数理解析研究所講究録
第 838 巻 1993 年 35-3935

A NOTE ON SPACES OF RATIONAL FUNCTIONSMARTIN GUEST
ANDRZEJ KOZLOWSKIMITUTAKA MURAYAMA
KOHHEI YAMAGUCHIThe University of Rochester
Toyama International University
Tokyo Institute of Technology
The University of Electro-Comnunications\S 1. INTRODUCTIONSpaces of holomorphic maps between complex manifolds have played an importantrole in topology and mathematical physics. In this note, we shall consider the topologyof spaces of holomorphic self-maps of the Riemann sphereand announce
the main results of [GKMY].
For a non-negative integer , let
be the space of all holomorphic maps fromtoconsisting of all maps whichof degree . Let
be the subspace ofpreserve a basepoint of. The corresponding space of continuous maps of degree$Map_{d}$, and the subspace of based maps byis denoted by. We call the map: $Xarrow Y$ a homotopy equivalence up to dimension if the induced homomorphism: $\pi_{j}(X)arrow\pi_{j}(Y)$ is bijective for all $j<d$ and surjective for $j=d$ . It is anelementary fact thatandare connected spaces. For $d=1$ , it is easy to see$Ho1_{1}\cong
PSL_{2}(\mathbb{C})$that. The following general results were obtained
andby Epshtein, J.D.S Jones and Segal:$S^{2}=\mathbb{C}\cup\{\infty\}$$d$$S^{2}$$S^{2}$$d$$Ho1_{d}$$Ho1_{d}$$Ho1_{d}^{*}$$S^{2}$$d$$Map_{d}^{*}$

$d$

$f$

$f_{*}$$Ho1_{d}$$Ho1_{d}^{*}$$Ho1_{1}^{*}\cong \mathbb{C}^{*}\cross \mathbb{C}$Theorem ([Ep], [Se]). Let be a positive in teger.
(1) $\pi_{1}(Ho1_{d})=\mathbb{Z}/2d$ .
.
(2)(3) The natural in usion maps
$d$

$0$

$\pi_{1}(Ho1_{d}^{*})=\mathbb{Z}$$cl$$i_{d}$:$Ho1_{d}^{*}arrow Map_{d}^{*}$

andar homotopy $eq$ uivalen ces up to dimen sion .
$d$

$e$

$j_{d}$: $Ho1_{d}arrow Map_{d}$$\square$

The stable homotopy type ofwas studied in [CCMM] and in this note we shallextend the above result by determing some further homotopy groups of the spaces
and. Our main results are as follows:$Ho1_{d}^{*}$$Ho1_{d}$$Ho1_{d}^{*}$

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Theorem 1.
(1) For $k\geq 2$ ,$\pi_{k}(Ho1_{d})=\int\pi_{k}(S_{3})\pi_{k}(S^{3})\oplus\pi_{k}(S^{2})$$d=2d=1$$(\mathbb{Z}/2$$d\geq 3,$$k=2$(2) If $k\geq 3$ and $d\geq 3$ , then $\pi_{k}(Ho1_{d})=\pi_{k}(Ho1_{d}^{*})\oplus\pi_{k}(S^{3})$ .
(3) In par ticular, if $d>k\geq 3$ , then $\pi_{k}(Ho1_{d})=\pi_{k+2}(S^{2})\oplus\pi_{k}(S^{3})$ .

Theorem 2. The space $Ho1_{2}may$ be identified with a homogeneo us space of the form
, where $H$ is isomorphic to. In this semi-direct product,$\sigma^{4}=1>is$given bythe action of:. In particular,for.
is $hom$ otopy $eq$ uivalen to$(SL_{2}(\mathbb{C})\cross SL_{2}(\mathbb{C}))/H$$\mathbb{C}^{*}\rangle 4\mathbb{Z}/4$$\sigma\cdot\alpha=\alpha^{-1}$$\mathbb{Z}/4=<\sigma$$Ho1_{2}$$\alpha\in \mathbb{C}^{*}$$(S^{3}\cross S^{3})/(S^{1}\aleph \mathbb{Z}/4)$$t$

Theorem 3.
(1) The universal cover of(2) The universal cover of$Ho1_{2}^{*}$$Ho1_{2}$form$hom$ otopy $eq$ uivalent tois homotopy equivalent to .
may be identified with a homogeneous space of the, where $D$ is isomorphic to. In particular, It is$S^{2}$$(SL_{2}(\mathbb{C})\cross SL_{2}(\mathbb{C}))/D$$\mathbb{C}^{*}$$S^{3}\cross S^{2}$.

In this note, we shall discuss only Theorem 2 and (1) of Theorem 3, referring to[GKMY] for the remaining results.

Acknowled.qements: The first, second and fourth authors are indebted to the Mathematics Department of Tokyo Institute of Technology for its hospitality. A gap in an earlier
version of the proof of Theorem 3 was filled thanks to a suggestion of Professor A. Kono.
The authors are grateful to Professors Cohen and Kono for their kind assistance.

\S 2. SKETCH PROOF OF THEOREM 2
From now on, we identify
with the space of functions $f=p_{1}/p_{2}$ , whereare coprime polynomials such that $\max\{\deg(p_{1}), \deg(p_{2})\}=d$ . The group
acts onby pre- and post-compositions: for $(A, B)\in Ho1_{1}\cross Ho1_{1}$ and $f\in Ho1_{d}$ we have
$Ho1_{d}$$p_{1},p_{2}$$Ho1_{1}$$Ho1_{d}$$(A, B)\cdot f(z)=A\langle f(B^{-1}(z))$.

The following proposition is well known:
Proposition 2.1. The group $Ho1_{1}\cross Ho1_{1}$ acts transitively on$Ho1_{2}$.

$\square$lt is well known that the map$SL_{2}(\mathbb{C})arrow Ho1_{1}$,$(\begin{array}{ll}a bc d\end{array})\frac{az+b}{cz+d}$

is a double covering and induces an isomorphism
the groupacts (transitively) on$PSL_{2}(\mathbb{C})\cong Ho1_{1}$$G=SL_{2}(\mathbb{C})\cross SL_{2}(\mathbb{C})$$Ho1_{2}$.

of Lie groups. Thus

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Lemma 2.2. Let $H$ denote the isotropy $su$ bgro up ofThen$H=\{(\pm(\begin{array}{ll}\alpha^{2} 00 \alpha^{-2}\end{array})$$(\begin{array}{ll}\alpha 00 \alpha^{-1}\end{array}))$$SL_{2}(\mathbb{C})\cross SL_{2}(\mathbb{C})$$(\pm(\begin{array}{ll}0 i\alpha^{2}i\alpha^{-2} 0\end{array})$

Proof. This follows by direct calculation.

at $z^{2}\in Ho1_{2}$ .

$(\begin{array}{ll}0 \alpha-\alpha^{-1} 0\end{array}))$:$\alpha\in \mathbb{C}^{*}\}$.

$\square$

It Is also easy to establish:

Lemma2.3. $LetK=\mathbb{C}^{*}\aleph \mathbb{Z}\prime 4bethegroupdeBnedbytheactionof\mathbb{Z}/4=<\sigma:\sigma^{4}=$$1>on$ “ byfor. Then $H$ and $K$ are isom orphic Lie groups.
$\mathbb{C}$$\sigma\cdot\alpha=\alpha^{-1}$$\alpha\in \mathbb{C}^{*}$$\square$

Proof of Theorem 2. The first part of the Theorem follows easily from Proposition 2.1,Lemma 2.2 and Lemna 2.3. The inclusion map of the maximal compact subgroup$SU(2)=S^{3}$ ofinduces the homotopy equivalence$(S^{3}\cross S^{3})/(S^{1}\rangle\triangleleft \mathbb{Z}/4)\simeq$$SL_{2}(\mathbb{C})$$Ho1_{2}$.

$\square$\S 3. THE UNIVERSAL COVER OF$Ho1_{2}^{*}$In this section, we shall show that the universal cover ofto$S^{2}$$Ho1_{2}^{*}$.

Proposition 3.1 ([CS]). There is a fi bration

is homotopy equivalent$\pi$$S^{1}arrow Ho1_{2}^{*}arrow \mathbb{R}P^{2}$.

Remark. R. Cohen and D.Shimamoto ([CS]) deduce this from results of Donaldson ([D])
and Atiyah and Hitchin ([AH]) on monopoles. Although we give an elementary proof
using the homogeneous structure ofin [GKMY], we shall give here an alternativedirect proof.
$Ho1_{2}$Proof. We may identify$\mathbb{C}\cup\{\infty\}$with the space of all holomorphic self maps of $S^{2}=$of degree 2 with basepoint condition $f(\infty)=0$ . Then any $f\in Ho1_{2}^{*}$ is of the$f$$Ho1_{2}^{*}$form$f(z)=(az+b)/(z^{2}+cz+d)$.
where the polynomials $az+b$ and $z^{2}+cz+d$ are coprime, $(a, b)\neq\{0,0$ ) and $a,$, let $Q_{m,n}(U, V)$ be the space of all disjoint
For each pair $(U, V)$ of subspaces ofpositive divisors of $Sp^{m}(U)\cross Sp^{n}(V)$ ,$b,$ $c,$ $d\in \mathbb{C}$$S^{2}$$Q_{m,n}(U, V)=\{(\xi, \eta)\in Sp^{m}(U)\cross Sp^{n}(V) : \xi\cap\eta=\emptyset\}$,where $Sp^{k}(X)$ denotes the k-th symmetric product of $X,$ $Sp^{k}(X)=X^{k}/\Sigma_{k}$ . (symmetric group on letters.)$k$$\Sigma_{k}$is the38
The action ofon
and the quotient spacethat.
Observe that(by multiplication) induces a free action on. It suffices to show
may be identified with$S^{2}=\mathbb{C}U\{\infty\}$$\mathbb{C}^{*}$$Ho1_{2}^{*}/\mathbb{C}^{*}$$Ho1_{2}^{*}$$Q_{1,2}(S^{2}, \mathbb{C})$$Q_{1,2}(S^{2}, \mathbb{C})\simeq \mathbb{R}P^{2}$$Q_{1,2}(S^{2}, \mathbb{C})\simeq Q_{1,2}(S^{2}, D_{-})=Q_{1,2}(\overline{D}_{+}, D_{-})\cup Q_{1,2}(\overline{D}_{-}, D_{-})$,whereare the open northern and southern hemispheres of $S^{2}=$ { $z\in R^{3}$ : llzll $=1$ }.
The map
given by$D_{\pm}$$u:Q_{1,2}(\overline{D}_{+}, D_{-})arrow Q_{1,2}(\overline{D}_{+}, \{0\})\cong\overline{D}_{+}$$(\alpha, \beta_{1}+\beta_{2})\mapsto(\alpha, 0+0)$

is a homotopy equivalence.
The map$v:Q_{1,2}(\overline{D}_{-}, D_{-})arrow \mathbb{C}^{*}$

given by$(\alpha, \beta_{1}+\beta_{2})\mapsto(\alpha-\beta_{1})(\alpha-\beta_{2})$is also a homotopy equivalence. Regarding the intersection$Q_{1,2}(\overline{D}+, D_{-})\cap Q_{1,2}(\overline{D}_{-}, D_{-})=Q_{1,2}(S^{1}, D_{-})$,we have the homotopy commutative diagram$Q_{1,2}(S^{1}, D_{-})arrow Q_{1,2}(S^{1}, D_{-})$$u\downarrow$$v\downarrow$$Q_{1,2}(S^{1}, \{0\})arrow^{f}$

It follows from the definition of the map$v$$C^{*}$

that the bottom map$Q_{1,2}(S^{1}, \{0\})\cong S^{1}arrow \mathbb{C}^{*};$ $S^{1}\ni z\mapsto z^{2}\in \mathbb{C}^{*}$$f$is given by the map.

Hence.

$Q_{1,2}(S^{2}, \mathbb{C})\simeq Q_{1,2}(S^{2}, D_{-})\simeq \mathbb{C}^{*}\bigcup_{fou}Q_{1,2}(\overline{D}_{+}, D_{-})\simeq S^{1}\bigcup_{2}e^{2}\simeq \mathbb{R}P^{2}$$\square$

Proof of (1) of Theorem 3.
. Let $p:S^{2}arrow RP^{2}$ and :Consider the fibration
be the universal coverings. Since
is simply connected, there Is a lift :such that. It follows fron diagram chasing that :is$j\geq
2$. Since both spaces are simply connected, Is a homotopy
an isomorphism for all$S^{1}arrow Ho1_{2}^{*}arrow\pi \mathbb{R}P^{2}$$q$$H^{\sim}o1_{2}^{*}$$p\circ\theta=\pi\circ q$$\theta$$\theta_{*}$$\square$$H^{\sim}o1_{2}^{*}arrow S^{2}$$\pi_{j}(H^{\sim}o1_{2}^{*})arrow\pi_{j}(S^{2})$$\theta$equivalence.

$H^{\sim}o1_{2}^{*}arrow Ho1_{2}^{*}$

39-operad structure on Hol*
Remark. Using Theorem 3, we can prove that thegiven by [BM] is not compatible with that onup to homotopy. See[GKMY] for the details. 口$=$$C_{2}$$\Omega^{2}S^{2}$$\coprod_{d\geq 0}Ho1_{d}^{*}$REFERENCES[AH] M.F. Atiyah and N.J. Hitchin, The Geometry and Dynamics of Magnetic Monopoles, Princeton
Univ. Press, 1988.
[BM] C.P. Boyer and B.M. Mann, Monopoles, non-linear models and two-fold loop spaces, Commun.
Math. Phys. 115 (1988), 571-594.
[CCMM] F.R. Cohen, R.L. Cohen, B.M. Mann and R.J. Milgram, The topology of rational functions
and divisors of surfaces, Acta Math. 166 (1991), 163-221.
[CS] R.L. Cohen and D.H. Shimamoto, Rational functions,configurations and Hilbert schemes,J. Lond. Math. Soc. 43 (1991), 509-528.
[D] S.K. Donaldson, Nahm’s equations and the classification of monopoles, Commun. Math. Phys. 96(1984), 387-407.
[E] S.I. Epshtein, Fundamental groups of spaces of coprime polynomials, Functional Analysis and its
Applications 7 (1973), 82-83.
[GKMY] M.A. Guest, A. Kozlowski, M. Murayama and K.Yamaguchi, The homotopy type of spacesof rational functions, preprint (1992).
[S] G.B. Segal, The topology of spaces of rational functi\‘ons, Acta Math. 143 (1979), 39-72.
$\sigma$$\iota_{abe}u_{ed}$