EVALUATION FIBER SEQUENCE AND HOMOTOPY COMMUTATIVITY
MITSUNOBU TSUTAYAAbstract. We review the relation between the evaluation fiber sequence of mapping spaces and thevarious higher homotopy commutativities.

1. Introduction
Suppose that A is a non-degenerately based space and X a based space. We denote the spaceof continuous maps between A and X by Map(A, X). The subspace of based maps is denoted as
Map0 (A, X). These spaces are often called mapping spaces. For a based map φ : A → X, we denotethe path-components Map0 (A, X; φ) ⊂ Map0 (A, X) and Map(A, X; φ) ⊂ Map(A, X) consisting of maps
freely homotopic to φ which are often considered to be based at the point φ. In order to guarantee theexponential law for any space, we will always assume that spaces are compactly generated.
Study of homotopy theory of mapping spaces is important in various situations. For example, see
the survey by S. B. Smith [Smith10]. The study of them is often difficult since the geometry of themare very complicated in general. To connect with known objects, one of the easiest way is considering
the evaluation map
ev : Map(A, X) → X
such that ev( f ) = f (∗) or its associated homotopy fiber sequence· · · → ΩX → Map0 (A, X) → Map(A, X) → X
called the evaluation fiber sequence. They are not only easy to define but also have rich homotopy theoretic structures. As an example among them, here we consider its relations with (higher) homotopy
commutativity.
The author thank Doctors Sho Hasui and Takahito Naito for organizing the workshop and giving
him an opportunity to talk there. This note is written considerably from the author’s interest and may
contain many incomplete explanations.
2. Recognition of homotopy fiber sequence
The “delooping” of a loop space in various sense has been considered. The classifying space ofa topological group is the most well-known example. J. P. May [May73] and others studied when aspace is delooped n-times. Such a type of result is now called a recognition principle. We review
recognitions of maps in a homotopy fiber sequence from the viewpoint of topological groups.
Let f : X → Y be a based map. Taking homotopy fibers repeatedly, the map f generates thehomotopy fiber sequence
Ωi

Ωf

∂

i

f

· · · → ΩF −→ ΩY −−→ ΩX →
− F→
− Y→
− X,
where i : F → Y and ∂ : ΩX → F are the inclusion of homotopy fibers of f and i, respectively. We
note that the map Ω f is a loop map and that ∂ is a fiber inclusion of a principal fibration. In thissection, we study how to recover whole this sequence by ∂ or Ω f with some extra data.
1

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MITSUNOBU TSUTAYAFirst we set the notation. The i-dimensional simplex ∆i is given as∆i = {(t0 , . . . , ti ) ∈ [0, 1]×(i+1) | t0 + · · · + ti = 1}.
There are the boundary maps ∂k : ∆i−1 → ∆i and the degeneracy maps σk : ∆i+1 → ∆i for k = 0, 1, . . . , isuch that∂k (t0 , . . . , ti ) = (t0 , . . . , tk−1 , 0, tk , . . . , ti ),σk (t0 , . . . , ti+1 ) = (t0 , . . . , tk−1 , tk + tk+1 , tk+2 , . . . , ti+1 ).
Unless otherwise stated, we assume that every topological monoid is based at its identity element.
Definition 2.1. For a topological monoid G, let X be a right G-space and Y a left G-space. The bar
construction B(X, G, Y) of the triple (X, G, Y) is defined by∞,a i∆ × X × G×i × Y  ∼,B(X, G, Y) = i=0where the identification ∼ is described as(t; xg1 , g2 , . . . , gi , y)
k=0



(t; x, g1 , . . . , gk gk+1 , . . . , gi , y) 0 < k < i(∂k t; x, g1 , . . . , gi , y) ∼  (t; x, g1 , . . . , gi−1 , gi y)
k = i,(t; x, g1 , . . . , gk−1 , ∗, gk+1 , . . . , gi ) ∼ (σk t; x, g1 , . . . , gk−1 , gk+1 , . . . , gi ).
`n i
The image of i=0 ∆ × X × G×i × Y in B(X, G, Y) is denoted by Bn (X, G, Y).
Lemma 2.2. Let G be a topological monoid and X a right G-space. Then, the obvious inclusion
X → X × G = B0 (X, G, G) → B(X, G, G) is a homotopy equivalence and its homotopy inverse
B(X, G, G) → X is given by[t; x, g1 , . . . , gi , gi+1 ] 7→ xg1 · · · gi gi+1 .
Proof. Take a deformation H : [0, 1] × B(X, G, G) → B(X, G, G) of B(X, G, G) into X as
H(s, [t; x, g1 , . . . , gi , gi+1 ]) = [(1 − s)t, s; x, g1 , . . . , gi , gi+1 , ∗].
For the proof of the next theorem, see [May75, Theorem 8.2].
Theorem 2.3. Let G be a non-degenerately based topological group. Then, for a right G-space X
and a left G-space Y, the projections
B(X, G, Y) → B(∗, G, Y) andB(X, G, Y) → B(X, G, ∗)are fiber bundles with fiber X and Y, respectively. In particular, B(X, G, G) → B(X, G, ∗) is a principal
G-bundle.
Suppose that a topological group G is non-degenerately based. Let us denote EG = B(∗, G, G)
and BG = B(∗, G, ∗). Since EG is contractible by Lemma 2.2, the principal G-bundle EG → BG isuniversal. Moreover, the square
B(X, G, G)/ B(∗, G, G)/ B(∗, G, ∗)B(X, G, ∗)EVALUATION FIBER SEQUENCE AND HOMOTOPY COMMUTATIVITY

3

is pullback and thus the map B(X, G, ∗) → B(∗, G, ∗) = BG induced by the G-equivariant map X → ∗is the classifying map of the principal bundle B(X, G, G) → B(X, G, ∗).
Corollary 2.4. Let G be a non-degenerately based topological group and X a right G-space. Then,the following sequence is a homotopy fiber sequence:
G → X → B(X, G, ∗) → BG,
where the left arrow is given by g 7→ g∗ for the basepoint ∗ ∈ X.
Corollary 2.5. Let G and H be non-degenerately based topological groups. For a homomorphism
f : G → H, the following sequence is a homotopy fiber sequence:
f

Bf

G→
− H → B(H, G, ∗) → BG −−→ BH.
Proof. The left four term is a homotopy fiber sequence by Corollary 2.4. Moreover, the square
B(H, G, ∗)B(id, f,∗)

/ B(H, H, ∗)B(∗, G, ∗) B(∗, f,∗) / B(∗, H, ∗)is a homotopy pullback. Then the right three term is also a homotopy fiber sequence.



Corollary 2.6. Let G and H be non-degenerately based topological groups, where H is a CW complex.
Suppose that a homomorphism G → H is surjective and has the homotopy lifting property. Denotethe kernel by K. Then the sequence
K → G → H → BK → BG → BH
is a homotopy fiber sequence, where the right two maps are induced by the homomorphisms K → G
and G → H, and H → BK is given as'

H←− B(G, K, ∗) → BK.
Proof. By the commutative diagram/ B(G, K, K)

K/ B(G, K, ∗)

'


/H

/G

K

where each row is a homotopy fiber sequence, the map B(G, K, ∗) → H is a weak equivalence.
Consider the commutative diagram
B(G, K, ∗)'

B(H, G, G)

/ BK

/ BG


/ B(H, G, ∗)/ BG

of which the rows are homotopy fiber sequences. Then the map BK → B(H, G, ∗) is a homotopy
equivalence. Combining these equivalences, we obtain the desired homotopy fiber sequence.

Example 2.7. The space BnG = Bn (∗, G, ∗) is called the n-projective space of G.
• For a topological monoid G, the “projective line” B1G is naturally homeomorphic to the reduced suspension ΣG.

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MITSUNOBU TSUTAYA• According as G = O(1), U(1), Sp(1), the n-th projective space BnG is homeomorphic to
RPn , CPn , HPn , respectively.
Lemma 2.8. Let G be a non-degenerately based topological group. Then the compositeEζ: G −→ ΩΣG  ΩB1G → ΩBGof the suspension E and the loop of the inclusion is a weak homotopy equivalence. Moreover, the mapζ is an H-map.
Proof. Consider a contracting homotopy H : [0, 1] × EG → BG
H(s, [t; ∗, g1 , . . . , gi , gi+1 ]) = [(1 − s)t, s; ∗, g1 , . . . , gi , gi+1 , ∗].
This induces a map EG → PBG covering the identity on BG. Then the restriction G → ΩBG on thefiber is the considered map above. Since G → EG → BG is a fiber bundle and EG is contractible, it
is a weak homotopy equivalence.
Let g1 , g2 ∈ G. Then we can compute the composite of ζ(g1 ) and ζ(g2 ) as(
[1 − 2t, 2t, 0; ∗, g1 , g2 , ∗]0 ≤ t ≤ 1/2
ζ(g1 )ζ(g2 )(t) =[0, 2 − 2s, 2s − 1; ∗, g1 , g2 , ∗] 1/2 ≤ t ≤ 1.
Obviously, this path is canonically homotopic toζ(g1 g2 )(t) = [1 − t, 0, t; ∗, g1 , g2 , ∗].
Therefore, ζ is an H-map.

The multiplication on a based loop space ΩX is not a group unless the path-component of the basepoint is just a point. But ΩX have the good an identity element, associativity and homotopy inversion
in an appropriate sense. Then, naively, we expect the existence of some “group model” for ΩX. Thiscan be carried out with use of the Kan’s simplicial loop group. For details, see [Neisendorfer10,
Section 6.8].
Proposition 2.9. For a path-connected based space X, there is a principal bundle G X → T X → X̄functorial about X such that G X , T X , X̄ are CW complex, there is a natural weak equivalence X̄ → X
and T X is contractible.
This means that the based loop space ΩX has a group model which is a CW complex. Moreprecisely, there is a commutative diagram
GX/ EG X/ BG X




/ PX


/X

ΩX

such that all the vertical arrows are weak homotopy equivalences and the left vertical map is an H-map
by Lemma 2.8 (in fact, is an A∞ -map).
Let f : X → Y be a based map. Then, by Proposition 2.9, the homotopy fiber sequence associatedto f is equivalent to the sequence
Gf

BG f

· · · → G X −−→ GY → B(GY , G X , ∗) → BG X −−−→ BGY .

EVALUATION FIBER SEQUENCE AND HOMOTOPY COMMUTATIVITY

5

In this sense, the maps in a homotopy fiber sequence is recognized as follows:ΩXhomomorphism of topological groupsΩYaction of ΩY on F



F


X


based map

Y.
Conversely, each of these structures recovers the whole homotopy fiber sequence by Corollary 2.4
and 2.5.
3. Connecting map of evaluation fiber sequence
Suppose that G is a non-degenerately based topological group. Let us denote the conjugation byα(g)(x) = gxg−1 and the left translation by λ(g)(x) = gx. The conjugation induces a bundle map
Eα(g) = B(∗, α(g), λ(g)) : EG → EG covering Bα(g) = B(∗, α(g), ∗) : BG → BG. Then G naturally
acts on BG.
For a non-degenerately based space A, take a based map φ : A → BG and the induced principal
bundle P = φ∗ EG → A. The mapping space Map0 (A, BG; φ) is equipped with the right action by
G induced by the inverse conjugation on BG as well. The following theorem is essentially due to
Kishimoto–Kono [KK10, Section 6]. More detailed proof can be found in [Tsutaya, Section 7].
Theorem 3.1. Suppose that G is a non-degenerately based topological group and A is a based CW
complex. Take a based map φ : A → BG. Then the evaluation fiber sequenceevΩBG → Map0 (A, BG; φ) → Map(A, BG; φ) −→ BG
is weakly equivalent to the homotopy fiber sequence
G → Map0 (A, BG; φ) → Map0 (A, BG; φ) ×G EG → BG
induced by the conjugation action on Map0 (A, BG; φ) by G. More precisely, for some CW complex
M, there is a commutative diagram
Map0 (A, BG; φ)O/ Map(A, BG; φ)O

M ×BG EG

/ M

/ BG




/ B(Map (A, BG; φ), G, ∗)/ BG

Map0 (A, BG; φ)0such that all the vertical arrows are weak equivalences.

ev

/ BG

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MITSUNOBU TSUTAYA

Remark 3.2. In the above statement, the mapping spaces Map(A, BG; φ) and Map0 (A, BG; φ) are notnecessarily homotopy equivalent to CW complexes and then one cannot expect the existence of adirect weak equivalence between Map(A, BG; φ) and B(Map0 (A, BG; φ), G, ∗). This is the reason why
the weak equivalence between them is given indirectly. For the existence of the direct weak equivalence, it is sufficient to suppose that G is a CW complex and A is a finite CW complex.
Lemma 3.3. Let G be a non-degenerately based topological group and A a CW complex. Take abased map φ : ΣA → BG. Then there is a natural weak homotopy equivalence
Map0 (A, G) → Map0 (ΣA, BG)
which is G-equivariant with respect to the conjugation actions on G and BG.
Proof. Note that the suspension E : G → ΩΣG and the inclusion ΣG → BG are G-equivariant. Then,by Lemma 2.8, the map ζ : G → ΩBG is a G-equivariant weak homotopy equivalence. Combining
the G-equivariant homeomorphism
Map0 (ΣA, BG)  Map0 (A, ΩBG),

we obtain the desired result.

Corollary 3.4. Let G be a non-degenerately based topological group and A a CW complex. Take abased map φ : ΣA → BG. Then for the connecting map ∂∗ : ΩBG → Map0 (ΣA, BG; φ), there is anatural commutative diagram
δG/ Map (A, G)
0



ΩBG

∂∗


/ Map (ΣA, BG),
0

where the vertical arrows are weak homotopy equivalences and δ(g) = α(g−1 ) ◦ φ.
Example 3.5 (free loop space). The evaluation fiber sequenceΩBG → LBG → BG
on the free loop space LBG is equivalent to the fiber bundle
G → ad EG → BG,
where the central term is the associated bundle ad EG = EG ×G G with respect to the conjugation on
G. The bundle ad EG is a fiberwise topological group.
This fact follows from Lemma 3.3 for A = S 0 and Theorem 3.1.
4. Homotopy commutativity I: mapping space as centralizer
Let Γ be a discrete group. Then, by the fiber sequenceΓ → EΓ → BΓ,
we obtain that πi (BΓ) = 0 for i , 1. By using Lemma 2.8, we fix the identification
π1 (BΓ)  π0 (ΩBΓ)  Γ.
We investigate the mapping spaces Map(X, BΓ) and Map0 (X, BΓ) for a connected CW complex X
and a discrete group Γ. Their homotopy types had been determined by D. H. Gottlieb [Gottlieb69a].
By standard obstruction argument, we obtain that πi (Map(X, BΓ)) = 0 for i ≥ 2 and πi (Map0 (X, BΓ)) =0 for i ≥ 1 with respect to an arbitrary basepoint.

EVALUATION FIBER SEQUENCE AND HOMOTOPY COMMUTATIVITY

7

Lemma 4.1. Let X be a connected CW complex and Γ a discrete group. Then, the fundamental groupfunctor induces a bijection
π1 : [X, BΓ]0 → Hom(π1 (X), π1 (BΓ))  Hom(π1 (X), Γ).
Moreover, this map is Γ-equivariant with respect to the conjugation on BΓ and Γ.
Proof. Being bijective follows from standard obstruction argument using the van Kampen’s theorem.
This map is Γ-equivariant since the identification Γ  π0 (Γ)  π0 (ΩBΓ)  π1 (BΓ) is Γ-equivariant. 
Corollary 4.2. Let X be a connected CW complex and Γ a discrete group. Then there is a naturalbijection[X, BΓ]  Hom(π1 (X), Γ)/Γ,
where the quotient on the right hand side is taken by the conjugation.
Proof. Since X and BΓ are connected CW complexes, the map[X, BΓ]0 → [X, BΓ]
is surjective. Then, by Lemma 4.1, we have bijections[X, BΓ]  [X, BΓ]/π1 (BΓ)  [X, BΓ]/Γ  Hom(π1 (X), Γ)/Γ.

Now, all we have to determine is the fundamental group of each component of Map(X, BΓ).
Proposition 4.3. Let X be a connected CW complex, Γ a discrete group and f : X → BΓ a based map.
Then the fundamental group of Map(X, BΓ; f ) is computed as
π1 (Map(X, BΓ; f )) = ZΓ ( f∗ (π1 (X))),
where ZΓ (Λ) denotes the centralizer of a subgroup Λ ⊂ Γ.
Proof. By the evaluation fiber sequence, we have the following exact sequence of sets:
0 → π1 (Map(X, BΓ; f )) → Γ → Hom(π1 (X), Γ),
where the right map is the composition γ 7→ αγ−1 ◦ f∗ with the conjugation αγ−1 . Note that αγ−1 ◦ f∗ = f∗
if and only if γ ∈ ZΓ ( f∗ (π1 (X))). Thus we can compute π1 (Map(X, BΓ; f )) as above.

Theorem 4.4 (Gottlieb). Let X be a connected CW complex, Γ a discrete group and f : X → BΓ abased map. Then the mapping space Map(X, BΓ; f ) has the weak homotopy type of the classifying
space BZΓ ( f∗ (π1 (X))). In particular, Map(BΓ, BΓ; id) has the weak homotopy type of the classifying
space BZ(Γ) of the center Z(Γ).
If G is a Lie group, the mapping space Map(X, BG; φ) is complicated. But a certain variant of theabove theorem is proved by W. G. Dwyer and C. Wilkerson [DW95]. Note that the multiplication
map Z(G) × G → G induces the classifying map BZ(G) × BG  B(Z(G) × G) → BG and its adjoint
BZ(G) → Map(BG, BG; id).
Theorem 4.5 (Dwyer–Wilkerson). Let G be a compcat Lie group. Then the above map
BZ(G) → Map(BG, BG; id)
induces an isomorphism on homology with any finite coefficient.
Remark 4.6. This map does not induce an isomorphism on rational homology in general. For example,
if G = SU(2), there are homotopy equivalences BZ(SU(2))(0) ' ∗ and Map(B SU(2), B SU(2); id)(0) 'B SU(2)(0) .

8MITSUNOBU TSUTAYA

5. Homotopy commutativity II: Whitehead product and Samelson product
The Whitehead product on homotopy groups of spheres is introduced by J. H. C. Whitehead[Whitehead41]. M. Arkowitz [Arkowitz62] generalized it for maps from a general suspension toa base space.
Let α : ΣA → X and β : ΣB → X be maps from the reduced suspensions ΣA and ΣB to a based
space X. The map w : Σ(A × B) → X defined byw = (α ◦ Σp1 ) + (β ◦ Σp2 ) − (α ◦ Σp1 ) − (β ◦ Σp2 )determines the map [α, β] : Σ(A ∧ B) → X uniquely up to homotopy such that, for the quotient map
q : A × B → A ∧ B, [α, β] ◦ Σq = w in [Σ(A × B), X]0 . The uniqueness is verified by the splitting Puppe
sequence(Σq)∗

0 → [Σ(A ∧ B), X]0 −−−→ [Σ(A × B), X]0 → [ΣA ∨ ΣB, X]0 → 0.
The map [α, β] or its homotopy class is called the Whitehead product of α and β.
The next theorem is due to M. Arkowitz [Arkowitz62, Section 4].
Theorem 5.1. Let A and B be based spaces. Denote the inclusions ιA : ΣA → ΣA ∨ ΣB and ιB : ΣB →
ΣA ∨ ΣB. Then the mapping cone inclusion of [ιA , ιB ] : Σ(A ∧ B) → ΣA ∨ ΣB is equivalent to the
obvious inclusion ΣA ∨ ΣB → ΣA × ΣB.
Let α : A → ΩX and β : B → ΩX be maps from the based spaces A and B to the based loop spaceΩX. Like the Whitehead product, the map s : A × B → ΩX
w = (α ◦ p1 ) + (β ◦ p2 ) − (α ◦ p1 ) − (β ◦ p2 )determines the map hα, βi : A ∧ B → ΩX uniquely up to homotopy such that hα, βi ◦ q = s in
[A × B, ΩX]0 . The map hα, βi or its homotopy class is called the Samelson product of α and β.
Example 5.2. Note that SU(2) is homeomorphic to the 3-dimensional sphere S 3 . The Samelson
product hidS 3 , idS 3 i of two copies of the identity idS 3 : S 3 → S 3 with respect to the loop structure
S 3 ' ΩB SU(2) is known to have order 12. This generates the homotopy group π6 (S 3 )  Z/12Z.
Two maps corresponding to each other by the adjunction[ΣA, X]0  [A, ΩX]0
are said to be mutually adjoint. The next proposition says that the Samelson product is adjoint to theWhitehead product in some sense.
Proposition 5.3. Let α : ΣA → X and β : ΣB → X be maps from the reduced suspensions ΣA and ΣB
to a based space X. Denote the adjoint maps of α and β by α0 : A → ΩX and β0 : B → ΩX. Then, theSamelson product hα0 , β0 i is adjoint to the Whitehead product [α, β].
Proof. The adjoint α0 is defined by α0 (a)(t) = α(t, a). Then the desired adjointness immediately
follows from definitions.
Suppose that X is a non-degenerately based space and A and B are CW complexes. Then, taking'
the group model f : G X →− ΩX, note that the Samelson product [α, β] of α ∈ [A, ΩX] and β ∈ [B, ΩX]
is equal to the image of the commutator g∗ (α̃β̃α̃−1 β̃−1 ), where g∗ (α̃) = α and g∗ (β̃) = β.
Let A and X be based CW complexes and φ : ΣA → X a based map. Let us consider the following
evaluation fiber sequence:∂∗ev· · · → ΩX −→ Map0 (ΣA, X; φ) → Map(ΣA, X; φ) −→ X.
The next theorem is due to G. E. Lang [Lang73].

EVALUATION FIBER SEQUENCE AND HOMOTOPY COMMUTATIVITY

9

Theorem 5.4. Under the above setting, the adjoint ΣΩX ∧ A → X of the composite
∂

−φ

ΩX →
− Map0 (ΣA, X; φ) −−→ Map0 (ΣA, X; ∗)is homotopic to the Whitehead product [−e, φ] of the evaluation e : ΣΩX → X and φ.
Proof. Taking the group model G X ' ΩX, the corresponding map G X → Map0 (A, G X ; ∗) is the commutator map g 7→ (a 7→ g−1 φgφ−1 ) by Corollary 3.4. Then the adjoint ΩX ∧ A → ΩX is the Samelson
product [− idGX , φ0 ], where φ0 is the adjoint of φ. By Proposition 5.3, the adjoint is the Whitehead
product [−e, φ].
An H-space X having the homotopy inversion is said to be homotopy commutative if the commutator map
X × X → X,
(x, y) → ((xy)x−1 )y−1
is null-homotopic.
Since the Whitehead product is the adjoint of the Samelson product, if ΩX is homotopy commutative, then all the generalized Whitehead products in X vanish. This implies the following corollary.
Corollary 5.5. Under the setting of Theorem 5.4, if ΩX is homotopy commutative, then the connecting map ΩX → Map0 (A, X; φ) is null-homotopic. In particular, when X is a CW complex, ΩX is
homotopy commutative if and only if the connecting map ΩX → Map0 (ΣΩX, X; e) for A = ΣΩX isnull-homotopic.
Remark 5.6. If X is a CW complex, then ΣΩX has the homotopy type of a CW complex.
6. Homotopy commutativity III: Higher homotopy commutativity
In this section, we outline the results on higher homotopy commutativity by the author [Tsutaya].
To state it, it is convenient to consider grouplike topological monoids while we have considered only
topological groups so far. A topological monoid G is said to be grouplike if the induced monoid π0 (G)is a group.
In the previous section, we saw that the Samelson product defined by commutator map is the adjointof the Whitehead product. There are many notion of higher homotopy commutativity. We collectvarious higher homotopy commutativities and describe them as a generalization of the Whitehead
product. Though each of them have the description in terms of higher homotopy, we do not explainthem in such a way.
The simplest one is defined by F. D. Williams [Williams69], where the following description is due
to Saümell [Saümell95].
Definition 6.1. Let G be a non-degenerately based grouplike topological monoid. Then G is said tobe a Williams Cn -space if the n-fold wedge sum
∨n
ι∨n
→ BG
1 : (ΣG)of the inclusion ι1 : ΣG  B1G → BG extends to the map (ΣG)×n → BG from the product (ΣG)×n .
The obstructions to extending the map ι∨n
1 is called the higher Whitehead products. This definition
requires that all the i-th Whitehead products for i ≤ n on BG vanish.
A stronger homotopy commutativity was defined by M. Sugawara [Sugawara60] before Williams.
His definition is generalized by C. A. McGibbon [McGibbon89].

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MITSUNOBU TSUTAYADefinition 6.2. Let G be a non-degenerately based grouplike topological monoid. Then G is said tobe a Sugawara C n -space if the wedge sumιn ∨ ιn : BnG ∨ BnG → BG
of the inclusion ιn : BnG → BG extends over the union of products[
BiG × B jG.
i+ j=nEquivalently, G is a Sugawara C n -space if the multiplication map G × G → G is an An -map in thesense of Stasheff [Stasheff63].
Further higher homotopy commutativities are considered by Hemmi–Kawamoto [HK11] and by
Kishimoto–Kono [KK10].
Definition 6.3. Let G be a non-degenerately based grouplike topological monoid.
(i) G is said to be a Ck (n)-space if the wedge sum ιk ∨ιn extends over the union of products BiG×B jG
for i + j = n and i ≤ k.
(ii) G is said to be a C(k, `)-space if the wedge sum ιk ∨ ι` extends over the product BkG × B`G.
For 1 ≤ n ≤ ∞ and 1 ≤ k ≤ n, it is not difficult to see the implications
Sugawara C n -space ⇒ Ck (n)-space ⇒ C(k, n − k)-space ⇒ Williams Cn -space.
In particular, we have the equivalences
Sugawara C 2 -space ⇔ Williams C2 -space ⇔ homotopy commutativeandSugawara C ∞ -space ⇔ the classifying space is an H-space.
Remark 6.4. A topological group G is said to be a double loop space if the classifying space BG ishomotopy equivalent to a based loop space ΩX for some space X. A double loop space has much better
homotopy commutativity than Sugawara C ∞ -spaces since the classifying space BG of a Sugawara
C ∞ -space G is just an H-space.
In relation with free loop spaces, J. Aguadé [Aguadé87] proved the next proposition.
Proposition 6.5 (Aguadé). Let X be a connected CW complex. Then the free loop fibrationΩX → LX → X
is trivial if and only if the based Moore loop space ΩM X is a C1 (∞)-space.
Jim Stasheff considered the intermediate conditions between a loop map and a based map with no
extra structure.
Definition 6.6 (Stasheff). A based map f : G → G0 between non-degenerately based grouplike topological monoids is said to be an An -map if the adjoint
f 0 : B1G  ΣG → BG0
of f extends over BnG.
The author [Tsutaya, Lemma 7.3] proved that there is a G-equivariant weak equivalence
An (G, G; id) → Map0 (BnG, BG; ιn )from a grouplike topological monoid equipped with G-action induced from a homomorphism G →An (G, G; id). This implies that the evaluation fiber sequence Map0 (BnG, BG; ιn ) → BG extends toEVALUATION FIBER SEQUENCE AND HOMOTOPY COMMUTATIVITY

11

the right by Corollary 2.5 since the map G → Map0 (BnG, BG; ιn ) is the connecting map. For higher
homotopy commutativity, he proved that the following.
Theorem 6.7 (T). Let G be a non-degenerately based grouplike topological monoid which is a CW
complex. Then G is a C(k, `)-space if and only if the homomorphism G → A` (G, G; id) is nullhomotopic as an Ak -map.
Remark 6.8. The author does not know An (G, G; id) is non-degenerately based or not. Then weshould replace it by an appropriate non-degenerately based approximation.
7. Another application: Gottlieb groups
The 1st Gottlieb group was introduced by Jiang [Jiang64] and applied to the Nielsen fixed point
theory. Later, Gottlieb defined the higher Gottlieb groups in [Gottlieb69b].
Definition 7.1. Let X be a based space. The n-th Gottlieb group Gn (X) ⊂ πn (X) is the image under
the homomorphismev∗ : πn (Map(X, X; idX )) → πn (X).
Proposition 7.2. Let X be a non-degenerately based space and α ∈ πn (X). Then the following three
conditions are equivalent:(i) α ∈ Gn (X),(ii) the map (α, id) : S n ∨ X → X extends to a map S n × X → X,(iii) in the homotopy exact sequence of the evaluation fiber sequence Map0 (X, X; id) → Map(X, X; id) →X, ∂α = 0 in πn−1 (Map0 (X, X; id)).
Proposition 7.3. Let X be a based CW complex. Then the following properties hold.
(i) The group G1 (X) is contained in the center of π1 (X).
(ii) For the universal covering p : X̃ → X and α ∈ π1 (X), α ∈ G1 (X) if and only if the covering transformation X̃ → X̃ corresponding to the loop α is (basepoint-freely) homotopic to theidentity id : X̃ → X̃.
Proof. (i) Consider a commutative diagram∂ΩX/ Map (X, X; id)
0

e∗

ΩX

∂

ΩX

∂


/ Map (ΣΩX, X; ev)0/ Map (ΩX, ΩX; id).
0Then, the bottom horizontal arrow is the adjoint of the conjugation ΩX ×ΩX → ΩX. This implies thatthe kernel of ∂ : π1 (X) → π0 (Map0 (ΣΩX, X; ev)) is just the center of π1 (X). Thus G1 (X) is containedin the center of π1 (X).
(ii) Every extension S 1 × X → X of (α, id) lifts to the desired homotopy of maps X̃ → X̃.

Example 7.4. It is obvious that if X is an H-space, then Gn (X) = πn (X) for any n.
Gottlieb [Gottlieb65] applied G1 (X) as follows.
Theorem 7.5. Let X be a compact connected polyhedron, to which the Lefschetz fixed point theorem
can be applied. Then, if the Euler characteristic of X is not zero, we have G1 (X) = 0.

12

MITSUNOBU TSUTAYA

Outline of the proof. We use the Nielsen theory. All of the fixed points of id : X → X is contained in
the class of the identity id : X̃ → X̃. Then the Lefschetz number is equal to the fixed point index ofthis class. By the Lefschetz fixed point theorem, it is equal to the Euler characteristic of X. For any
α ∈ G1 (X), the class of the corresponding covering map is freely homotopic to the identity id : X̃ → X̃.
As is well-known, this implies that their fixed point indices are the same. Thus we conclude that α = 0or the Euler characteristic is zero.
Jiang applied G1 (X) to the Nielsen fixed point theory. For details, see [Jiang83].
8. Another application: Universal fibration
Let X be a based CW complex. Let us denote the subspaces of homotopy equivalences by Eq0 (X) ⊂Map0 (X, X) and Eq(X) ⊂ Map(X, X), which are considered to be based at the identity map. Since themap Eq(X) → X comes from the action of Eq(X) on X, the evaluation fiber sequence extends to theright with appropriate replacement of spaces by Corollary 2.4:ev

ΩX → Eq0 (X) → Eq(X) −→ X → B Eq0 (X) → B Eq(X).
Gottlieb [Gottlieb73] proved that the last three terms classify the fibration with homotopy fiber X.
Theorem 8.1. Suppose that X is a based CW complex. Then, for any map E → B between based CW
complexes with homotopy fiber X, there exists a homotopy pullback squareE/ B Eq (X)/ B Eq(X),B

0where a map B → B Eq(X) admitting such a homotopy pullback is unique up to (basepoint-free)homotopy.
By this theorem, we have a commutative diagram
πn (X)/ πn (E)

πn (X)

/ πn−1 (Map (X, X; id)).
0This means that the connecting map πn (X) → πn−1 (Map0 (X, X; id)) of the evaluation fiber sequence isuniversal among the inclusions of homotopy fibers in some sense.
Corollary 8.2. Suppose that X is a based CW complex. Then, the n-th Gottlieb group Gn (X) is
characterized as the union of the kernel of the induced map πn (X) → πn (E) with respect to any
fibration X → E → B.
As a consequence of this corollary, if α ∈ πn (X) is not contained in the Gottlieb group Gn (X), thenthe image of α under the map πn (X) → πn (E) never vanishes for any fibration X → E → B.
References[Aguadé87] J. Aguadé, Decomposable free loop spaces, Canad. J. Math. 39 (1987), 938–955.
[Arkowitz62] M. Arkowitz, The generalized Whitehead product, Pacific J. Math. 12 (1962), 7–23.
[DW95] W. G. Dwyer and C. Wilkerson, The center of a p-compact group, Contemp. Math. 181 (1995), 119–157.
[Gottlieb65] D. H. Gottlieb, A certain subgroup of the fundamental group, Amer. J. Math. 87 (1965), 840–856.
[Gottlieb69a] D. H. Gottlieb, Covering transformations and universal fibrations, Illinois J. Math. 13, (1969), 432–437.

EVALUATION FIBER SEQUENCE AND HOMOTOPY COMMUTATIVITY

13[Gottlieb69b] D. H. Gottlieb, Evaluation subgroups of homotopy groups, Amer. J. Math. 91, (1969), 729–756.
[Gottlieb73] D. H. Gottlieb, The total space of universal fibrations, Pacific J. Math. 46 (1973), 415–417.
[HK11] Y. Hemmi and Y. Kawamoto, Higher homotopy commutativity and the resultohedra, J. Math. Soc. Japan 63(2011), 443–471.
[Jiang64] B. J. Jiang, Estimation of the Nielsen numbers, Acta Math. Sinica, Chinese Series, 14 (1964), 304–312.
[Jiang83] B. J. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14 (1983).
[KK10] D. Kishimoto and A. Kono, Splitting of gauge groups, Trans. Amer. Math. Soc. 362 (2010), 6715–6731.
[Lang73] G. E. Lang, The evaluation map and EHP sequences, Pacific J. Math. 44 (1973), 201–210.
[May73] J. P. May, The geometry of iterated loop spaces, Lecture Notes in Mathematics 271, Springer Berlin, 1972.
[May75] J. P. May, Classifying spaces and fibrations, Mem. Amer. Math. Soc. 155 (1975).
[McGibbon89] C. A. McGibbon, Higher forms of homotopy commutativity and finite loop spaces, Math. Z. 201 (1989),
363–374.
[Neisendorfer10] Joseph Neisendorfer, Algebraic Methods in Unstable Homotopy Theory, New Mathematical Monographs, Cambridge University Press, 2010.
[Saümell95] L. Saümell, Higher homotopy commutativity in localized groups, Math. Z. 219 (1995), 339-347.
[Smith10] S. B. Smith, The homotopy theory of function spaces: a survey, Contemp. Math. 519 (2010), 3–39.
[Stasheff63] J. D. Stasheff, Homotopy associativity of H-spaces. II, Trans. Amer. Math. Soc. 108 (1963), 293–312.
[Sugawara60] M. Sugawara, On the homotopy-commutativity of groups and loop spaces, Mem. College Sci. Univ. Kyoto
Ser. A Math. 33 (1960), 257–269.
[Tsutaya] M. Tsutaya, Mapping spaces from projective spaces, preprint, arXiv:1408.2010.
[Whitehead41] J. H. C. Whitehead, On adding relations to homotopy groups, Ann. of Math. 42 (1941), 409–428.
[Williams69] F. D. Williams, Higher homotopy-commutativity, Trans. Amer. Math. Soc. 139 (1969), 191–206.
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
E-mail address: tsutaya@math.kyoto-u.ac.jp