h1. Then we define the map φα′ as a C(S2m)-linear isomorphism of the form
[TABLE]
where C(S2m)⊗On+1 is identified with the algebra
[TABLE]
The map τθα′∘φα′ is the Busby invariant of a unital essential extension of C(S2m)⊗On+1. The following lemma says that τα′∘φα′∼w.u.eτ=idC(S2m)⊗τ0 where τ0 is the Busby invariant in Definition 2.5.
Lemma 3.7**.**
Let m≥1 be a natural number and let α′ be an element of Map(S2m−1,AuteEn+1) which is homotopic to l in Map(S2m−1,EndeEn+1). Let τα′∘φα′ and τ be as above. Let i:C(S2m)∋(f1,f2)↦(f1,f2)∈Bα′ be the canonical unital embedding and let j:C(S2m)⊗K⊂θα′(Bα′) be the inclusion map. Then the following hold.
(1)* (θα′∘i)∗:K0(C(S2m))≅K0(θα′(Bα′)).*
We denote g1:=(θα′∘i)∗([1C(S2m)]0) and g2:=(θα′∘i)∗(b1), where b1 is a generator of K0C(S2m).
(2)* We have j∗([1C(S2m)⊗e]0)=−ng1, and there exists a generator b2∈K0(C(S2m)⊗K) with j∗(b2)=−ng2.
In particular, it follows that [τα′∘φα′]=[τ] in Ext(C(S2m)⊗On+1,C(S2m)⊗K).*
Proof.
We identify the sphere S2m with the space (D∘2m∪S2m−1∪D∘2m) where D∘2m is the interior of the 2m-disc, and we identify C0(D∘2m) with the algebra {F∈C0[0,1)⊗C(S2m−1)∣F(0)∈C1C(S2m−1)}.
Let x0∈S2m−1 be the base point of S2m and S2m−1.
The map ι0:C0(D∘2m)∋F↦(F,0)∈C0(S2m,x0) induces an isomorphism of K-groups.
An element b1 is the generator of K0(C0(Sk+1,x0)).
Let ι:(C0(D∘2m)⊗En+1)⊕2∋(F1,F2)↦(F1,F2)∈Bα′ be an embedding, and let r:Bα′∋(F1,F2)↦F1(1)∈C(S2m−1)⊗En+1 be the restriction map.
First, we show (1).
We have the following commutative diagram
[TABLE]
From the KK-equivalence of En+1 and C, the vertical maps (idC0(D∘2m)⊗1En+1)⊕2 and idC(S2m−1)⊗1En+1 induce isomorphisms of K-groups.
Therefore the map K0(i):Ki(C(S2m))→Ki(Bα′) is an isomorphism by 6-term exact sequences and the 5-lemma.
Second, we find b2.
We denote by ι1 the inclusion C0(D∘2m)⊗En+1∋F1↦(F1,0)∈(C0(D∘2m)⊗En+1)⊕2.
We consider the following commutative diagram
[TABLE]
Since θα′∘ι∘ι1=ι0⊗idK and K0(ι0) is an isomorphism of K-groups, from diagram chasing we can find a generator b2′∈K0(C0(D∘2m)⊗K) that sent to −nb1∈K0(C0(S2m,x0)) by the map K0(θα′)−1∘K0(j)∘K0(θα′∘ι∘ι1). Hence we have b2:=K0(θα′∘ι∘ι1)(b2′).
Third, we show j∗([1⊗e]0)=−ng1.
From the assumptions, there exists the map h′:[0,1]×Sk→EndeEn+1 with h1′=α′, h0′=l. We have the unital ∗-homomorphism
[TABLE]
which sends (e,e)∈Bα′ to (e,e)∈Bl=C(S2m)⊗En+1.
We have θα′−1(j(1C(S2m)⊗e))=(e,e)∈Bα′ and (e,e)=1C(S2m)⊗e∈Bl=C(S2m)⊗En+1, and the following commutative diagram holds
[TABLE]
We have
[TABLE]
Since i∗ is an isomorphism, the map η∗ is also an isomorphism, and we have [j(1C(Sk+1)⊗e)]0=−n[1θα′(Bα′)]0=−ng1.
Finally, we show that [τα′∘φα′]=[τ].
By Theorem 2.3, we identify Ext(C(S2m)⊗On+1,C(S2m)⊗K) with ExtZ1(K0(C(S2m)⊗On+1),K0(C(S2m)⊗K)) because k is an odd number. The element [τα′∘φα′] is identified with the class of extension
[TABLE]
By the computation above, it is equal to the class
[Z⊕2−nZ⊕2→Zn⊕2]=[τ].
∎
We have the Busby invariants of two extensions of On+1 by C(S2m)⊗K :
[TABLE]
From the lemma above, we have
[TABLE]
in Ext(On+1,C(S2m)⊗K).
Hence there exists a unitary wα′ in UQ(C(S2m)⊗K) satisfying σα′=Adwα′∘σ by Lemma 2.14.
For a unital essential extension ν:On+1→Q(C(S2m)⊗K), we take a unitary Vν introduced in [24, Section 1] :
[TABLE]
We denote (S1,⋯,Sn+1) by S. We claim that, for the above σ, we have ind([Vσ]1)=−[1C(S2m)⊗e]0∈K0(C(S2m)⊗K).
Indeed, there is a unitary lift
[TABLE]
of Vσ⊕Vσ∗,
where T=(T1,⋯,Tn+1) and the element w is of the form 1C(S2m)⊗w0 and w0:H→H⊕n+1 is a unitary operator.
The element w∈Mn+1,1(M(C(S2m)⊗K)) is a partial isometry with ww∗=1n+1 and w∗w=1 in Mn+1(M(C(S2m)⊗K)).
From direct computation of the index map, we have ind([Vσ]1)=−[1C(S2m)⊗e]0∈K0(C(S2m)⊗K). Direct computation yields
[TABLE]
Hence we have [Vσα′]1−[Vσ]1=−n[wα′]1 in K1(Q(C(S2m)⊗K)).
We show [wα′]1=0∈K1(Q(C(S2m)⊗K)) in Theorem 3.11,
and we need the following three lemmas for that.
Recall the path h:[0,1]×S2m−1→AutOn+1 from l~=h0 to α′~−1=h1 in Definition 3.6.
Here and subsequently, we write a unitary ∑i=1n+1h(1⊗Si)1⊗Si∗∈U(C([0,1]×S2m−1)⊗On+1) by v where we denote 1C([0,1]×S2m−1)⊗Si by 1⊗Si for simplicity.
We denote
[TABLE]
By the definition of τθα′ and φα′, the following lemma holds.
Lemma 3.8**.**
Let yα′ be an element of the form
[TABLE]
where we write 1C([0,1]×S2m−1)⊗S simply by S.
Then we have
[TABLE]
In the lemma below, we regard an element x∈C([0,1]×S2m−1)⊗En+1 as a C(S2m−1)⊗En+1 valued continuous function on [0,1] and denote by xt,t∈[0,1], and frequently write 1C(S2m−1)⊗En+1 by 1C(S2m−1) for simplicity.
Lemma 3.9**.**
Let V∈UC([0,1],C(S2m−1)⊗En+1) be a unitary with V0=1C(S2m−1)⊗En+1.
Then we can choose a unitary V∈UC([0,1],C(S2m−1)⊗K)∼ satisfying the following
[TABLE]
Proof.
There is a partition 0=t0<t1<⋯<tm=1 satisfying,
[TABLE]
We construct the unitary V by induction.
For t∈[t0,t1], we have a polar decomposition
[TABLE]
for t∈[t0,t1],
and there exists a unitary
[TABLE]
with V00=1C(S2m−1).
Since π(Vt(1C(S2m−1)⊗(1−e))Vt∗(1C(S2m−1)⊗(1−e)))=1C(S2m−1)⊗On+1 and (1),
we have 1C(S2m−1)−Vt0∈C(S2m−1)⊗K.
The unitary V0∗V∈UC([0,1],C(S2m−1)⊗En+1) satisfies the following
[TABLE]
The condition (2) is satisfied by the computation below
[TABLE]
Let l be a number with m−1≥l≥0.
Assume that there exist unitaries V0,⋯,Vl satisfying
[TABLE]
where we denote Utl:=Vt∗Vt0⋯Vtl.
Now we construct a unitary Vl+1 satisfying
[TABLE]
where we write 1C(S2m−1)⊗e simply by 1⊗e.
By the assumption (3), we have a partial isometry wl+1 of a polar decomposition
[TABLE]
Let Vl+1 be a unitary of the form
[TABLE]
Since π((1C(S2m−1)−Utl∗(1⊗e)Utl)(1C(S2m−1)−Utl+1l∗(1⊗e)Utl+1l))=1C(S2m−1)⊗On+1 and (7), we have (4).
By the construction of Vl+1, we have
[TABLE]
and Vl+1 satisfies (5).
For every k, m−1≥k≥0, direct computation yields
[TABLE]
and the condition (6) is satisfied by (3).
Now we have a sequence of unitaries V0,⋯,Vm−1 by induction,
and a unitary V:=V0⋯Vm−1 satisfies the assertion of the lemma.
∎
In the sequal,
we denote by uα′−1 the element ∑i=1n+1α′−1(Ti)Ti∗.
The element uα′−1 is in UC(S2m−1)⊗(1−e)En+1(1−e) because α′∈Map(S2m−1,AuteEn+1).
We remark that
[TABLE]
We also note v∈U0(C([0,1]×S2m−1)⊗On+1) because v0=1C(S2m−1).
Lemma 3.10**.**
There exists a unitary V∈UC([0,1]×S2m−1)⊗En+1 satisfying the following
[TABLE]
In particular, an element Yα′ of the form
[TABLE]
is send to yα′ by the quotient map πα′:Bα′→Aα′ where we write 1C([0,1]×S2m−1)⊗T and 1C([0,1]×S2m−1)⊗e by T and e respectively for simplicity.
Proof.
Since v∈U0(C([0,1]×S2m−1)⊗On+1), one has a unitary lift V′∈U0(C(S2m−1)⊗En+1) of v with V0′=1.
By Lemma 3.9, we may assume the following
[TABLE]
Now we show that we can get the unitary V by a compact perturbation of V′.
By (8), an element uα′−1V1′∗ is a unitary in U(C(S2m−1)⊗(1−e)K(1−e))∼ with π(uα′−1V1′∗)=1C(S2m−1)⊗On+1.
Since α′ is homotopic to l in Map(S2m−1,EndeEn+1), the unitary uα′−1 is in U0(C(S2m−1)⊗(1−e)En+1(1−e)).
Hence we have uα′−1+1C(S2m−1)⊗e∈U0(C(S2m−1)⊗En+1).
Recall that the map K1(C(S2m−1)⊗K)↪K1(C(S2m−1)⊗En+1) is injective because Tor(K1(C(S2m−1)),Zn)=0 and the map
[TABLE]
is an isomorphism.
Therefore we have [uα′−1V1′∗]1=0 in K1(C(S2m−1)⊗(1−e)K(1−e)),
and we get a continuous path c:[0,1]→U(C(S2m−1)⊗(1−e)K(1−e))∼ from uα′−1V1′∗ to 1C(S2m−1)⊗(1−e) by the K1-injectivity of C(S2m−1)⊗(1−e)K(1−e).
For every t∈[0,1], we have λ(t)∈S1 with
[TABLE]
by (9) and π(uα′−1V1′∗)=1C(S2m−1)⊗On+1,
and the function λ:[0,1]→S1 is continuous.
Now we get the element V:=(λˉc+1C([0,1]×S2m−1)⊗e)V′∈U(C([0,1]×S2m−1)⊗En+1) satisfying the assertion of the lemma.
Since V1(1C(S2m−1)⊗T)=uα′−1(1C(S2m−1)⊗T)=α′−1(1C(S2m−1)⊗T) and α′(1C(S2m−1)⊗e)=(1C(S2m−1)⊗e),
direct computation yields
[TABLE]
Hence Yα′ is an element of M2n+4(Bα′) that is sent to yα′ by the quotient map πα′:Bα′→Aα′.
∎
We remark that Yα′ is a partial isometry.
We have θα′(Yα′)θα′(Yα′)∗=1⊕0n+1⊕01⊕1n+1 and θα′(Yα′)∗θα′(Yα′)=01⊕1n+1⊕1⊕0n+1.
Recall that W is a partial isometry with WW∗=01⊕1n+1⊕1⊕0n+1 and W∗W=1⊕0n+1⊕01⊕1n+1.
Therefore an element W+θα′(Yα′) is a unitary in UM2n+4(M).
Theorem 3.11**.**
Let m≥1 be a natural number.
Let α′ be an element of Map(S2m−1,AuteEn+1) that is homotopic to l in Map(S2m−1,EndeEn+1).
Let wα′,Yα′ and Vσα′ be as mentioned above.
Then we have [wα′]1=0 in K1(Q(C(S2m)⊗K)).
so W+θα′(Yα′) is a unitary lift of Vσα′⊕Vσα′∗.
Let P be a projection of the form
[TABLE]
We have ind[Vσα′Vσ∗]1=ind[Vσα′]1−ind[Vσ]1=[P]0+[1C(S2m)⊗e]0−[1n+2]0∈K0((C(S2m)⊗K)∼) and we show that the index is 0.
Recall Vt(1C(S2m−1)⊗e)=(1C(S2m−1)⊗e)Vt=(1C(S2m−1)⊗e) and Ut(1C(S2m−1)⊗e)=(1C(S2m−1)⊗e)Ut=(1C(S2m−1)⊗e) by Definition 3.5.
Direct computation yields
[TABLE]
Now we have
P=(1−e)⊕1n+1⊕0n+2
and get ind[Vσα′Vσ∗]1=[P]0+[1C(S2m)⊗e]0−[1n+2]0=0.
Therefore we have −n[wα′]1=[Vσα′Vσ∗]1=0 and this proves the theorem because Tor(K1(Q(C(S2m)⊗K)),Zn)=0.
∎
Corollary 3.12**.**
For the above α′, we have [α′]=0 in [S2m−1,AutEn+1].
Proof.
Since [wα′]1=0, there is a unitary Wα′ in UM(C(S2m)⊗K) that is a lift of wα′.
Therefore we have C(S2m)-linear isomorphism AdWα′:Bl→Bα′, and Pl≅Pα′ from Lemma 2.12.
By Lemma 3.1, we have [α′]=[l]=0.
∎
Lemma 3.13**.**
Let m≥1 be a natural number. Let α be an element in Map(S2m−1,AuteEn+1). If α∼hl in Map(S2m−1,End0En+1), then there exists α′ in Map(S2m−1,AuteEn+1) satisfying the following :
[TABLE]
Proof.
It follows from Lemma 2.29 that there is an exact sequence
[TABLE]
and by Remark 2.31 the map [S2m−1,EndeEn+1]→[S2m−1,UEn+1] which sends [α] to [uα]:=[e+∑i=1n+1αx(Ti)Ti∗]∈[S1,UEn+1] is an isomorphism. Since BS1 is K(Z,2) space, if m≥2 and α∼hl in Map(S2m−1,End0En+1), then we have [α]=0 in [S2m−1,EndeEn+1] because [S2m−1,EndeEn+1]=[S2m−1,End0En+1], m≥2.
Hence it is sufficient to show the claim in the case of m=1.
Let α be an element of Map(S1,AuteEn+1) with α∼hl in Map(S1,End0En+1).
Computation in Theorem 2.36 yields that there exists d∈Z with [uα]=−nd∈[S1,UEn+1]=Z. We define ρd by
[TABLE]
By Lemma 2.10, there is a continuous path from (zˉdT1T1∗+(1−T1T1∗)) to zˉd in UC(S1)⊗En+1, and ρd∼hAdzˉd=l in Map(S1,AutEn+1).
We have [uρdα]1=[ρd(uα)]1+[uρd]1 in K1(C(S1)⊗En+1)=[S1,UEn+1].
By Lemma 2.24, it follows that [ρd(uα)]1=[uα]1. Hence we have [uρdα]1=−nd+[uρd]1. The following computations yield [uρd]1=nd :
[TABLE]
[TABLE]
Therefore we have [uρdα]=0 in [S1,UEn+1].
By Remark 2.31, we have the isomorphism [S1,EndeEn+1]∋ρdα↦[uρdα]∈[S1,UEn+1], and α′:=ρdα satisfies all assumptions of the lemma.
∎
We show the weak homotopy equivalence.
Theorem 3.14**.**
The inclusion map AutEn+1→End0En+1 is a weak homotopy equivalence.
Proof.
By Lemma 3.3 and Theorem 2.36, we consider only the case of odd homotopy groups.
Let k be an odd number.
First, we show the map [Sk,AutEn+1]→[Sk,End0En+1] is injective.
If α in Map(Sk,AutEn+1) is homotopic to l in Map(Sk,End0En+1), we may assume that there exists α′∈Map(Sk,AuteEn+1) homotopic to α by Remark 2.33.
From Lemma 3.13, we may assume that α′∼hl in Map(Sk,EndeEn+1), and we have [α′]=[α]=0 in [Sk,AutEn+1] by Corollary 3.12.
Therefore the map [Sk,AutEn+1]→[Sk,End0En+1] is injective.
Second, we show the surjectivity.
The following commutative diagram holds
[TABLE]
In the case of k=1, we have [S1,End0En+1]=[S1,EndOn+1]=Zn because the generators of the both groups are constructed from canonical gauge actions of S1 that are of the form λz:Ti↦zTi and λ~z:Si↦zSi. Therefore the surjectivity follows from Lemma 3.4.
In the case of k≥3, the map [Sk,UEn+1]→[Sk,End0En+1]=Z in Theorem 2.36 is an isomorphism.
Therefore the map Z=[Sk,End0En+1]→[Sk,EndOn+1]=[Sk,UOn+1]=Zn is the quotient by nZ.
Hence the image of the map [Sk,AutEn+1]→[Sk,End0En+1]=Z contains an element nd+1 for some d∈Z by Lemma 3.4.
On the other hand, we show that the image contains nZ.
For every V∈UC(Sk)⊗En+1, there exists V′∈UC(Sk)⊗En+1 with V′(1C(Sk)⊗e)=(1C(Sk)⊗e)V′=(1C(Sk)⊗e) which is homotopic to V in UC(Sk)⊗En+1 by Remark 2.31.
Since the isomorphism [Sk,UEn+1]→[Sk,End01En+1] sends
[TABLE]
to [AdV′]=[AdV],
the subset
[TABLE]
is mapped onto the subset
[TABLE]
Therefore the image contains nd+1 and nZ, and we have the conclusion.
∎
3.2 An exact sequence of homotopy sets
We have the principal AuteEn+1-bundle
AuteEn+1iAutEn+1ηBS1. We denote by f the classifying map of the bundle and denote by r the restriction map AutEn+1→AutK. In this section, we show the following theorem.
Theorem 3.15**.**
Let X be a compact CW-complex. Then we have the following exact sequence of the pointed set where first 4-terms gives the exact sequence of the groups :
[TABLE]
It follows that Imη∗⊂Tor(H2(X),Zn) and ImBr∗⊂Tor(H3(X),Zn).
The following lemma is well known in the homotopy theory.
We refer to [20, Chap 3, Section 6]
Lemma 3.16**.**
Let X be a CW-complex.
Let G be a topological group and let H be a subgroup of G such that H→G→G/H is a principal H-bundle.
Suppose that G/H has a homotopy type of a CW-complex.
Let f:G/H→BH be its classifying map. Then we have the exact sequence of pointed sets :
[TABLE]
Since BS1 has a homotopy type of a CW-complex, we can apply the above lemma to AuteEn+1→AutEn+1→BS1.
Lemma 3.17**.**
Let X be a CW-complex.
The following sequence of pointed sets is exact :
[TABLE]
Proof.
The group AuteK is identified with the group UM(K) by the map taking the implementing unitary Uα=∑i=1α(ei1)e1i for α∈Aute11K. Hence it is contractible, and [X,BAuteK]={pt}.
From the commutative diagram below,
[TABLE]
Br∗∘Bi∗ is trivial.
Therefore it is sufficient to prove that for every P∈Map(X,BAutEn+1) with the trivial associated bundle P×AutEn+1AutK, the structure group of P is reduced to AuteEn+1.
Let P∈Map(X,BAutEn+1) be a principal AutEn+1-bundle with the trivial associated bundle P×AutEn+1AutK.
We take an open covering {Ui} of X giving a local trivialization of P, and denote by ϕji:Uj∪Ui→AutEn+1 the transition function.
By the assumption, there exists the map hi:Ui×AutK→Ui×AutK that is compatible with the transition functions, and is equivariant with respect to the right multiplication of AutK.
The diagram below holds
[TABLE]
We also denote by ϕji the map
[TABLE]
We denote hi(x):=Pri(hi(x,id)) where Pri:Ui×AutK→AutK. Since hi is equivariant, we have hi−1(x)=hi(x)−1.
We have hj(x)r(ϕji(x))hi−1(x)(e)=e because hj∘ϕji∘hi−1(x,id)=(x,id) for every x∈Uj∩Ui.
If we take an appropriate refinement of {Ui}, we may assume that for every i, there exists xi∈Ui satisfying ∣∣hi−1(x)(e)−hi−1(xi)(e)∣∣<1,x∈Ui. There is a unitary Vi′(x) that is the sum of partial isometries constructed from the polar decomposition of hi−1(x)(e)hi−1(xi)(e) and (1−hi−1(x)(e))(1−h−1(xi)(e)), and Vi′(x)hi−1(xi)(e)Vi′(x)∗=hi−1(x)(e) holds. We fix a unitary Wi∈UK1 with WieWi∗=hi−1(xi)(e). Then we have a unitary Vi(x)=Vi′(x)Wi∈UK∼ with Vi(x)eVi(x)∗=hi−1(x)(e). The correction of the map
[TABLE]
gives the following :
[TABLE]
where ϕ~ji is of the form
[TABLE]
We have the transition function
[TABLE]
by the computation below :
[TABLE]
Therefore the structure group of P is reduced to AuteEn+1.
∎
Lemma 3.18**.**
Let X be a compact Hausdorff space.
The map AuteEn+1∋α↦e+∑i=1n+1α(Ti)Ti∗∈UEn+1 induces a group isomorphism [X,AuteEn+1]→[X,UEn+1]=K1(X).
Proof.
By Remark 2.35 and Theorem 3.14, the map is bijective.
So we show that it is a group homomorphism.
Let α and β be elements of Map(X,AuteEn+1), and we denote uα:=1C(X)⊗e+∑i=1n+1α(1C(X)⊗Ti)1C(X)⊗Ti∗∈UC(X)⊗En+1. We show [uαβ]1=[uα]1+[uβ]1.
Since α and β fix e, direct computation yields
[TABLE]
By Lemma 2.24, we have [α(uβ)]1=K1(α)([uβ]1)=[uβ]1.
∎
We need the following fact to determine the second cohomology group of AutEn+1. See Allen Hatcher’s unpublished book [15, Proposition 5.11].
Proposition 3.19**.**
Let X be a path connected space with finite homotopy groups. Then its homology group Hn(X) is finite for all n>0.
Lemma 3.20**.**
We have the following cohomology groups :
[TABLE]
Proof.
Two spaces AutEn+1 and AutOn+1 are path connected and the map AutEn+1→AutOn+1 gives
[TABLE]
So we have
[TABLE]
by Whitehead’s theorem (see [5, Corollary 6.69]). By the universal coefficient theorem, we have
[TABLE]
where free(H2(AutEn+1)) is the free part of the homology group. By Proposition 3.19, the homology groups of AutOn+1 are finite, and free(H2(AutEn+1))=0. So we have H2(AutEn+1)≅Tor(H1(AutOn+1)). Hurewicz’ theorem ( [5, Theorem 6.66]) yields H1(AutOn+1)=π1(AutOn+1)=Zn.
Similarly, we have H3(BAutEn+1)=Zn.
∎
By Lemma 3.16 and the long exact sequences of the principal bundle AuteEn+1iAutEn+1ηBS1, we have an exact sequence of pointed set where first 4-terms gives the exact sequence of the groups :
[TABLE]
By Lemma 3.18, and Lemma 3.17, we have the exact sequence :
[TABLE]
where we identify H3(X) with [X,BAutK] because BAutK is the K(Z,3)-space.
We identify [AutEn+1,BS1] with H2(AutEn+1).
For every [α]∈[X,AutEn+1], it follows that η∗([α])=α∗([η]), and the element α∗([η]) is in the image of the map
[TABLE]
Therefore we have Imη∗⊂Tor(H2(X),Zn) from Lemma 3.20.
Similar argument yields ImBr∗⊂Tor(H3(X),Zn).
∎
Acknowledgements
The author would like to show his greatest appreciation to his supervisor Prof. Masaki Izumi who gave many insightful comments and suggestions, and patiently checked his arguments.
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