数理解析研究所講究録
第 1922 巻 2014 年 166-169

166

ON EVALUATION FIBER SEQUENCES
MITSUNOBU TSUTAYAABSTRACT. We denote the n-th projective space of a topological monoid by
and the classifying
space by $BG$ . Our main result states that, if is a topological group, then the evaluation fiber sequence$Map_{0}(B_{n}G, BG)arrow Map(B_{n}G, BG)arrow BG$ extends to the right. This theorem is proved by the techniqueof -maps.
$G$$B_{n}G$$G$$A_{n}$

1. INTRODUCTION
The aim of this note is to present the result of the author’s paper [Tsu].
First of all, let us review the history of -theory. $H$ -space was introduced by J.-P. Serre, which isnamed after H. Hopf. A pointed space is said to be an $H$-space if a continuous binary operation: $G\cross Garrow G$ of which the identity element is the basepoint is given. Every topological monoid isof courese an $H$ -space. But the converse does not hold.
J. F. Adams [Ada60] proved that an -dimensional sphere admits a structure of an $H$-space if andonly if $n=0$ , 1, 3 or 7. But I. M. James [Jam57] proved thatnever admit a homotopy associative$H$ -structure. An $H$-space $(G, m)$ is said to be homotopy associative if the maps $m\circ(id_{G}\cross m)$ andare homotopic. So, there is some diﬀerence between $H$ -spaces and topological monoids.
Then, how about the diﬀerence between homotopy associative $H$ -spaces and topological monoids?
J. D. Stasheﬀ $[Sta63a]$ introduced the notion of -spaces. $H$ -space and homotopy associative$H$ -space are nothing but-spaces and -spaces, respectively. Every topological monoid is an-space is an $H$-space with some higher homotopy associativity of orderspace. For generalin some sense. This homotopy associativity data is called -form. His work revealed that being a
topological monoid is much far from being an $H$ -space.
The same thing can be said for maps between topological monoids. A map $f$ : $Garrow G’$ is said to be
an $H$ -map if the maps
and $(g_{1},g_{2})\mapsto f(g_{1})f(g_{2})$ are homotopic. Every continuous
homomorphism between topological monoids is an $H$ -map but the converse is false.
M. Sugawara [Sug60] given a condition for that a map between topological monoid induces amap between their classifying spaces. As is well-known, a homomorphism $Garrow G’$ induces a map$Bf$ : $BGarrow BG’$ between the classifying spaces. He introduced the notion strongly multiplicative
map and proved that a strongly multiplicative map induces a map between the classifying spaces.
After that, Stasheﬀ $[Sta63b]$ proved the converse for path-connected topological monoids. Being astrongly multiplicative map requires a map to preserve the infinitely higher homotopy associativityof topological monoids. Stasheﬀ weakened the condition and defined -map between topologicalmonoids. -map is also defined by the data of preservation of higher homotopy associativity, called$A_{n}$$G$$m$$S^{n}$$n$$S^{7}$$m\circ(m\cross id_{G})$$A_{n}$$A_{2}$$n,$$A_{\infty}-$$A_{3}$$A_{n}$$n$$A_{n}$$(g_{1},g_{2})\mapsto f(g_{1}g_{2})$$A_{n}$$A_{n}$$A_{n}$-form.

By definition of -map, one may expect that -map is generalized for morphism betweenmaps. Stasheﬀ [Sta70] described the condition only for $n\leq 4$ because it needs combinatoriallycomplicated cell complex, called multiplihedra. Abstractly, this was done by Boardman-Vogt [BV73]$A_{n}$$A_{n}$

Key words and phrases. mapping space, homotopy fiber sequence,$A_{n^{-}}$$A_{n}$-space,$A_{n}$-map, gauge group.

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MITSUNOBU TSUTAYA

using their”’ -constmction”’. Independently, N. Iwase also constructed those complexes in his masterthesis [Iwa83]. He described their combinatorial structure as well.
$W$

2. SPACE 0F$A_{n}$-MAps-maps between topologicalmonoids. Consider a path-connected topological monoid . Let us denote the n-th projective space
and $BG=B_{\infty}G$ , respectively. We remark that there is the canonicaland the classifying space bywith the reduced suspension.
for $n_{1}<n_{2}$ and the homeomorphism
inclusion$EG$$EGarrow
BG$
The restriction $E_{n}Garrow$contractible.
withThere is a universal principal G-fibration
overis also a principal G-fibration. Using the canonical homotopy cofiber sequence
Now, to state our result, we get back to the work of Stasheﬀ about$A_{n}$$G$$B_{n}G$$B_{1}G\cong\Sigma G$$B_{n_{1}}G\subset B_{n_{2}}G$$B_{n}G$$B_{n}G$$E_{n}Garrow B_{n}Garrow B_{n+1}G$

and the homotopy fiber sequence$E_{n}Garrow B_{n}Garrow BG,$

Stasheﬀ’s result [ $Sta63b$ , Theorem 4.5] is rephrased as follows.
Theorem 2.1 (Stasheﬀ, 1963). Letand $G’$ be connected topological monoids which are $CW$
$Garrow
G’$ is an$f$-map if and only if the reduced suspension
complex. Then, a pointed map::
can be extended to a map $B_{n}Garrow BG’.$$G$$A_{n}$$\Sigma f$$\Sigma Garrow\Sigma G’\subset BG’$

Theorem 2.1 states nothing about the correspondence of -forms. Our result refines this point.
Let us denote the space of continuous maps between and by Map $(X, Y)$ and that of pointed
ones by $Map_{0}(X, Y)$ . To guarantee the exponential law, we always work in the category of compactly
generated spaces and the mapping spaces are considered in the manner of compactly generated spaces.
is denoted by $ﬄ_{n}(G, G’)$ .
The space of -maps with -forms between and$A_{n}$$X$$A_{n}$$G$$A_{n}$$G$$Y$’Theorem 2.2 (T). Let be a well-pointed topological monoid of homotopy type of a $CW$ complex and$G’$ a well-pointed grouplike topological monoid. Then the following composite is a weak equivalence.
$G$$\ovalbox{\tt\small REJECT}_{n}(G,G’)arrow Map_{0}(B_{n}G, B_{n}G’)B_{n}arrow Map_{0}(B_{n}G, BG(\iota_{n})_{\#}$A pointed space is said to be well-pointed if the basepoint of has the homotopy extension
is a group with respect to theis said to be grouplike if
property. A topological monoid
multiplication induced from that of . The map is given by Sugawara’s construction [Sug60] andis the composition with the inclusion : $B_{n}Garrow BG$ . In [Tsu], the author constructsthe mapas aa topological categoryof topological monoids and -maps between them and realizes
continuous functor fromto the category of (compactly generated) pointed spaces.
$X$

$X$

$G$

$\pi_{0}(G)$$G$$(\iota_{n})_{\#}$$B_{n}$$\iota_{n}$$\ovalbox{\tt\small REJECT} l_{n}$$A_{n}$$B_{n}$$ﬄ_{n}$

3. EVALUATION FIBER SEQUENCE
Next, we explain the main result on evaluation fiber sequences. If $X$ is well-pointed, then theevaluation

Map $(X, Y)arrow Y$at the basepoint is a Hurewicz fibration of which the fiber over the basepoint is $Map_{0}(X, Y)$ . This
fiber sequence is called the evaluation fiber sequence. Roughly, our second result states that thisfiber sequence extends to the right if $X=B_{n}G$ and $Y=BG$ . One may think that this is strange.
Because it implies not only that the mapping space $Map_{0}(B_{n}G, BG)$ is equivalent to a topologicalmonoid, but also the connecting map $Garrow Map_{0}(B_{n}G, BG)$ is equivalent to a homomorphism between

168
ON EVALUATION FIBER SEQUENCEStopological monoids. But Theorem 2.2 claims that $Map_{0}(B_{n}G, BG)$ is equivalent to the topologicalmonoid $fl_{n}(G,G)$ .
There is a well-known adjunction$Map_{0}(\Sigma X, Y)\cong Map_{0}(X, \Omega Y)$

for pointed spaces and , whereis the space of based loops in
For a topological monoid , define the subspace $\overline{Map}_{0}(B_{n}G, BG)\subset Map_{0}(B_{n}G, BG)$ consistingof maps $B_{n}Garrow BG$ which restricts to a map $\Sigma Garrow BG$ with the adjoint $Garrow\Omega BG\simeq G$ is ahomotopy equivalence. We also define the subspace $\overline{Map}(B_{n}G, BG)\subset Map(B_{n}G, BG)$ as the unionof the path-components that intersect with.
For a topological group , the composition of the conjugation $Garrow G$ by an element of to an-map from the left defines the left action of on. In particular, the composition to theidentity map defines a map :.
Note that there is a natural well-pointed replacement $tWGarrow G$ of a topological monoid , which
is a homomorphism between topological monoids and is a weak equivalence.
$X$

$Y$

$\Omega Y$

$Y.$

$G$

$\overline{Map}_{0}(B_{n}G, BG)$

$G$

$G$

$G$

$A_{n}$$\delta$$\ovalbox{\tt\small REJECT}_{n}(G, G)$$Garrow\ovalbox{\tt\small REJECT}_{n}(G, G)$$G$

Theorem 3.1 (T). Let
Consider a map$G$be a well-pointed topological group of homotopy type of a $CW$ complex.
defined by the composite$BGarrow B’W\ovalbox{\tt\small REJECT}_{n}(G, G; eq)$

$BGarrow\simeq KWGarrow KW\ovalbox{\tt\small REJECT}_{n}(G, G;KW\delta eq)$.

Then the sequence$\overline{Map}_{0}(B_{n}G, BG)arrow$

Map $(B_{n}G, BG)arrow BGarrow KW\ovalbox{\tt\small REJECT}_{n}(G, G; eq)$ ,is a homotopy fiber sequence.
When $n=\infty$ , the above sequence can be extended as$\overline{Map}_{0}(BG, BG)arrow\overline{Map}(BG, BG)arrow eBGarrow B^{t}W\overline{Map}_{0}(BG, BG)arrow B^{t}W\overline{Map}(BG, BG)$,

where the subspaces $\overline{Map}_{0}(BG, BG)\subset Map_{0}(BG, BG)$ and Map $(BG, BG)\subset Map(BG, BG)$ are exactly those of homotopy equivalences $BGarrow BG$ . For details, see Gottlieb’s paper [Got73].

4.

$A_{n}-$lYPES OF GAUGE GROUPS

Theorem 2.2 and 3.1 may be applied in many situations. As an application of them, we obtain theresult for -types of gauge groups. For a principal -bundle $Parrow B$ , the topological groupconsisting of -equivariant maps $Parrow P$ that induces the identity on is called the gauge group of $P.$
The associated bundle $adP=P\cross_{G}G$ with respect to the adjoint action of is called the adjoint bundle
and becomes a bundle of topological groups. The space of sections $\Gamma(adP)$ is naturally isomorphic tothe gauge group. For example, the classification of the -types of the gauge groups of principal$SU(2)$ -bundles overis investigated by Kono [Kon91], by Crabb-Sutherland [CS00], by Tsukuda[TsuOl] and by the author [Tsu12]. For homotopy types, there are many related works.
$G$$A_{n}$$\mathcal{G}(P)$$G$

$B$

$G$

$A_{n}$$\mathcal{G}(P)$$S^{4}$

Theorem 4.1 (Kishimoto-Kono, 2010 and T). Let be a well-pointed topological group and bea pointed space, both of which have the pointed homotopy types of $CW$ complexes. For a principal-bundle over classified by : $Barrow BG$, the following conditions are equivalent:
$B$

$G$

$G$

$P$

$B$

(i) ad $P$ is(ii) the map$A_{n}$$\epsilon$-trivial: $B\vee B_{n}Garrow BG$ extends over the product$(\epsilon, \iota_{n})$(iii) the composite$Barrow BG\epsilonarrow\theta W\ovalbox{\tt\small REJECT}_{n}(G, G; eq )$$B\cross B_{n}G,$is null-homotopic.

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MITSUNOBU TSUTAYA

A bundle of topological monoids $Earrow B$ is said to be -trivial if there exist a topological monoid
and a“fiberwise -equivalence”’ $B\cross Garrow adP$ . The -triviality of ad implies the -equivalence
ofand Map $(B, G)$ . The equivalence of (i) and (ii) have already been known by Kishimoto-Kono$A_{n}$$G$$A_{n}$$A_{n}$$P$$A_{n}$$\mathcal{G}(P)$[KK10].
In [Tsu], the author applied Theorem 2.2 and 3.1 to higher homotopy commutativity and to cyclicmaps as well.
5. FUTURE W0RK
It seems that Theorem 3.1 has several applications to homotopy theory of gauge groups and higher
homotopy commutativity. Another direction is generalizations of Theorem 2.2. The author is trying
to generalize Theorem 2.2 for the-functors between small topological categories.
$A_{\infty}$REFERENCES[Ada60] J. F. Adams, On the non-existence of elements of Hopfinvariant one, Ann. of Math. 72 (1960), 20-104.
[BV73] J. M. Boardman and R. M. Vogt, Homotopy invariant structures on topological spaces, Lecture Notes in Mathe-matics 347, Springer-Verlag, Berlin, 1973.
[CS00] M. C. Crabb and W. A. Sutherland, Counting homotopy types of gauge groups, Proc. London Math. Soc. 81(2000), 747-768.
[Got73] D. H. Gottlieb, The total space of universal fibrations, Pacific J. Math. 46 (1973), 415-417.
[Iwa83] N. Iwase, On the ring structure of $K^{*}(XP^{n})$ (Japanese), Master Thesis, Kyushu Univ., 1983.
[Jam57] I. M. James, Multiplication on spheres. II, Trans. Amer. Math. Soc. 84 (1957), 545-558.
[KK10] D. Kishimoto and A. Kono, Splitting of gauge groups, Trans. Amer. Math. Soc. 362 (2010), 6715-6731.
[Kon91] A. Kono, A note on the homotopy type of certain gauge groups, Proc. Roy. Soc. Edinburgh: Sect. A117 (1991),295-297.
[Sta63a] J. D. Stasheﬀ, Homotopy associativity of $H$-spaces. I, Trans. Amer. Math. Soc. 108 (1963), 275-292.
[Sta63b] J. D. Stasheﬀ, Homotopy associativity of $H$-spaces. II, Trans. Amer. Math. Soc. 108 (1963), 293-312.
[Sta70] J. D. Stasheﬀ, $H$ -spaces from a homotopy point of view, Lecture Notes in Mathematics 161, 1970.
[Sug60] M. Sugawara, On the homotopy-commutativity of groups and loop spaces, Mem. College Sci. Univ. Kyoto Ser.
A Math. 33 (1960), 257-269.
[TsuOl] S. Tsukuda, Comparing the homotopy types of the components of Map $(S^{4}, BSU(2))$ , J. Pure and Appl. Algebra
161 (2001), 235-247.
[Tsu12] M. Tsutaya, Finiteness of -equivalence types of gauge groups, J. London Math. Soc. 85 (2012), 142-164.
[Tsu] M. Tsutaya, Mapping spaces from projective spaces, arXiv:1408.2010, preprint.
$A_{n}$

DEPARTMENT 0F MATHEMATICS, KYOTO UNIVERSITY, KYOTO606-8502, JAPAN-mail address: tsutaya@math.kyoto-u.ac.jp$E$