The homotopy type of the space of algebraic loops on a toric variety
aa r X i v : . [ m a t h . A T ] S e p The homotopy type of the space ofalgebraic loops on a toric variety
Andrzej Kozlowski ∗ and Kohhei Yamaguchi † Abstract
We investigate the homotopy type of the space of tuples of poly-nomials inducing base-point preserving algebraic maps from the circle S to a toric variety X Σ . In particular, we prove a homotopy stabilityresult for this space by combining the Vassiliev spectral sequence [27]and the scanning map [26]. For two topological spaces X and Y with base-points, let Map ∗ ( X, Y ) de-note the space of all continuous based maps (i.e. base-point preserving maps) f : X → Y with compact-open topology. When these two spaces have someadditional structure, e.g. that of a complex or symplectic manifold or an al-gebraic variety, it is natural to consider the subspace S ( X, Y ) ⊂ Map ∗ ( X, Y )of all based maps f which preserve this structure and to ask whether theinclusion map i : S ( X, Y ) → Map ∗ ( X, Y ) is a homotopy or homology equiv-alence up to some dimension. Early examples of this type of phenomenoncan be found in [9]. In many cases of interest the infinite dimensional space S ( X, Y ) has a filtration by finite dimensional subspaces, given by some kindof “degree of maps”, and the topology of these finite dimensional spacesapproximates the topology of the entire space of continuous maps; the ap-proximation becoming more accurate as the degree increases. A great deal ofattention has been devoted to problems of the above kind in the case where ∗ Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2,02-097 Warsaw, Poland (E-mail: [email protected]) † Department of Mathematics, University of Electro-Communications, Chofu, Tokyo182-8585, Japan (E-mail: [email protected]); The second author is supported by JSPSKAKENHI Grant Number 18K03295.2010
Mathematics Subject Clasification.
Primary 55P10; Secondly 55R80, 55P35, 14M25.
Theorem 1.1 (G. Segal, [26]) . If d ∈ N , the inclusion map i d : Hol ∗ d ( S , C P n ) → Map ∗ d ( S , C P n ) = Ω d C P n ≃ Ω S n +1 is a homotopy equivalence through dimension (2 n − d − . Here, we denoteby N the space of all positive integers and the space Hol ∗ d ( S , C P n ) ( resp. Ω d C P n ) denotes the space consisting of all based holomorphic ( resp. basedcontinuous ) maps f : S → C P n of degree d . Remark 1.2.
Recall that a map g : V → W is called a homology ( resp.homotopy ) equivalence through dimension N if the induced homomorphism g ∗ : H k ( V ; Z ) → H k ( W ; Z ) (resp. g ∗ : π k ( V ) → π k ( W )) is an isomorphismfor all k ≤ N .Segal conjectured that this theorem should have analogues for holomor-phic maps from the Riemann sphere S (or even any compact Riemann sur-face) to various complex manifolds Y. He also suggested that some analoguesmay exists for higher dimensional complex manifolds in place of S . All ofSegal’s conjectures have been shown to be true, although no fully satisfactorygeneral explanation of these phenomena has been found.Guest [11] was the first one to consider Segal’s problem for the space ofholomorphic maps from S to a compact toric variety. His result was verygeneral but the stability dimension he obtained was rather low. A muchbetter stability bound was obtained by Mostovoy and Munguia-Villanueva[24]. Using methods developed by Vassiliev and Mostovoy [23], they wereable to obtain a Segal type theorem for the case of spaces of holomorphicmaps from C P n for n ≥ C P n as source, for technical reasons the targetspace has to be compact. That leaves out some interesting toric varieties forwhich Segal’s type theorems still hold (see for example [17]). By replacingMostovoy’s approach (which relies on the Stone-Weierstrass Theorem) wewere able to prove the Mostovoy and Munguia-Villanueva result for a largerclass of target varieties but with C P = S as the source space and this isstated as follows. Theorem 1.3 ([20]) . Let X Σ be a simply connected non-singular toric va-riety associated to the fan Σ such that the condition ( ) (see below) issatisfied. Then if D = ( d , · · · , d r ) ∈ N r and P rk =1 d k n k = r , the inclusion The stablization dimension (2 n − d − n ≥ n − d + 1) − If X Σ is compact, the condition (2.14.1) is always satisfied (see Remark 2.8). ap i D,hol : Hol ∗ D ( S , X Σ ) ⊂ −→ Ω D X Σ is a homotopy equivalence through dimension d ∗ ( D, Σ) if r min (Σ) ≥ and ahomology equivalence through dimension d ∗ ( D, Σ) = d min − if r min (Σ) = 2 .Here, Ω D X Σ (resp. Hol ∗ D ( S , X Σ )) denotes the space of based continuous(resp. based holomorphic) maps from S to X Σ of degree D , the number r min (Σ) is defined in (2.24) and d ∗ ( D, Σ) is the number given by (1.1) d ∗ ( D, Σ) = (2 r min (Σ) − d min − , where d min = min { d , · · · , d r } . In his seminal work Segal observed that a real analogue of his theoremalso holds. In fact, it was this real case that inspired the interest in the entirearea. He considered the space of real rational maps, that is pairs of monicpolynomials of the same degree with real coefficients and without common(complex) roots (see [26, Proposition 1.4]).Later in [13], [21] and [1], a different ‘real version’ of Seqal’s theory wasintroduced; it involves looking at tuples of real polynomials of the samedegree without common real roots. One such case was considered in [14],where the analogue of the Mostovoy and Munguia-Villanueva theorem wasproved for real rational maps from R P n to a compact complex toric variety.The method used in it was the same as in [24] and it was limited only to caseof compact toric varieties. By imposing a condition involving common real roots rather than complex ones, one can also study spaces of rational mapsfrom S to complex toric varieties.An important difference between them is that tuples of polynomials with-out common real roots no longer correspond to algebraic maps. For example,a pair ( p ( z ) , q ( z )) of monic polynomials of the same degree, without a com-mon real root defines a unique rational (algebraic) map S → C P but bymultiplying both polynomials by the same monic polynomial without realroots, we obtain the same rational map. Thus it is necessary to distinguishbetween spaces of tuples of polynomials and that of rational maps - theirimage in the space of continuous maps.Now we can state the main result of this paper (precise definitions of allthe terms used will be given in § Theorem 1.4 (Theorem 2.16, Theorem 2.20) . Let D = ( d , · · · , d r ) ∈ N r be an r -tuple of positive integers and let X Σ be a simply connected non-singular toric variety associated to the fan Σ such that the condition (2.14.1)is satisfied. Then the natural map j D : Pol ∗ D ( S , X Σ ) → Ω U ( K Σ ) ≃ Ω Z K Σ s a homotopy equivalence through dimension d ( D, Σ) , where d min denotesthe positive integer d min = min { d , · · · , d r } and the number d ( D, Σ) is givenby (1.2) d ( D, Σ) = 2( r min (Σ) − d min − . Here the space U ( K Σ ) (resp. Z K Σ ) is the complement U ( K Σ ) of the coordi-nate subspaces of type K Σ (resp. the moment-angle complex of the underly-ing simplicial complex K Σ ) and Pol ∗ D ( S , X Σ ) denotes the space of r -tuples ( p ( z ) , . . . , p r ( z )) ∈ C [ z ] r of monic polynomials of degrees d , . . . , d r satisfy-ing the condition (2.19.1). The image of this space under j D is contained inthe space Alg ∗ ( S , X Σ ) of based algebraic loops on X Σ . Corollary 1.5 (Corollary 2.17, Corollary 2.21) . Under the same assumptionsas Theorem 1.4, there is a natural map i D : Pol ∗ D ( S , X Σ ) → Ω X Σ which induces an isomorphism on homotopy groups ( i D ) ∗ : π k (Pol ∗ D ( S , X Σ )) ∼ = −→ π k (Ω X Σ ) ∼ = π k +1 ( X Σ ) for any ≤ k ≤ d ( D, Σ) . This paper is organized as follows. In § §
9. We also consider some example (Example 2.25).In § §
4, we recall the definitions of the non-degenerate simplicial resolutionand the associated truncated simplicial resolutions. In §
5, we construct theVassiliev spectral sequence and compute its E -terms. In § j D when P rk =1 d k n k = n .In § § § In this section we shall recall several basic definitions and facts for describingthe main results. 4 ans and toric varieties
A convex rational polyhedral cone σ in R n is asubset of R n of the form σ = Cone( S ) = Cone( m , · · · , m s ) = n s X k =1 λ k m k : λ k ≥ k o (2.1)for a finite set S = { m k } sk =1 ⊂ Z n . A convex rational polyhedral cone σ is called strongly convex if σ ∩ ( − σ ) = { n } , and its dimension dim σ is thedimension of the smallest subspace in R n which contains σ . A face τ of σ isa subset τ ⊂ σ of the form(2.2) τ = σ ∩ { x ∈ R n : L ( x ) = 0 } for some linear form L on R n , such that L ( x ) ≥ x ∈ σ . If { k : L ( m k ) = 0 , ≤ k ≤ s } = { i , · · · , i t } , we easily see that τ =Cone( m i , · · · , m i t ). Thus, a face τ is also a strongly convex rational poly-hedral cone if σ is so.A finite collection Σ of strongly convex rational polyhedral cones in R n iscalled a fan in R n if every face τ of σ ∈ Σ belongs to Σ and the intersectionof any two elements of Σ is a face of each.An n dimensional irreducible normal variety X (over C ) is called a toricvariety if it has a Zariski open subset T n C = ( C ∗ ) n and the action of T n C onitself extends to an action of T n C on X . The most significant property of atoric variety is the fact that it is characterized up to isomorphism entirely byits associated fan Σ. We denote by X Σ the toric variety associated to a fanΣ (see [7] for the details).Since the fan of T n C is { n } and the case X Σ = T n C is trivial, we alwaysassume that any fan Σ in R n satisfies the condition { n } $ Σ . Definition 2.1.
Let Σ be a fan in R n such that { n } $ Σ and let(2.3) Σ(1) = { ρ , · · · , ρ r } denote the set of all one dimensional cones in Σ. For each integer 1 ≤ k ≤ r ,we denote by n k ∈ Z n the primitive generator of ρ k , such that(2.4) ρ k ∩ Z n = Z ≥ · n k . Note that ρ k = Cone( n k ) = R ≥ · n k for each 1 ≤ k ≤ r . When S is the emptyset ∅ , we set Cone( ∅ ) = { n } and we may also regard it as oneof strongly convex rational polyhedral cones in R n , where we denote by n the zero vectorin R n defined by n = (0 , · · · , ∈ R n . olyhedral products Next, recall the definition of polyhedral products.
Definition 2.2.
Let K be a simplicial complex on the vertex set [ r ] = { , , · · · , r } , and let ( X, A ) = { ( X , A ) , · · · , ( X r , A r ) } be a set of pairs ofbased spaces such that A i ⊂ X i for each 1 ≤ i ≤ r .(i) Let Z K ( X, A ) denote the polyhedral product of (
X, A ) with respect to K given by the union Z K ( X, A ) = [ σ ∈ K ( X, A ) σ , where we set(2.5) ( X, A ) σ = { ( x , · · · , x r ) ∈ X × · · · × X r : x k ∈ A k if k / ∈ σ } . When ( X i , A i ) = ( X, A ) for each 1 ≤ i ≤ r , we write Z K ( X, A ) = Z K ( X, A ) . (ii) For a subset σ ⊂ [ r ], let L σ denote the coordinate subspace of C r defined by(2.6) L σ = { ( x , · · · , x r ) ∈ C r : x i = 0 if i ∈ σ } . Define the complement U ( K ) of the coordinate subspaces of type K by U ( K ) = C r \ [ σ ∈ I ( K ) L σ where we set I ( K ) = { σ ⊂ [ r ] : σ / ∈ K } . (2.7)(iii) For a fan Σ in R n , let K Σ denote the underlying simplicial complexof Σ defined by(2.8) K Σ = n { i , · · · , i s } ⊂ [ r ] : Cone( n i , n i , · · · , n i s ) ∈ Σ o . Note that K Σ is a simplicial complex on the vertex set [ r ]. Remark 2.3. (i) It is easy to see that the following equality holds:(2.9) U ( K ) = Z K ( C , C ∗ ) . (ii) The fan Σ is completely determined by the pair ( K Σ , { n k } rk =1 ) . Indeed, if we set C ( σ ) = Cone( n i , · · · , n i s ) for σ = { i , · · · , i s } ⊂ [ r ] and C ( ∅ ) = { n } , then it is easy to see that Σ = { C( σ ) : σ ∈ K Σ } . Let K be some set of subsets of [ r ]. Then the set K is called an abstract simplicialcomplex on the vertex set [ r ] if the following condition holds: if τ ⊂ σ and σ ∈ K , then τ ∈ K . In this paper by a simplicial complex K we always mean an an abstract simplicialcomplex , and we always assume that a simplicial complex K contains the empty set ∅ . omogenous coordinates Recall the homogenous coordinates on toricvarieties. Let Σ be a fan in R n as in Definition 2.1. Definition 2.4. (i) Let G Σ ⊂ T r C = ( C ∗ ) r denote the multiplicative subgroupof T r C defined by(2.10) G Σ = { ( µ , · · · , µ r ) ∈ T r C : r Y k =1 ( µ k ) h n k , m i = 1 for all m ∈ Z n } , where h , i denotes the standard inner product on R n given by h u , v i = P nk =1 u k v k for u = ( u , · · · , u n ) and v = ( v , · · · , v n ) ∈ R n .(ii) Consider the natural G Σ -action on Z K Σ ( C , C ∗ ) given by coordinate-wise multiplication, i.e.(2.11) µ · x = ( µ x , · · · , µ r x r )for ( µ, x ) = (( µ , · · · , µ r ) , ( x , · · · , x r )) ∈ G Σ × Z K Σ ( C , C ∗ ). We denote by(2.12) Z K Σ ( C , C ∗ ) /G Σ = U ( K Σ ) /G Σ the corresponding orbit space and let(2.13) q Σ : Z K Σ ( C , C ∗ ) → Z K Σ ( C , C ∗ ) /G Σ = U ( K Σ ) /G Σ denote the canonical projection.The following theorem, which plays a crucial role in the proof of our mainresult, states that in a toric variety, under certain mild conditions, one canconstruct “homogeneous coordinates” similar to those in a projective space. Theorem 2.5 ([5], Theorem 2.1) . If the set { n k } rk =1 of all primitive gener-ators spans R n ( i.e. P rk =1 R · n k = R n ) , there is a natural isomorphism (2.14) X Σ ∼ = Z K Σ ( C , C ∗ ) /G Σ = U ( K Σ ) /G Σ . By using the above result we can obtain the following result whose proofis postponed in the last part of this section.
Lemma 2.6 (cf. [6], Theorem 3.1) . Suppose that the set { n k } rk =1 of allprimitive generators spans R n , and let f k ∈ C [ z , · · · , z m ] be a homogenouspolynomial of the degree d ∗ k for each ≤ k ≤ r such that the polynomials { f k } k ∈ σ have no common real root except m +1 ∈ R m +1 for each σ ∈ I ( K Σ ) .Then there is a unique map f : R P m → X Σ such that the following diagram R m +1 \ { } ( f , ··· ,f r ) −−−−−→ U ( K Σ ) = Z K Σ ( C , C ∗ ) γ m y q Σ y R P m f −−−→ U ( K Σ ) /G Σ = X Σ s commutative if and only if P rk =1 d ∗ k n k = n , where γ m : R m +1 \ { } → R P m denotes the canonical double covering and the map q Σ is the canonicalprojection induced from the identification ( ) . Remark 2.7.
We call the map f determined by an r -tuple ( f , · · · , f r ) ofhomogenous polynomials as an algebraic map and we write f = [ f , · · · , f r ].Note that two different such r -tuples of polynomials can determine the samemaps. For example, suppose that ( f , · · · , f r ) is the r -tuple of homogenouspolynomials in C [ z , · · · , z m ] of degree d ∗ , · · · , d ∗ r satisfying the conditiongiven in Lemma 2.6. Then if ( a , · · · , a r ) ∈ N r is the r -tuple of positiveintegers and it satisfies the condition P rk =1 d ∗ k n k = P rk =1 a k n k = n , we caneasily see that f = [ f , · · · , f r ] = [ g a f , · · · , g a r f r ] = [ h a f , · · · , h a r f r ] for g = P mk =0 z k and h = ( z + z ) + P mk =2 z k . Assumptions
Let Σ be a fan in R n satisfying the condition (2.3) as inDefinition 2.1. From now on, we assume that the following two conditionshold.(2.14.1) There is an r -tuple D ∗ = ( d ∗ , · · · , d ∗ r ) ∈ N r of positive integers suchthat P rk =1 d ∗ k n k = n . (2.14.2) The set { n k } rk =1 of primitive generators spans Z n over Z . Remark 2.8. (i) Note that X Σ is a compact iff S σ ∈ Σ σ = R n [7, Theorem3.4.1]. Note also that X Σ is simply connected if and only if P rk =1 Z · n k = Z n (see Lemma 3.4 below). Hence, the condition (2.14.2) always holds if X Σ iscompact or simply connected. On the other hand, if the condition (2.14.2)holds, one can easily see that the set { n k } rk =1 spans R n over R , and there isan isomorphism (2.14) for the space X Σ .(ii) We know that the condition (2.14.1) holds if X Σ is compact and non-singular [6, Theorem 3.1].(iii) Let Σ denote the fan in R given by Σ = {{ } , Cone( n ) , Cone( n ) } for the standard basis n = e = (1 , , n = e = (0 , X Σ of Σ is C which has trivial homogenous coordinates. It is clearlya (simply connected) smooth toric variety, and the condition (2.14.1) alsoholds. However, in this case, P k =1 d ∗ k n k = iff ( d ∗ , d ∗ ) = (0 , R P m → X Σ = C other than the constant maps. Assuming the condition (2.14.1) guaranteesthe existence of non-trivial algebraic maps R P m → X Σ . Of course, it wouldbe sufficient to assume that D = ( d , . . . , d r ) = (0 , . . .
0) but if d i = 0 forsome i , then the number d ( D, Σ) (defined in (2.25)) is not a positive integerand our assertion (Theorem 2.16 below) is vacuous. For this reason, we willassume the condition d ∗ k ≥ ≤ k ≤ r in (2.14.1).8 paces of tuples of polynomials which define algebraic maps Let X Σ be a non-singular toric variety and make the identification X Σ = U ( K Σ ) /G Σ .Let z , · · · , z m be variables.Now we consider the space of all tuples of polynomials which define basedalgebraic maps. Definition 2.9. (i) For each d, m ∈ N , let H dm ( C ) denote the space of allhomogenous polynomials f ( z , · · · , z m ) ∈ C [ z , · · · , z m ] of degree d .(ii) For each r -tuple D = ( d , · · · , d r ) ∈ N r , let Pol ∗ D ( R P m , X Σ ) denotethe space of r -tuples f = ( f ( z , · · · , z m ) , · · · , f r ( z , · · · , z m )) ∈ H d m ( C ) × · · · × H d r m ( C )of homogenous polynomials satisfying the following two conditions:(2.15.1) f ( x ) = ( f ( x ) , · · · , f r ( x )) ∈ U ( K Σ ) for any point x = ( x , · · · , x m ) ∈ R m +1 \ { m +1 } .(2.15.2) f ( e ) = ( f ( e ) , · · · , f r ( e )) = (1 , , · · · , z ) d k in f k ( z , · · · , z m ) is 1 for each 1 ≤ k ≤ r , where we write e = (1 , , · · · , ∈ R m +1 . Definition 2.10.
We always assume the identification X Σ = U ( K Σ ) /G Σ ,denote by [ y , · · · , y r ] the point in X Σ represented by ( y , · · · , y r ) ∈ U ( K Σ ) , and choose the two points [1 : 0 : · · · : 0] ∈ R P m and ∗ = [1 , · · · , ∈ X Σ asthe base-points of R P m and X Σ respectively.Let D = ( d , · · · , d r ) ∈ N r be an r -tuple of positive integers such that P rk =1 d k n k = n . Then by using Lemma 2.6, for each r -tuple f = ( f ( z , · · · , z m ) , · · · , f r ( z , · · · , z m )) ∈ Pol ∗ D ( R P m , X Σ )one can define based algebraic map[ f ] = [ f , · · · , f r ] : ( R P m , [ e ]) → ( X Σ , ∗ ) by(2.15) [ f ]([ x ]) = [ f ( x ) , · · · , f r ( x )](2.16)for [ x ] = [ x : · · · : x m ] ∈ R P m , where x = ( x , · · · , x m ) ∈ R m +1 \ { m +1 } .Hence, we obtain the natural map(2.17) i D,m : Pol ∗ D ( R P m , X Σ ) → Map ∗ D ( R P m , X Σ ) given by(2.18) i D,m ( f ) = [ f ] = [ f , · · · , f r ]for f = ( f ( z , · · · , z m ) , · · · , f r ( z , · · · , z m )) ∈ Pol ∗ D ( R P m , X Σ ), where wedenote by Map ∗ D ( R P m , X Σ ) the path-component of Map ∗ ( R P m , X Σ ) whichcontains all algebraic maps of degree D .9 emark 2.11. When m = 1, we make the identification R P = S = R ∪ ∞ and choose the points ∞ as the base-point of R P . Then, by setting z = z z ,we can view a homogenous polynomial f ( z , z ) ∈ C [ z , z ] of degree d as amonic polynomial f k ( z ) ∈ C [ z ] of degree d .Thus, when m = 1, one can redefine the space Pol ∗ D ( S , X Σ ) as follows. Definition 2.12. (i) Let P d ( C ) denote the space of all monic polynomials f ( z ) = z d + a z d − + · · · + a d − z + a d ∈ C [ z ] of degree d , and let(2.19) P D = P d ( C ) × P d ( C ) × · · · × P d r ( C ) . Note that there is a homeomorphism φ : P d ( C ) ∼ = C d given by φ ( z d + P dk =1 a k z d − k ) = ( a , · · · , a d ) ∈ C d . (ii) For any r -tuple D = ( d , · · · , d r ) ∈ N r , let Pol ∗ D ( S , X Σ ) denote thespace of all r -tuples ( f ( z ) , · · · , f r ( z )) ∈ P D of monic polynomials satisfyingthe following condition ( † ):( † ) The polynomials f i ( z ) , · · · , f i s ( z ) have no common real root for any σ = { i , · · · , i s } ∈ I ( K Σ ), i.e. ( f i ( α ) , · · · , f i s ( α )) = s for any α ∈ R .When the condition P rk =1 d k n k = n holds, by identifying X Σ = U ( K Σ ) /G Σ and R P = S = R ∪ ∞ , one can define a natural map(2.20) i D = i D, : Pol ∗ D ( S , X Σ ) → Map ∗ ( S , X Σ ) = Ω X Σ by(2.21) i D ( f ( z ) , · · · , f r ( z ))( α ) = ( [ f ( α ) , · · · , f r ( α )] if α ∈ R [1 , , · · · ,
1] if α = ∞ for ( f ( z ) , · · · , f r ( z )) ∈ Pol ∗ D ( S , X Σ ) and α ∈ S = R ∪ ∞ , where we choosethe points ∞ and [1 , , · · · ,
1] as the base-points of S and X Σ .Note that Pol ∗ D ( S , X Σ ) is simply connected (which will be proved inProposition 5.9) and that the map Ω q Σ : Ω U ( K Σ ) → Ω X Σ is a universalcovering (which will be shown in Lemma 3.4). Thus, when P rk =1 d k n k = n ,the map i D lifts to the space Ω Z K Σ and there is a map(2.22) j D : Pol ∗ D ( S , X Σ ) → Ω U ( K Σ ) ≃ Ω Z K Σ such that(2.23) Ω q Σ ◦ j D = i D . emark 2.13. (i) Note that Pol ∗ D ( S , X Σ ) is path-connected (see (ii) ofRemark 7.4), and that X Σ is simply connected if the condition (2.14.2) issatisfied (see (i) of Lemma 3.4).(ii) Even if P rk =1 d k n k = n we can define the two maps i D : Pol ∗ D ( S , X Σ ) → Ω X Σ , j D : Pol ∗ D ( S , X Σ ) → Ω U ( K Σ )and this will be done in § r min (Σ) and d ( D, Σ).
Definition 2.14.
Let Σ be a fan in R n as in Definition 2.1.(i) We say that a set S = { n i , · · · , n i s } is primitive in Σ if Cone( S ) / ∈ Σbut Cone( T ) ∈ Σ for any proper subset T $ S .(ii) For D = ( d , · · · , d r ) ∈ N r define integers r min (Σ) and d ( D, Σ; m ) by(2.24) r min (Σ) = min { s ∈ N : { n i , · · · , n i s } is primitive in Σ } ,d ( D, Σ; m ) = (2 r min (Σ) − m − d min − , where d min = min { d , · · · , d r } . In particular, when m = 1 we also define the positive integer d ( D, Σ) by(2.25) d ( D, Σ) = d ( D, Σ; 1) = (2 r min (Σ) − d min − . Now recall the following result.
Theorem 2.15 ([14]) . Let m ≥ be a positive integer, X Σ be a compactsmooth toric variety and D = ( d , · · · , d r ) ∈ N r be an r -tuple of positiveintegers such that P rk =1 d k n k = n . Then the natural map i D,m : Pol ∗ D ( R P m , X Σ ) → Map ∗ D ( R P m , X Σ ) is a homology equivalence through dimension d ( D, Σ; m ) . Note that the above result does not hold for the case m = 1. For example,this can be seen in [13] for the case X Σ = C P n . In fact, the main purpose ofthis paper is to investigate the result corresponding to this theorem for thecase m = 1. The main results
More precisely the main results of this paper are asfollows. 11 heorem 2.16.
Let D = ( d , · · · , d r ) ∈ N r be an r -tuple of positive integerssatisfying the condition P rk =1 d k n k = n , and let X Σ be a simply connectednon-singular toric variety such that the condition (2.14.1) holds.Then the map j D : Pol ∗ D ( S , X Σ ) → Ω U ( K Σ ) ≃ Ω Z K Σ is a homotopy equivalence through dimension d ( D, Σ) . Corollary 2.17.
Under the same assumption as in Theorem 2.16, the map i D : Pol ∗ D ( S , X Σ ) → Ω X Σ induces an isomorphism ( i D ) ∗ : π k (Pol ∗ D ( S , X Σ )) ∼ = −→ π k (Ω X Σ ) ∼ = π k +1 ( X Σ ) for any ≤ k ≤ d ( D, Σ) . Corollary 2.18.
Let D = ( d , · · · , d r ) ∈ N r be an r -tuple of positive integerssatisfying the condition P rk =1 d k n k = n , and let X Σ be a simply connectedcompact non-singular toric variety. Let Σ(1) denote the set of all one dimen-sional cones in Σ , and Σ any fan in R n such that Σ(1) ⊂ Σ $ Σ . (i) Then X Σ is a non-singular open toric subvariety of X Σ and the map j D : Pol ∗ D ( S , X Σ ) → Ω U ( K Σ ) ≃ Ω Z Σ is a homotopy equivalence through dimension d ( D, Σ ) . (ii) Moreover, the map i D : Pol ∗ D ( S , X Σ ) → Ω X Σ induces the isomor-phism ( i D ) ∗ : π k (Pol ∗ D ( S , X Σ )) ∼ = −→ π k (Ω X Σ ) ∼ = π k +1 ( X Σ ) for any ≤ k ≤ d ( D, Σ ) . Remark 2.19. If X Σ is compact, the condition (2.14.1) is satisfied (see (ii)of Remark 2.8) and one can prove the above two results (Theorem 2.16,Corollary 2.17) by using [14, Theorem 6.2]. However, the proof given in [14]uses the Stone-Weierstrass theorem and it cannot be applied when X Σ is notcompact.Finally consider the r -tuple D = ( d , · · · , d r ) ∈ N r such that P rk =1 d k n k = n . Then the map i D is not well-defined (see Lemma 2.6). However, even inthis situation we have a map j D : Pol ∗ D ( S , X Σ ) → Ω Z K Σ and the followingresult holds. 12 heorem 2.20. Let D = ( d , · · · , d r ) ∈ N r be an r -tuple of positive integerssuch that P rk =1 d k n k = n , and let X Σ be a simply connected non-singulartoric variety such that the condition (2.14.1) holds.Then the map defined by (6.16) j D : Pol ∗ D ( S , X Σ ) → Ω U ( K Σ ) ≃ Ω Z K Σ is a homotopy equivalence thorough dimension d ( D, Σ) . Corollary 2.21.
Under the same assumption as Theorem 2.20, the map i D = (Ω q Σ ) ◦ j D : Pol ∗ D ( S , X Σ ) → Ω X Σ induces an isomorphism ( i D ) ∗ : π k (Pol ∗ D ( S , X Σ )) ∼ = −→ π k (Ω X Σ ) ∼ = π k +1 ( X Σ ) for any ≤ k ≤ d ( D, Σ) . Since X Σ is compact and Σ(1) ⊂ Σ $ Σ, the condition (2.14.1) holdsfor the fan Σ and we obtain the following result by using Theorem 2.20 andCorollary 2.21. Corollary 2.22.
Let D = ( d , · · · , d r ) ∈ N r be an r -tuple of positive integers,and let X Σ be a simply connected compact non-singular toric variety. Let Σ(1) denote the set of all one dimensional cones in Σ , and Σ any fan in R n such that Σ(1) ⊂ Σ $ Σ . (i) Then X Σ is a non-singular open toric subvariety of X Σ and the map j D : Pol ∗ D ( S , X Σ ) → Ω U ( K Σ ) ≃ Ω Z Σ is a homotopy equivalence through dimesnion d ( D, Σ ) . (ii) Furthermore, the map i D : Pol ∗ D ( S , X Σ ) → Ω X Σ induces the iso-morphism ( i D ) ∗ : π k (Pol ∗ D ( S , X Σ )) ∼ = −→ π k (Ω X Σ ) ∼ = π k +1 ( X Σ ) for any ≤ k ≤ d ( D, Σ ) . Remark 2.23.
The map i D is well-defined only when P rk =1 d k n k = n .Hence, when P rk =1 d k n k = n , there is no map j D : Pol ∗ D ( S , X Σ ) → Ω Z K Σ satisfying the condition (2.23). However, even if P rk =1 d k n k = n we candefine the map j D : Pol ∗ D ( S , X Σ ) → Ω Z K Σ by using the map j D (for some D = ( m d ∗ , · · · , m d ∗ r ) ∈ N r ) and the stabilization map s D,D . In fact, Inthis case the maps j D and i D are defined by j D = j D ◦ s D,D and i D = Ω q Σ ◦ j D (see (6.16) and (9.1) in detail). So the condition (2.23) still holds.13 xamples Since the case X Σ = C P n of Corollary 2.18 was treated in [15],consider the case that X Σ is the Hirzerbruch surface H ( k ). Definition 2.24.
For an integer k ∈ Z , let H ( k ) be the Hirzerbruch surface defined by H ( k ) = (cid:8) ([ x : x : x ] , [ y : y ]) ∈ C P × C P : x y k = x y k (cid:9) ⊂ C P × C P . Since there are isomorphisms H ( − k ) ∼ = H ( k ) for k = 0 and H (0) ∼ = C P × C P , without loss of generality we can assume that k ≥
1. Let Σ k denotethe fan in R given byΣ k = (cid:8) Cone( n i , n i +1 ) (1 ≤ i ≤ , Cone( n , n ) , Cone( n j ) (1 ≤ j ≤ , { } (cid:9) , where we set n = (1 , , n = (0 , , n = ( − , k ) , n = (0 , − . It is easy to see that Σ k is the fan of H ( k ) and that H ( k ) is a compactnon-singular toric variety. Note that Σ k (1) = { Cone( n i ) : 1 ≤ i ≤ } . Since { n , n } and { n , n } are only primitive in Σ k , r min (Σ k ) = 2.Moreover, for D = ( d , d , d , d ) ∈ N the equality P k =1 d k n k = holds iff ( d , d ) = ( d , kd + d ). Thus, if P k =1 d k n k = , we have d min =min { d , d , d , d } = min { d , d } . By Corollary 2.18 and Corollary 2.22 we have:
Example 2.25.
Let D = ( d , d , d , d ) ∈ N , k ∈ N , and Σ be a fan in R such that Σ k (1) = { Cone( n i ) : 1 ≤ i ≤ } ⊂ Σ ⊂ Σ k as in Definition 2.24. (i) X Σ is a non-singular open toric subvariety of H ( k ) if Σ $ Σ k . (ii) If P k =1 d k n k = , the equality ( d , d ) = ( d , kd + d ) holds and themap j D : Pol ∗ D ( S , X Σ ) → Ω Z K Σ is a homotopy equivalence through dimen-sion { d , d } − . Moreover, the map i D : Pol ∗ D ( S , X Σ ) → Ω X Σ inducesan isomorphism ( i D ) ∗ : π k (Pol ∗ D ( S , X Σ )) ∼ = −→ π k (Ω X Σ ) ∼ = π k +1 ( X Σ ) for any ≤ k ≤ { d , d } − . (iii) If P k =1 d k n k = , the map j D : Pol ∗ D ( S , X Σ ) → Ω Z K Σ is a ho-motopy equivalence through dimension { d , d , d , d } − , and the map i D : Pol ∗ D ( S , X Σ ) → Ω X Σ induces an isomorphism ( i D ) ∗ : π k (Pol ∗ D ( S , X Σ )) ∼ = −→ π k (Ω X Σ ) ∼ = π k +1 ( X Σ ) for any ≤ k ≤ { d , d , d , d } − . emark 2.26. (i) There are 15 non isomorphic (as varieties) non-compactsubvarieties X Σ of H ( k ) which satisfy the assumption of Corollary 2.25.(ii) Note that there is an isomorphism(2.26) π ( X Σ ) ∼ = Z r − n (see Lemma 3.4 below) , in general. So ( r − n ) of the r positive integers { d k } rk =1 can be chosen freely.For example, in Example 2.24, ( r, n ) = (4 ,
2) and r − n = 4 − d , d can be chosen freely and the otherintegers d and d are determined by uniquely as ( d , d ) = ( d , kd + d ) . Finally in this section we give the proof of Lemma 2.6.
Proof of Lemma 2.6.
It follows from the assumptions that we can identify X Σ = U ( K Σ ) /G Σ and that the map F = ( f , · · · , f r ) : R m +1 \ { m +1 } → U ( K Σ ) is well-defined. Thus, it suffices to show that F ( λ x ) = F ( x ) up to G Σ -action for any ( λ, x ) ∈ R ∗ × ( R m +1 \ { m +1 } ) iff P rk =1 d k n k = n . Since f k is a homogenous polynomial of degree d k , by using (2.11) we have F ( λ x ) = ( f ( λ x ) , · · · , f r ( λ x )) = ( λ d f ( x ) , · · · , λ d r f r ( x ))= ( λ d , · · · , λ d r ) · ( f ( x ) , · · · , f r ( x )) . Hence, it remains to show that ( λ d , · · · , λ d r ) ∈ G Σ for any λ ∈ R ∗ iff P rk =1 d k n k = n . However, ( λ d , · · · , λ d r ) ∈ G Σ for any λ ∈ R ∗ iff r Y k =1 ( λ d k ) h n k , m i = λ h P rk =1 d k n k , m i = 1 for any m ∈ Z n ⇔ r X k =1 d k n k = n and this completes the proof. In this section, we recall several known results from toric topology.
Definition 3.1 ([3]; Example 6.39) . Let K be a simplicial complex on thevertex set [ r ].(i) Then we denote by Z K and DJ ( K ) the moment-angle complex of K and the Davis-Januszkiewicz space of K defined by(3.1) Z K = Z K ( D , S ) , DJ ( K ) = Z K ( C P ∞ , ∗ ) . (ii) Let ι K : Z K = Z K ( D , S ) → DJ ( K ) = Z K ( C P ∞ , ∗ ) denote thenatural map induced from the following composite of maps(3.2) ( D , S ) pin −→ ( S , ∗ ) = ( C P , ∗ ) ⊂ −→ ( C P ∞ , ∗ ) , where pin : ( D , S ) → ( S , ∗ ) denotes the natural pinching map.15 emma 3.2 ([3], [20]) . Let K be a simplicial complex on the vertex set [ r ] . (i) The space Z K is -connected and there is an T r -equivariant deforma-tion retraction (3.3) r C : U ( K ) = Z K ( C , C ∗ ) ≃ −→ Z K , where T r = ( S ) r . (ii) There is a fibration sequence (up to homotopy) (3.4) Z K ι K −→ DJ ( K ) ⊂ −→ ( C P ∞ ) r (iii) If X Σ is a simply connected non-singular toric variety satisfying thecondition (2.14.1), there is a fibration sequence (up to homotopy) (3.5) T n C = ( C ∗ ) n −→ X Σ p Σ −→ DJ ( K Σ ) . Proof.
The assertions follow from [3, Theorem 6.33, Theorem 8.9], [3, Theo-rem 6.29, Corollary 6.30] and [20, Proposition 4.4].
Lemma 3.3 ([25]; (6.2) and Proposition 6.7) . Let X Σ be a non-singular toricvariety such that the condition (2.14.2) holds. Then there is an isomorphism (3.6) G Σ ∼ = T r − n C = ( C ∗ ) r − n , and the group G Σ acts on the space Z K Σ ( C , C ∗ ) freely. So there is a principal G Σ -bundle (3.7) q Σ : U ( K Σ ) = Z K Σ ( C , C ∗ ) → X Σ . Lemma 3.4. (i)
The space X Σ is simply connected if and only if the condition(2.14.2) is satisfied. (ii) If X Σ is a simply connected non-singular toric variety, π ( X Σ ) = Z r − n and the map (3.8) Ω q Σ : Ω Z K Σ → Ω X Σ is a universal covering projection with fiber Z r − n .Proof. The assertion (i) easily follows from [7, Theorem 12.1.10], and it suf-fices to show (ii). By Lemma 3.2 and (3.3), Z K Σ ( C , C ∗ ) ≃ Z K Σ is 2-connected.Then by using the homotopy exact sequence of the principal G Σ -bundle (3.7)and the isomorphism (3.6), we easily see that π ( X Σ ) = Z r − n and that Ω q Σ is a universal covering (up to homotopy). Remark 3.5. If X Σ is a simply connected non-singular toric variety satisfy-ing the condition (2.14.1), one can show that there is a homotopy equivalence(3.9) Ω X Σ ≃ Ω Z K Σ × T r − n . Although this can be proved by using (3.3), (3.6) and (3.7), we do not needthis fact and omit the detail. 16
Simplicial resolutions
In this section, we summarize the definitions of the non-degenerate simplicialresolution and the associated truncated simplicial resolutions ([22], [27]).
Definition 4.1. (i) For a finite set v = { v , · · · , v l } ⊂ R N , let σ ( v ) denotethe convex hull spanned by v . Let h : X → Y be a surjective map such that h − ( y ) is a finite set for any y ∈ Y , and let i : X → R N be an embedding.Let X ∆ and h ∆ : X ∆ → Y denote the space and the map defined by(4.1) X ∆ = (cid:8) ( y, u ) ∈ Y × R N : u ∈ σ ( i ( h − ( y ))) (cid:9) ⊂ Y × R N , h ∆ ( y, u ) = y. The pair ( X ∆ , h ∆ ) is called the simplicial resolution of ( h, i ). In particular, itis called a non-degenerate simplicial resolution if for each y ∈ Y any k pointsof i ( h − ( y )) span ( k − R N .(ii) For each k ≥
0, let X ∆ k ⊂ X ∆ be the subspace of the union of the( k − y in Y given by(4.2) X ∆ k = (cid:8) ( y, u ) ∈ X ∆ : u ∈ σ ( v ) , v = { v , · · · , v l } ⊂ i ( h − ( y )) , l ≤ k (cid:9) . We make the identification X = X ∆1 by identifying x ∈ X with the pair( h ( x ) , i ( x )) ∈ X ∆1 , and we note that there is an increasing filtration(4.3) ∅ = X ∆0 ⊂ X = X ∆1 ⊂ X ∆2 ⊂ · · · ⊂ X ∆ k ⊂ · · · ⊂ ∞ [ k =0 X ∆ k = X ∆ . Since the map h ∆ : X ∆ → Y is a proper map, it extends to the map h ∆+ : X ∆+ → Y + between the one-point compactifications, where X + denotes theone-point compactification of a locally compact space X .(iii) A space X ⊂ R n is called semi-algebraic if it is a subspace of the form X = S si =1 T r i j =1 { ( x , · · · , x n ) ∈ R n : f ij ∗ ij } , where f ij ∈ R [ X , · · · , x n ]and ∗ ij is either < or =, for i = 1 , · · · s and j = 1 , · · · , r i . Similarly, when X ⊂ R n and Y ⊂ R m are semi-algebraic spaces, a map f : X → Y is asemi-algebraic map if the graph of f is semi-algebraic. Lemma 4.2 ([27], [28] (cf. Lemma 3.3 in [18])) . Let h : X → Y be asurjective map such that h − ( y ) is a finite set for any y ∈ Y, and let i : X → R N be an embedding. (i) If X and Y are semi-algebraic spaces and the two maps h , i are semi-algebraic maps, then the map h ∆+ : X ∆+ ≃ → Y + is a homotopy equivalence. (ii) There is an embedding j : X → R M such that the associated simplicialresolution ( ˜ X ∆ , ˜ h ∆ ) of ( h, j ) is non-degenerate. If there is an embedding j : X → R M such that the associated simpli-cial resolution ( ˜ X ∆ , ˜ h ∆ ) of ( h, j ) is non-degenerate, the space ˜ X ∆ is uniquelydetermined up to homeomorphism. Moreover, there is a filtration preservinghomotopy equivalence q ∆ : ˜ X ∆ ≃ → X ∆ such that q ∆ | X = id X . Remark 4.3.
In this paper we only need the weaker assertion that the map h ∆+ is a homology equivalence. One can easily prove this result by the sameargument as used in the second revised edition of Vassiliev’s book [27, Proofof Lemma 1 (page 90)]. Remark 4.4 ([27], [28]) . Even for a surjective map h : X → Y which isnot finite to one, it is still possible to construct an associated non-degeneratesimplicial resolution. Recall that it is known that there exists a sequence ofembeddings { ˜ i k : X → R N k } k ≥ satisfying the following two conditions foreach k ≥ y ∈ Y , any t points of the set ˜ i k ( h − ( y )) span ( t − R N k if t ≤ k .(ii) N k ≤ N k +1 and if we identify R N k with a subspace of R N k +1 , then˜ i k +1 = ˆ i ◦ ˜ i k , where ˆ i : R N k ⊂ → R N k +1 denotes the inclusion.In this situation, in fact, a non-degenerate simplicial resolution may be con-structed by choosing a sequence of embeddings { ˜ i k : X → R N k } k ≥ satisfyingthe above two conditions for each k ≥ X ∆ k = (cid:8) ( y, u ) ∈ Y × R N k : u ∈ σ ( v ) , v = { v , · · · , v l } ⊂ ˜ i k ( h − ( y )) , l ≤ k (cid:9) . Then by identifying naturally X ∆ k with a subspace of X ∆ k +1 , define thenon-degenerate simplicial resolution X ∆ of h as X ∆ = [ k ≥ X ∆ k . Definition 4.5.
Let h : X → Y be a surjective semi-algebraic map betweensemi-algebraic spaces, j : X → R N be a semi-algebraic embedding, andlet ( X ∆ , h ∆ : X ∆ → Y ) denote the associated non-degenerate simplicialresolution of ( h, j ).Let k be a fixed positive integer and let h k : X ∆ k → Y be the map definedby the restriction h k := h ∆ |X ∆ k . The fibers of the map h k are ( k − h ∆ and, in general, always fail to be simplices overthe subspace Y k = { y ∈ Y : card( h − ( y )) > k } . Let Y ( k ) denote the closureof the subspace Y k . We modify the subspace X ∆ k so as to make all the fibersof h k contractible by adding to each fibre of Y ( k ) a cone whose base is thisfibre. We denote by X ∆ ( k ) this resulting space and by h ∆ k : X ∆ ( k ) → Y thenatural extension of h k . 18ollowing [23], we call the map h ∆ k : X ∆ ( k ) → Y the truncated ( afterthe k -th term ) simplicial resolution of Y . Note that that there is a naturalfiltration X ∆0 ⊂ X ∆1 ⊂ · · · ⊂ X ∆ l ⊂ X ∆ l +1 ⊂ · · · ⊂ X ∆ k ⊂ X ∆ k +1 = X ∆ k +2 = · · · = X ∆ ( k ) , where X ∆0 = ∅ , X ∆ l = X ∆ l if l ≤ k and X ∆ l = X ∆ ( k ) if l > k . Lemma 4.6 ([23], cf. Remark 2.4 and Lemma 2.5 in [16]) . Under the sameassumptions and with the same notation as in Definition 4.5, the map h ∆ k : X ∆ ( k ) ≃ −→ Y is a homotopy equivalence. In this section, we always assume that D = ( d , · · · , d r ) ∈ N r is an r -tuple ofpositive integers and let us write d min = min { d , · · · , d r } . First, we constructthe Vassiliev spectral sequence. Definition 5.1. (i) For a space X and a positive integer k ≥
1, let F ( X, k )denote the ordered configuration space of k -distinct points in X given by(5.1) F ( X, k ) = { ( x , · · · , x k ) ∈ X k : x i = x j if i = j } . The symmetric group S k of k letters acts on F ( X, k ) freely by the usualcoordinate permutation and we denote by C k ( X ) the unordered configurationspace of k -distinct points in X given by the orbit space(5.2) C k ( X ) = F ( X, k ) /S k . (ii) Let Σ D denote the discriminant of Pol ∗ D ( S , X Σ ) in P D given by thecomplementΣ D = P D \ Pol ∗ D ( S , X Σ )= { ( f ( z ) , · · · , f r ( z )) ∈ P D : ( f ( x ) , · · · , f r ( x )) ∈ L (Σ) for some x ∈ R } , where(5.3) L (Σ) = [ σ ∈ I ( K Σ ) L σ = [ σ ⊂ [ r ] ,σ / ∈ K Σ L σ . (iii) Let Z D ⊂ Σ D × R denote the tautological normalization of Σ D con-sisting of all pairs (( f ( z ) , . . . , f r ( z )) , x ) ∈ Σ D × R satisfying the condition( f ( x ) , · · · , f r ( x )) ∈ L (Σ). Projection on the first factor gives a surjectivemap π D : Z D → Σ D . 19 emark 5.2. Let σ k ∈ [ r ] for k = 1 ,
2. It is easy to see that L σ ⊂ L σ if σ ⊃ σ . Letting P r (Σ) = { σ = { i , · · · , i s } ⊂ [ r ] : { n i , · · · , n i s } is a primitive in Σ } , we see that(5.4) L (Σ) = [ σ ∈ P r (Σ) L σ and by using (2.25) we obtain the equality(5.5) dim L (Σ) = 2( r − r min (Σ)) . Our goal in this section is to construct, by means of the non-degenerate simplicial resolution of the discriminant, a spectral sequence converging tothe homology of Pol ∗ D ( S , X Σ ). Definition 5.3.
Let ( X D , π ∆ D : X D → Σ D ) be the non-degenerate simplicialresolution associated to the surjective map π D : Z D → Σ D with the naturalincreasing filtration as in Definition 4.1, ∅ = X D ⊂ X D ⊂ X D ⊂ · · · ⊂ X D = ∞ [ k =0 X Dk . By Lemma 4.2, the map π ∆ D : X D ≃ → Σ D is a homotopy equivalencewhich extends to a homotopy equivalence π ∆ D + : X D + ≃ → Σ D + , where X + denotes the one-point compactification of a locally compact space X . Since X Dk + / X Dk − ∼ = ( X Dk \ X Dk − ) + , we have a spectral sequence (cid:8) E k,st ; D , d t : E k,st ; D → E k + t,s +1 − tt ; D (cid:9) ⇒ H k + sc (Σ D ; Z ) , where E k,s D = H k + sc ( X Dk \ X Dk − ; Z ) and H kc ( X ; Z ) denotes the cohomologygroup with compact supports given by H kc ( X, Z ) = ˜ H k ( X + ; Z ) . Let N ( D ) denote the positive integer given by(5.6) N ( D ) = r X k =1 d k . Since there is a homeomorphism P D ∼ = C N ( D ) , by Alexander duality there isa natural isomorphism 205.7) ˜ H k (Pol ∗ D ( S , X Σ ); Z ) ∼ = H N ( D ) − k − c (Σ D ; Z ) for any k. By reindexing we obtain a spectral sequence (cid:8) E t ; Dk,s , ˜ d t : E t ; Dk,s → E t ; Dk + t,s + t − (cid:9) ⇒ H s − k (Pol ∗ D ( S , X Σ ); Z ) , (5.8)where E Dk,s = H N ( D )+ k − s − c ( X Dk \ X Dk − ; Z ) . Let L k ;Σ ⊂ ( R × L (Σ)) k denote the subspace defined by(5.9) L k ;Σ = { (( x , s ) , · · · , ( x k , s k )) : x j ∈ R , s j ∈ L (Σ) , x i = x j if i = j } . The symmetric group S k on k letters acts on L k ;Σ by permuting coordinates.Let C k ;Σ denote the orbit space(5.10) C k ;Σ = L k ;Σ /S k . By (5.5) we know that C k ;Σ is a cell-complex of dimension(5.11) dim C k ;Σ = (1 + 2 r − r min (Σ)) k. Lemma 5.4. If ≤ k ≤ d min = min { d , · · · , d r } , X Dk \X Dk − is homeomorphicto the total space of a real affine bundle ξ D,k over C k ;Σ with rank l D,k =2 N ( D ) − rk + k − .Proof. The argument is exactly analogous to the one in the proof of [1,Lemma 4.4]. Namely, an element of X Dk \ X Dk − is represented by ( F, u ) =(( f , · · · , f r ) , u ), where F = ( f , · · · , f r ) is an r -tuple of monic polynomials inΣ D and u is an element of the interior of the span of the images of k distinctpoints { x , · · · , x k } ∈ C k ( R ) such that F ( x j ) = ( f ( x j ) , · · · , f r ( x j )) ∈ L (Σ)for each 1 ≤ j ≤ k , under a suitable embedding. Since the k distinct points { x j } kj =1 are uniquely determined by u , by the definition of the non-degeneratesimplicial resolution, there are projection maps π k,D : X Dk \X Dk − → C k ;Σ givenby (( f , · · · , f r ) , u )
7→ { ( x , F ( x )) , . . . , ( x k , F ( x k )) } .Now suppose that 1 ≤ k ≤ d min and let c = { ( x j , s j ) } kj =1 ∈ C k ;Σ ( x j ∈ R , s j ∈ L (Σ)) be any fixed element. Consider the fibre π − k,D ( c ). For each1 ≤ j ≤ k , we set s j = ( s ,j , · · · , s r,j ) and consider the condition(5.12) F ( x j ) = ( f ( x j ) , · · · , f r ( x j )) = s j ⇔ f t ( x j ) = s t,j for 1 ≤ t ≤ r. In general, the condition f t ( x j ) = s t,j gives one linear condition on the coef-ficients of f t , and determines an affine hyperplane in P d t ( C ). For example, if21e set f t ( z ) = z d t + P d t − i =0 a i,t z i , then f t ( x j ) = s t,j for any 1 ≤ j ≤ k if andonly if(5.13) x x · · · x d t − x x · · · x d t − ... . . . . . . . . . ...1 x k x k · · · x d t − k · a ,t a ,t ... a d t − ,t = s t, − x d t s t, − x d t ... s t,k − x d t k Since 1 ≤ k ≤ d min ( D ) and { x j } kj =1 ∈ C k ( R ), it follows from the properties ofVandermonde matrices that the condition (5.13) gives exactly k independentconditions on the coefficients of f t ( z ). Thus the space of polynomials f t ( z )in P d t ( C ) which satisfies (5.13) is the intersection of k affine hyperplanes ingeneral position and has codimension k in P d t ( C ). Hence, the fibre π − k,D ( c )is homeomorphic to the product of an open ( k − P ri =1 ( d i − k ) = 2 N ( D ) − rk . It is now easyto show that π k,D is a (locally trivial) real affine bundle over C k ;Σ of rank l D,k = 2 N ( D ) − rk + k − Lemma 5.5. If ≤ k ≤ d min , there is a natural isomorphism E Dk,s ∼ = H rk − sc ( C k ;Σ ; ± Z ) , where the twisted coefficients system ± Z comes from the Thom isomorphism.Proof. Suppose that 1 ≤ k ≤ d min . By Lemma 5.4, there is a homeomorphism( X Dk \X Dk − ) + ∼ = T ( ξ D,k ) , where T ( ξ D,k ) denotes the Thom space of ξ D,k . Since(2 N ( D ) + k − s − − l D,k = 2 rk − s, by using the Thom isomorphism thereis a natural isomorphism E Dk,s ∼ = H rk − sc ( C k ;Σ ; ± Z ) . Definition 5.6. (i) Let X ∆ denote the truncated (after the d min -th term)simplicial resolution of Σ D with the natural filtration ∅ = X ∆0 ⊂ X ∆1 ⊂ · · · ⊂ X ∆ d min ⊂ X ∆ d min +1 = X ∆ d min +2 = · · · = X ∆ , where X ∆ k = X Dk if k ≤ d min and X ∆ k = X ∆ d min +1 if k ≥ d min + 1.(ii) Let { e i : 1 ≤ i ≤ r } denote the standard R -basis of R r given by(5.14) e = (1 , , , · · · , e = (0 , , , · · · , · · · , e r = (0 , · · · , , , ∈ R r . For each 1 ≤ i ≤ r , let us consider(5.15) D + e i = ( d , · · · , d i − , d i + 1 , d i +1 , · · · , d r )22nd let Y ∆ denote the truncated (after the d min -th term) simplicial resolutionof Σ D + e i with the natural filtration ∅ = Y ∆0 ⊂ Y ∆1 ⊂ · · · ⊂ Y ∆ d min ⊂ Y ∆ d min +1 = Y ∆ d min +2 = · · · = Y ∆ , where Y ∆ k = X Dk if k ≤ d min and Y ∆ k = Y ∆ if k ≥ d min + 1.Then by using the same argument as in (5.8) and Lemma 4.6, we havethe following two truncated spectral sequences (cid:8) E tk,s , d t : E tk,s → E tk + t,s + t − (cid:9) ⇒ H s − k (Pol ∗ D ( S .X Σ ); Z )(5.16) (cid:8) ′ E tk,s , ′ d t : ′ E tk,s → ′ E tk + t,s + t − (cid:9) ⇒ H s − k (Pol ∗ D + e i ( S , X Σ ); Z ) , where E k,s = H N ( D )+ k − s − c ( X ∆ k \ X ∆ k − ; Z ) , E k,s = H N ( D )+ k − s +1 c ( Y ∆ k \ Y ∆ k − ; Z ) . Lemma 5.7. (i) If k < or k ≥ d min + 2 , E k,s = ′ E k,s = 0 for any s . (ii) E , = ′ E , = Z and E ,s = ′ E ,s = 0 if s = 0 . (iii) If ≤ k ≤ d min , there are isomorphisms E k,s ∼ = ′ E k,s ∼ = H rk − sc ( C k ;Σ ; ± Z ) . (iv) If ≤ k ≤ d min , E k,s = ′ E k,s = 0 for any s ≤ (2 r min (Σ) − k − . (v) E d min +1 ,s = ′ E d min +1 ,s = 0 for any s ≤ (2 r min (Σ) − d min − .Proof. Let us write r min = r min (Σ). Since the proofs of both cases are almostidentical, it suffices to prove the assertions for the case E k,s . Since X ∆ k = X Dk for any k ≥ d min + 2, the assertions (i) and (ii) are clearly true. Since X ∆ k = X Dk for any k ≤ d min , (iii) easily follows from Lemma 5.5. Thus itremains to prove the assertions (iv) and (v).By using (5.11), we see that 2 rk − s > dim C k ;Σ if and only if s ≤ (2 r min (Σ) − k −
1. Thus H rk − sc ( C k ;Σ ; ± Z ) = 0 if s ≤ (2 r min (Σ) − k − E d min +1 ,s . Then by Lemma [23, Lemma 2.1],dim( X ∆ d min +1 \ X ∆ d min ) = dim( X Dd min \ X Dd min − ) + 1 = ( l D,d min + dim C d min ;Σ ) + 1= 2 N ( D ) + 2 d min − r min d min . Since E d min +1 ,s = H N ( D )+ d min − sc ( X ∆ d min +1 \ X ∆ d min ; Z ) and 2 N ( D ) + d min − s > dim( X ∆ d +1 \ X ∆ d ) ⇔ s ≤ (2 r min − d min −
1, we see that E d min +1 ,s = 0 for any s ≤ (2 r min − d min −
1. Hence, (v) is proved.23 efinition 5.8.
Define the integer C D (Σ) by(5.17) C D (Σ) = 2( r min (Σ) − d min − . Note that r min (Σ) ≥ d min ( D ) ≥ D ∈ N r . Thus, we see that C D (Σ) ≥ D ∈ N r . Proposition 5.9.
The space
Pol ∗ D ( S , X Σ ) is C D (Σ) -connected. Thus, thespace Pol ∗ D ( S , X Σ ) is simply connected for any D ∈ N r .Proof. it follows from Lemma 5.7 that we can easily check that E k,s = 0 if s − k ≤ C D (Σ) = 2( r min (Σ) − d min − . Hence by using the first spectralsequence given in (5.16), we see that H i (Pol ∗ D ( S , X Σ ); Z ) = 0 for any 1 ≤ i ≤ C D (Σ).On the other hand, by using the string representation as in [12, Appendix],one can show that π (Pol ∗ D ( S , X Σ )) is an abelian group and one know thatthere is an isomorphism π (Pol ∗ D ( S , X Σ )) ∼ = H (Pol ∗ D ( S , X Σ ); Z ) = 0 . Thus,the space Pol ∗ D ( S , X Σ ) is simply connected and it follows from the Hurewicztheorem that the space Pol ∗ D ( S , X Σ ) is C D (Σ)-connected. Definition 6.1.
For an r -tuple D = ( d , · · · , d r ) ∈ N r of positive integers, let N ( D ) denote the positive integer given by N ( D ) = P rk =1 d k as in (5.6), andlet us write U D = { w ∈ C : Re( w ) < N ( D ) } . For each D = ( d , · · · , d r ) ∈ N r let us choose any any homeomorphism(6.1) ϕ D : C ∼ = −→ U D such that ϕ D ( R ) = { w ∈ R : w < N ( D ) } = ( −∞ , N ( D )) and fix it.(i) For a monic polynomial f ( z ) = d Y k =1 ( z − α k ) ∈ P d ( C ), let ϕ D ( f ( z ))denote the monic polynomial of the same degree d defined by(6.2) ϕ D ( f ( z )) = d Y k =1 ( z − ϕ ( α k )) . (ii) Now we choose a point x D ∈ C \ U D = { w ∈ C : Re( w ) > N ( D ) } freely and fix it. Then for each 1 ≤ i ≤ r , let us recall D + e i = ( d , · · · , d i − , d i + 1 , d i +1 , · · · , d r )24nd define the stabilization map (6.3) s D,i : Pol ∗ D ( S , X Σ ) → Pol ∗ D + e i ( S , X Σ ) by(6.4) s D,i ( f ) = ( ϕ D ( f ( z )) , · · · , ϕ D ( f i ( z ))( z − x D ) , · · · , ϕ D ( f r ( z )))for f = ( f ( z ) , · · · , f r ( z )) ∈ Pol ∗ D ( S , X Σ ) . (iii) Let D = ( d , · · · , d r ) ∈ N r and D ′ = ( d ′ , · · · , d ′ r ) ∈ N r be r -tuples ofpositive integers such that D ′ − D = ( d ′ − d , · · · , d ′ r − d r ) ∈ ( N ∪ { } ) r \ { r } . Now let us choose any point x D = ( x D, , · · · , x D,r ) ∈ F ( C \ U D , r ).Then define the stabilization map (6.5) s D,D ′ : Pol ∗ D ( S , X Σ ) → Pol ∗ D ′ ( S , X Σ ) by(6.6) s D ( f ) = (( ϕ D ( f ( z ))( z − x D, ) d ′ − d , · · · , ϕ D ( f r ( z ))( z − x D,r ) d ′ r − d r )for f = ( f ( z ) , · · · , f r ( z )) ∈ Pol ∗ D ( S , X Σ ) . In particular, when D ′ = D + e =( d + 1 , d + 1 , · · · , d r + 1), we write(6.7) s D = s D,D ′ = s D,D + e : Pol ∗ D ( S , X Σ ) → Pol ∗ D + e ( S , X Σ ) , where we set(6.8) e = (1 , , · · · ,
1) = e + e + · · · + e r . Remark 6.2.
Note that the definition of the map s D,i depends on the choiceof the pair ( ϕ D , x D ), but the homotopy class of the map s D,i does not dependon its choice. Similarly the definition of the map s D,D ′ also depends on thechoice of the pair ( ϕ d , x D ), but its homotopy class does not.Now recall the definition of the stabilization map s D,i : Pol ∗ D ( S , X Σ ) → Pol ∗ D + e i ( S , X Σ ) . By using the choice of the point x D ∈ C \ U D , one can easily see that itextends to an open embedding(6.9) s D,i : C × Pol ∗ D ( S , X Σ ) → Pol ∗ D + e i ( S , X Σ ) . It also naturally extends to an open embedding ˜ s D,i : P D → P D i and byrestriction we obtain an open embedding(6.10) ˜ s D,i : C × Σ D → Σ D + e i . s D + e i ) + : (Σ D i ) + → ( C × Σ D ) + = S ∧ Σ D + . Note that there is a commutative diagram(6.11) ˜ H k (Pol ∗ D ( S , X Σ ); Z ) s D,i ∗ −−−→ ˜ H k (Pol ∗ D + e i ( S , X Σ ); Z ) Al y ∼ = Al y ∼ = H N ( D ) − k − c (Σ D ; Z ) ˜ s ∗ Di + −−−→ H N ( D ) − k +1 c (Σ D + e i ; Z )where Al is the Alexander duality isomorphism and ˜ s ∗ D i + denotes the com-posite of the suspension isomorphism with the homomorphism (˜ s D + ) ∗ , H N ( D ) − k − c (Σ D ; Z ) ∼ = → H N ( D ) − k +1 c ( C × Σ D ; Z ) (˜ s Di + ) ∗ −→ H N ( D ) − k +1 c (Σ D + e i ; Z ) . By the universality of the non-degenerate simplicial resolution ([22, pages286-287]), the map ˜ s D,i also naturally extends to a filtration preserving openembedding ˜ s D,i : C × X D → X D + e i between non-degenerate simplicial resolutions. This induces a filtration pre-serving map (˜ s D,i ) + : X D + e i + → ( C × X D ) + = S ∧ X D + , and we obtain thehomomorphism of spectral sequences(6.12) { ˜ θ tk,s : E t ; Dk,s → E t ; D + e i k,s } , where (cid:8) E t ; Dk,s , ˜ d t : E t ; Dk,s → E t ; Dk + t,s + t − (cid:9) ⇒ H s − k (Pol ∗ D ( S , X Σ ); Z ) , (cid:8) E t ; D + e i k,s , ˜ d t : E t ; D + e i k,s → E t ; D + e i k + t,s + t − (cid:9) ⇒ H s − k (Pol ∗ D + e i ( S , X Σ ); Z ) , and the groups E Dk,s and E D i k,s are given by ( E Dk,s = H N ( D )+ k − − sc ( X Dk \ X Dk − ; Z ) ,E D + e i k,s = H N ( D )+ k +1 − sc ( X D + e i k \ X D + e i k − ; Z ) . Lemma 6.3. If ≤ i ≤ r and ≤ k ≤ d min , ˜ θ k,s : E Dk,s → E D + e i k,s is anisomorphism for any s .Proof. Since the case k = 0 is clear, suppose that 1 ≤ k ≤ d min . It followsfrom the proof of Lemma 5.4 that there is a homotopy commutative diagramof affine vector bundles C × ( X Dk \ X Dk − ) −−−→ C k ;Σ y kX D + e i k \ X D + e i k − −−−→ C k ;Σ E ,Dk,s −−−→ ∼ = H rk − sc ( C k ;Σ ; ± Z ) ˜ θ k,s y k E ,D + e i k,s −−−→ ∼ = H rk − sc ( C k ;Σ ; ± Z )and the assertion follows.Now we consider the spectral sequences induced by truncated simplicialresolutions.By using Lemma 4.6 and the same method as in [23, § §
3] (cf. [16,Lemma 2.2]), we obtain the following truncated spectral sequences (cid:8) E tk,s , d t : E tk,s → E tk + t,s + t − (cid:9) ⇒ H s − k (Pol ∗ D ( S , X Σ ); Z ) , (cid:8) ′ E tk,s , d t : ′ E tk,s → ′ E tk + t,s + t − (cid:9) ⇒ H s − k (Pol ∗ D + e i ( S , X Σ ); Z ) , where E k,s = H N ( D )+ k − − sc ( X ∆ k \ X ∆ k − ; Z ) , ′ E k,s = H N ( D )+ k +1 − sc ( Y ∆ k \ Y ∆ k − ; Z ) . By the naturality of truncated simplicial resolutions, the filtration preserv-ing map ˜ s D,i : C × X D → X D i gives rise to a natural filtration preservingmap ˜ s ′ D,i : C × X ∆ → Y ∆ which, in a way analogous to (6.12), induces ahomomorphism of spectral sequences(6.13) { θ tk,s : E tk,s → ′ E tk,s } . Lemma 6.4. If ≤ k ≤ d min , θ k,s : E k,s → ′ E k,s is an isomorphism for any s .Proof. Since ( X ∆ k , Y ∆ k ) = ( X Dk , X D i k ) for k ≤ d min , the assertion follows fromLemma 6.3. Theorem 6.5.
For each ≤ i ≤ r and D = ( d , · · · , d r ) ∈ N r , the stabiliza-tion map s D,i : Pol ∗ D ( S , X Σ ) → Pol ∗ D + e i ( S , X Σ ) is a homotopy equivalence through dimension d ( D, Σ) , where the number d ( D, Σ) is given by d ( D, Σ) = 2( r min (Σ) − d min − as in (2.25).Proof. We write r min = r min (Σ). Since Pol ∗ D ( S , X Σ ) and Pol ∗ D + e i ( S , X Σ )are simply connected, it suffices to show that the map s D,i is a homologyequivalence through dimension d ( D, Σ).27or this purpose, consider the homomorphism θ tk,s : E tk,s → ′ E tk,s oftruncated spectral sequences given in (6.13). By using the commutativediagram (6.11) and the comparison theorem for spectral sequences, it sufficesto prove that the positive integer d ( D, Σ) has the following property:( ∗ ) θ ∞ k,s is an isomorphism for all ( k, s ) such that s − k ≤ d ( D, Σ).By Lemma 5.7, E k,s = ′ E k,s = 0 if k <
0, or if k ≥ d min + 2, or if k = d min + 1with s ≤ (2 r min − d min −
1. Since (2 r min − d min − − ( d min + 1) =(2 r min − d min − d ( D, Σ), we see that:( ∗ ) if k < k ≥ d min + 1, θ ∞ k,s is an isomorphism for all ( k, s ) such that s − k ≤ d ( D, Σ).Next, we assume that 0 ≤ k ≤ d min , and investigate the condition that θ ∞ k,s is an isomorphism. Now recall that E k,s = ′ E k,s = 0 if 1 ≤ k ≤ d min with s ≤ (2 r min − k − k = 0 with s = 0. (by Lemma 5.7).Note that the groups E k ,s and ′ E k ,s are not known for ( u, v ) ∈ S = { ( d min + 1 , s ) ∈ Z : s ≥ (2 r min − d min } . By considering the differentials d ’sof E k,s and ′ E k,s , and applying Lemma 6.4, we see that θ k,s is an isomorphismif ( k, s ) / ∈ S ∪ S , where S = { ( u, v ) ∈ Z : ( u + 1 , v ) ∈ S } = { ( d min , v ) ∈ Z : v ≥ (2 r min − d min } . A similar argument shows that θ k,s is an isomorphism if ( k, s ) / ∈ S t =1 S t ,where S = { ( u, v ) ∈ Z : ( u + 2 , v + 1) ∈ S ∪ S } . Continuing in the samefashion, considering the differentials d t ’s on E tk,s and ′ E tk,s and applying theinductive hypothesis, we see that θ ∞ k,s is an isomorphism if ( k, s ) / ∈ S := [ t ≥ S t = [ t ≥ A t , where A t denotes the set A t = There are positive integers l , · · · , l t such that , ( u, v ) ∈ Z ≤ l < l < · · · < l t , u + P tj =1 l j = d min + 1 ,v + P tj =1 ( l j − ≥ (2 r min − d min . Note that if this set was empty for every t , then, of course, the conclusionof Theorem 6.5 would hold in all dimensions (this is known to be false ingeneral). If A t = ∅ , it is easy to see that a ( t ) = min { s − k : ( k, s ) ∈ A t } = (2 r min − d min − ( d min +1)+ t = d ( D, Σ)+ t +1 . Hence, we obtain that min { a ( t ) : t ≥ , A t = ∅} = d ( D, Σ) + 2 . Since θ ∞ k,s isan isomorphism for any ( k, s ) / ∈ S t ≥ A t for each 0 ≤ k ≤ d min , we have thefollowing: 28 ∗ ) If 0 ≤ k ≤ d min , θ ∞ k,s is an isomorphism for any ( k, s ) such that s − k ≤ d ( D, Σ) + 1 . Then, by ( ∗ ) and ( ∗ ) , we know that θ ∞ k,s : E ∞ k,s ∼ = → ′ E ∞ k,s is an isomorphismfor any ( k, s ) if s − k ≤ d ( D, Σ), and this completes the proof.
Theorem 6.6.
For each D = ( d , · · · , d r ) ∈ N r and D ′ = ( d ′ , · · · , d ′ r ) ∈ N r with D ′ − D ∈ ( Z ≥ ) r \ { r } , the stabilization map s D,D ′ : Pol ∗ D ( S , X Σ ) → Pol ∗ D ′ ( S , X Σ ) is a homotopy equivalence through dimension d ( D, Σ) , where the number d ( D, Σ) is given by d ( D, Σ) = 2( r min (Σ) − d min − as in (2.25).Proof. For each 1 ≤ k ≤ r , let us write m k = d ′ k − d k . Let D = D =( d , · · · , d r ), and for each 1 ≤ k ≤ r let us write D k = D + k X i =1 m i e i = ( d + m , d + m , · · · , d k + m k , d k +1 , d k +2 , · · · , d r )= ( d ′ , d ′ , · · · , d ′ k , d k +1 , d k +2 , · · · , d r ) . Note that D r = ( d ′ , · · · , d ′ r ) = D ′ for k = r . Now for each 1 ≤ i < r ,define the map f i : Pol ∗ D i − ( S , X Σ ) → Pol ∗ D i ( S , X Σ ) by the composite ofstabilization maps f i = ( s D i − +( m i − e i ,i ) ◦ ( s D i − +( m i − e i ,i ) ◦· · ·◦ ( s D i − +2 e i ,i ) ◦ ( s D i − + e i ,i ) ◦ ( s D i − ,i ) . Since the stabilization map s D i − + k e i ,i is a homotopy equivalence throughdimension d ( D i − + k e i , Σ) by Theorem 6.5 and d ( D i − + k e i , Σ) ≤ d ( D i − +( k + 1) e i , Σ) for each 0 ≤ k < r , the map f i is a homotopy equivalencethrough dimension d ( D i − , Σ). Furthermore, we can easily see that s D,D ′ = f r − ◦ f r − ◦ · · · ◦ f ◦ f (up to homotopy equivalence)Since d ( D, Σ) = d ( D , Σ) ≤ d ( D , Σ) ≤ d ( D , Σ) ≤ · · · ≤ d ( D r , Σ) = d ( D ′ , Σ) , the map s D,D ′ is a homotopy equivalence through dimension d ( D, Σ).
Definition 6.7.
Let D = ( d , · · · , d r ) ∈ N r be an r -tuple of positive integers.If the condition P rk =1 d k n k = n is satisfied, the map(6.14) j D : Pol ∗ D ( S , X Σ ) → Ω Z K Σ P rk =1 d k n k = n .Suppose that P rk =1 d k n k = n . By the assumption (2.14.1), there is an r -tuple D ∗ = ( d ∗ , · · · , d ∗ r ) ∈ N r such that P rk =1 d ∗ k n k = n . If we choose asufficiently large integer m ∈ N , then the following condition holds:(6.15) d k < m d ∗ k for each 1 ≤ k ≤ r. Now we set D = m D ∗ = ( m d ∗ , m d ∗ , · · · , m d ∗ r ) and define the map(6.16) j D : Pol ∗ D ( S , X Σ ) → Ω Z K Σ as the composite of maps(6.17) j D = j D ◦ s D,D : Pol ∗ D ( S , X Σ ) s D,D −→ Pol ∗ D ( S , X Σ ) j D −→ Ω Z K Σ . Similarly, for each r -tuple D = ( d , · · · , d r ) ∈ N r of positive integers, definethe map(6.18) i D : Pol ∗ D ( S , X Σ ) → Ω X Σ as the composite of maps(6.19) i D = q Σ ◦ j D . Remark 6.8.
Note that the maps j D and i D depend on the choice of thepair ( D ∗ , m ), but one can show that their homotopy classes do not dependon this pair. In this section we consider configuration spaces and the scanning map. Inparticular, we recall the definition of the horizontal scanning map and provethat it is a homotopy equivalence.
Definition 7.1.
For a positive integer d ≥ X, let SP d ( X )denote the d -th symmetric product of X defined as the orbit space(7.1) SP d ( X ) = X d /S d , where the symmetric group S d of d letters acts on the d -fold product X d inthe natural manner. 30 emark 7.2. (i) Note that an element η ∈ SP d ( X ) may be identified witha formal linear combination(7.2) η = s X k =1 n k x k , where { x k } sk =1 ∈ C s ( X ) and { n k } sk =1 ⊂ N with P sk =1 n k = d .(ii) For example, when X = C , we have the natural homeomorphism(7.3) ψ d : P d ( C ) ∼ = −→ SP d ( C )given by using the above identification(7.4) ψ d ( f ( z )) = ψ d ( s Y k =1 ( z − α k ) d k ) = s X k =1 d k α k for f ( z ) = Q sk =1 ( z − α k ) d k ∈ P d ( C ). Definition 7.3. (i) For a subspace A ⊂ X , let SP d ( X, A ) denote the quotientspace(7.5) SP d ( X, A ) = SP d ( X ) / ∼ where the equivalence relation ∼ is defined by ξ ∼ η ⇔ ξ ∩ ( X \ A ) = η ∩ ( X \ A ) for ξ, η ∈ SP d ( X ) . Thus, the points of A are ignored. When A = ∅ , by adding a point in A wehave the natural inclusionSP d ( X, A ) ⊂ SP d +1 ( X, A ) . Thus, when A = ∅ , one can define the space SP ∞ ( X, A ) by the union(7.6) SP ∞ ( X, A ) = [ d ≥ SP d ( X, A ) , where we set SP ( X, A ) = {∅} and ∅ denotes the empty configuration.(ii) From now on, we always assume that X ⊂ C . For each r -tuple D = ( d , · · · , d r ) ∈ N r , let SP D ( X ) = Q ri =1 SP d i ( X ), and define the spaces E Σ , R D ( X ) and E Σ D ( X ) by E Σ , R D ( X ) = { ( ξ , · · · , ξ r ) ∈ SP D ( X ) : ( \ k ∈ σ ξ k ) ∩ R = ∅ for any σ ∈ I ( K Σ ) } ,E Σ D ( X ) = { ( ξ , · · · , ξ r ) ∈ SP D ( X ) : \ k ∈ σ ξ k = ∅ for any σ ∈ I ( K Σ ) } . A ⊂ X is a subspace, define an equivalence relation “ ∼ ” onthe space E Σ , R D ( X ) (resp. the space E Σ D ( X )) by( ξ , · · · , ξ r ) ∼ ( η , · · · , η r ) if ξ i ∩ ( X \ A ) = η i ∩ ( X \ A ) for each 1 ≤ j ≤ r. Let E Σ , R D ( X, A ) and E Σ D ( X, A ) be the quotient spaces defined by E Σ , R D ( X, A ) = E Σ , R D ( X ) / ∼ , and E Σ D ( X, A ) = E Σ D ( X ) / ∼ . When A = ∅ , by adding points in A we have natural inclusions E Σ , R D ( X, A ) ⊂ E Σ , R D + e i ( X, A ) , E Σ D ( X, A ) ⊂ E Σ D + e i ( X, A )for each 1 ≤ i ≤ r, where D + e i = ( d , · · · , d i − , d i + 1 , d i +1 , · · · , d r ) asin (5.15). Thus, when A = ∅ , one can define the spaces E Σ , R ( X, A ) and E Σ ( X, A ) by the unions(7.7) E Σ , R ( X, A ) = [ D ∈ N r E Σ , R D ( X, A ) , E Σ ( X, A ) = [ D ∈ N r E Σ D ( X, A ) , where the empty configuration ( ∅ , · · · , ∅ ) is the base-point of E Σ , R ( X, A ) or E Σ ( X, A ). Remark 7.4. (i) If X ⊂ R , the following equalities hold: E Σ , R D ( X ) = E Σ D ( X ) and E Σ , R ( X, A ) = E Σ ( X, A ) . (ii) Let D = ( d , · · · , d r ) ∈ N r . Then by using the identification (7.3) weeasily obtain the homeomorphism(7.8) Pol ∗ D ( S , X Σ ) Ψ D −−−→ ∼ = E Σ , R D ( C )( f ( z ) , · · · , f r ( z )) −−−→ ( ψ d ( f ( z )) , · · · , ψ d r ( f r ( z )))(iii) Now let ϕ D : C ∼ = −→ U D and x D = ( x D, , · · · , x D,r ) ∈ F ( C \ U D , r )be the homeomorphism and the point for defining the stabilization map s D in Definition 6.1. Then define the map(7.9) s Σ D : E Σ , R D ( C ) → E Σ , R D + e ( C ) by s Σ D ( ξ , · · · , ξ r ) = ( ϕ D ( ξ ) + x D, , · · · , ϕ D ( ξ r ) + x D,r )for ( ξ , · · · , ξ r ) ∈ E Σ , R D , where we write ϕ D ( ξ ) = P sk =1 n k ϕ D ( x k ) if ξ = P sk =1 n k x k ∈ SP d ( C ) and ( n k , x k ) ∈ N × C with P sk =1 n k = d .32hen by using the above homeomorphism (7.8), we have the followingcommutative diagram(7.10) Pol ∗ D ( S , X Σ ) s D −−−→ Pol ∗ D + e ( S , X Σ ) Ψ D y ∼ = Ψ D + e y ∼ = E Σ , R D ( C ) s Σ D −−−→ E Σ , R D + e ( C )(iv) Note that E Σ , R D ( C ) is path-connected. Indeed, for any two points ξ , ξ ∈ E Σ D ( C ) , one can construct a path ω : [0 , → E Σ , R D ( C ) such that ω ( i ) = ξ i for i ∈ { , } by using the string representation used in [12, § Appendix]. Thus the space Pol ∗ D ( S , X Σ ) is also path-connected. Definition 7.5.
Define the stabilized space Pol Σ D + ∞ by the colimit(7.11) Pol Σ D + ∞ = lim k →∞ Pol ∗ D + k e ( S , X Σ ) , where the colimt is taken from the family of stabilization maps { s D + k e : Pol ∗ D + k e ( S , X Σ ) → Pol ∗ D +( k +1) e ( S , X Σ ) } k ≥ Now we are ready to define the scanning map. From now on, we identify C = R in a usual way. Definition 7.6.
For a rectangle X in C = R , let σX denote the union of thesides of X which are parallel to the y -axis, and for a subspace Z ⊂ C = R ,let Z be the closure of Z . From now on, let I denote the interval I = [ − , < ǫ < be any fixed real number.For each x ∈ R , let V ( x ) be the set defined by V ( x ) = { w ∈ C : | Re( w ) − x | < ǫ, | Im( w ) | < } (7.12) = ( x − ǫ, x + ǫ ) × ( − , , and let’s identify I × I = I with the closed unit rectangle { t + s √− ∈ C : − ≤ t, s ≤ } in C .For each D = ( d , · · · , d r ) ∈ N r , define the horizontal scanning map (7.13) sc D : E Σ , R D ( C ) → Ω E Σ , R ( I , ∂I × I ) = Ω E Σ , R ( I , σI )as follows. For each r -tuple α = ( ξ , · · · , ξ r ) ∈ E Σ , R D ( C ) of configurations, let sc D ( α ) : R → E Σ , R ( I , ∂I × I ) = E Σ , R ( I , σI ) denote the map given by R ∋ x ( ξ ∩ V ( x ) , · · · , ξ r ∩ V ( x )) ∈ E Σ , R ( V ( x ) , σV ( x )) ∼ = E Σ , R ( I , σI ) , V ( x ) , σV ( x )) ∼ = ( I , σI ) . Since lim x →±∞ sc D ( α )( x ) = ( ∅ , · · · , ∅ ), by setting sc D ( α )( ∞ ) = ( ∅ , · · · , ∅ )we obtain a based map sc D ( α ) ∈ Ω E Σ , R ( I , σI ) , where we identify S = R ∪ ∞ and we choose the empty configuration ( ∅ , · · · , ∅ ) as the base-pointof E Σ , R ( I , σI ). One can show that the following diagram is homotopycommutative:(7.14) E Σ , R D + k e ( C ) sc D + k e −−−−→ Ω E Σ , R ( I , σI ) s Σ D + k e y k E Σ , R D +( k +1) e ( C ) sc D +( k +1) e −−−−−−→ Ω E Σ , R ( I , σI )Thus by using the above diagram and by identifying Pol ∗ D + k e ( S , X Σ ) with E Σ , R D + k e ( C ), we finally obtain the stable horizontal scanning map (7.15) S = lim k →∞ sc D + k e : Pol Σ D + ∞ → Ω E Σ , R ( I , σI ) , where Pol Σ D + ∞ is defined in (7.11).(ii) If we identify ( I, ∂I ) = ( I × { } , ∂I × { } ), then we notice that( ξ ∩ R , · · · , ξ r ∩ R ) ∈ E Σ , R ( I, ∂I ) for ( ξ , · · · , ξ r ) ∈ E Σ , R ( I , σI ) . Thus onecan define the map R Σ : E Σ , R ( I , σI ) → E Σ , R ( I, ∂I ) by(7.16) R Σ ( ξ , · · · , ξ r ) = ( ξ ∩ R , · · · , ξ r ∩ R ) for ( ξ , · · · , ξ r ) ∈ E Σ , R ( I , σI ).(iii) Let S H : Pol Σ D + ∞ → Ω E Σ , R ( I, ∂I ) denote the horizontal scanningmap defined by the composite of maps(7.17) S H = (Ω R Σ ) ◦ S : Pol Σ D + ∞ S −→ Ω E Σ , R ( I , σI ) Ω R Σ −→ Ω E Σ , R ( I, ∂I ) , where S denotes the stable horizontal scanning map given by (7.15). Lemma 7.7.
The map R Σ : E Σ , R ( I , σI ) ≃ −→ E Σ , R ( I, ∂I ) = E Σ ( I, ∂I ) is adeformation retraction.Proof. We identify I = { a + b √− ∈ C : − ≤ a, b ≤ } ⊂ C as before. LetΠ ⊂ I denote the subspace defined by Π = { a + b √− ∈ I : b ∈ { , ± }} . For b ∈ R , let ǫ ( b ) = b | b | if b = 0 and ǫ (0) = 0. Now consider the homotopy ϕ : I × [0 , → I given by ϕ ( α, t ) = a + { (1 − t ) b + ǫ ( b ) t }√− α = a + b √− ∈ I ( a, b ∈ R ). By means of this homotopy, one can define adeformation retraction R : E Σ , R ( I , σI ) ≃ −→ E Σ , R (Π , ∂I × { , ± } ) . Next, by using the homotopy given by f t ( a + b √−
1) = ta + (1 − t ) + b √− b = ± and f t ( a + b √−
1) = a if b = 0, one can also define a deformationretraction ϕ : E Σ , R (Π , ∂I × { , ± } ) ≃ −→ E Σ , R ( I, ∂I ) . Since R Σ = ϕ ◦ R , themap R Σ is a deformation retraction. 34 heorem 7.8 ([10], [26]) . The horizontal scanning map S H : Pol Σ D + ∞ ≃ −→ Ω E Σ ( I, ∂I ) is a homotopy equivalence.Proof. Since R Σ is a homotopy equivalence and S H = (Ω R Σ ) ◦ S , it sufficesto show that S is a homotopy equivalence.The proof is analogous to the one given in [26, Prop. 3.2, Lemma 3.4] and[10, Prop. 2]. We identify C = R by means of the identification x + √− y ( x, y ) in the usual way. Let B and B ∗ denote the rectangles in R = C givenby B ∗ = [ − , × [ − ,
1] and B = (0 , × ( − , { V t : 0 < t < } be thefamily of open rectangles in B given by V t = ( t − ǫ ( t ) , t + ǫ ( t )) × ( − , ǫ ( t ) denotes the continuous function defined on the open interval (0 ,
1) suchthat 0 < ǫ ( t ) < min { t, − t } for any t ∈ (0 ,
1) with lim t → +0 ǫ ( t ) = lim t → − ǫ ( t ) = 0 . Let e sc HD : E Σ , R D ( B ) × [0 , → E Σ , R ( B, σB ) denote the map given by e sc HD (( ξ , · · · , ξ r ) , t ) = ( ξ ∩ V t , · · · , ξ r ∩ V t ) ∈ E Σ , R ( V t , σV t ) ∼ = E Σ , R ( B, σB ) , where we use the canonical identification ( V t , σV t ) ∼ = ( B, σB ) . Since lim t → +0 e sc HD ( ξ, t ) = lim t → − e sc HD ( ξ, t ) = ( ∅ , · · · , ∅ ) for any ξ ∈ E Σ , R D ( B ),the adjoint of e sc HD defines the map sc HD : E Σ , R D ( B ) → Ω E Σ , R ( B, σB ) . If s Σ D : E Σ , R D ( B ) → E Σ , R D + e ( B ) denotes the stabilization map defined by addingpoints from infinity as in (7.10), we obtain a homotopy commutative diagramPol ∗ D ( S , X Σ ) s D −−−→ Pol ∗ D + e ( S , X Σ ) ∼ = y ∼ = y E Σ , R D ( B ) s Σ D −−−→ E Σ , R D + e ( B )for each D ∈ N r . If E Σ , R D + ∞ ( B ) denotes the colimit E Σ , R D + ∞ ( B ) = lim k →∞ E Σ , R D + k e ( B )taken over the stabilization maps { s Σ D, + k e : E Σ , R D + k e ( B ) → E Σ , R D +( k +1) e ( B ) } ,there is a homotopy commutative diagramPol ∗ D + ∞ ( S , X Σ ) S −−−→ Ω E Σ , R ( I , σI ) ∼ = y ∼ = y E Σ , R D + ∞ ( B ) S ′ −−−→ Ω E Σ , R ( B, σB )where we set S ′ = lim k →∞ sc HD + k e . It suffices to prove that S ′ is a homotopyequivalence. Let E Σ , R D denote the subspace of E Σ , R ( B ∗ , σB ∗ ) consisting of35ll r -tuples ( ξ , · · · , ξ r ) ∈ E Σ , R ( B ∗ , σB ∗ ) such that ( ξ ∩ B, · · · , ξ r ∩ B ) ∈ E Σ , R D ( B ) . Let q D : E Σ , R D → E Σ , R ( B ∗ , σB ∗ ∪ B ) ∼ = E Σ , R ( B, σB ) denote the re-striction of the the quotient map E Σ , R ( B ∗ , σB ∗ ) → E Σ , R ( B ∗ , σB ∗ ∪ B ) ∼ = E Σ , R ( B, σB ) , where E Σ , R ( B, σB ) = E Σ , R ( B, σB ) × E Σ , R ( B, σB ). It is easyto see that the fiber of q D is homeomorphic to E Σ , R D ( B ).Next, define the scanning map ˜ S D : E Σ , R D → Map([0 , , E Σ , R ( B, σB )) by E Σ , R D × [0 , −−−→ E Σ , R ( B, σB )(( ξ , · · · , ξ r ) , t ) −−−→ ( ξ ∩ B t , · · · , ξ r ∩ B t ) ∈ E Σ , R ( B t , σB t ) ∼ = E Σ , R ( B, σB )where { B t : 0 < t < } denotes the family of the open rectangles in B ∗ defined by B t = (2 t − , t ) × ( − ,
1) and we use the canonical identification( B t , σB t ) ∼ = ( B, σB ) . We obtain a homotopy commutative diagram(7.18) E Σ , R D q D −−−→ E Σ , R ( B, σB ) S D y k Map([0 , , E Σ , R ( B, σB )) res −−−→ E Σ , R ( B, σB ) where res : Map([0 , , E Σ , R ( B, σB )) → Map( { , } , E Σ , R ( B, σB )) ∼ = E Σ , R ( B, σB ) denotes the restriction map.Note that res is a fibration with fibre Ω E Σ , R ( B, σB ), but the map q D is not a fibration although every fiber of q D is homeomorphic to E Σ , R ( B ).However, once the map q D is stabilized, it becomes a quasifibration.To see this, let J = (0 ,
1) and for each D ∈ N r let us choose the r -tuple c D = ( c D, , · · · , c D,r ) ∈ ( J × J ) r such that c D,j = c D ′ ,k if ( D, j ) = ( D ′ , k )with lim D →∞ c D,k = ( ,
0) for each 1 ≤ k ≤ r .Let [ E Σ , R ( B ∗ , σB ∗ ) denote the space of r -tuples ( ξ , · · · , ξ r ) of formal in-finite divisors in ( B ∗ , σB ∗ ) satisfying the following two conditions:( † ) ( T rk =1 ξ k ) ∩ R ∩ ( B ∗ \ σB ∗ ) = ∅ . ( † ) Each divisor ξ k is represented as the formal infinite sum of the form ξ k = P D ξ k,D such that ξ k,D ∈ SP ( B ∗ , σB ∗ ) for each D ∈ N r , andit almost coincides (except finite sums) with ξ ∗ k = P D c D,k for each1 ≤ k ≤ r . 36ote that c D,k ∈ B for any ( D, k ) ∈ N r × [ r ]. Thus one can define the mapˆ q : [ E Σ , R ( B ∗ , σB ∗ ) → E Σ , R ( B, σB ) to be the natural quotient map [ E Σ , R ( B ∗ , σB ∗ ) → E Σ , R ( B ∗ , σB ∗ ∪ B ) ∼ = E Σ , R ( B, σB ) . By using the Dold-Thom argument exactly as in [26, Lemma 3.4] togetherwith the fact that E Σ , R D ( C ) is simply connected (by Proposition 5.9 and (7.8)),one can show that ˆ q is a quasifibration.Now define stabilization maps f D : E Σ , R D → E Σ , R D + e by f D ( ξ , · · · , ξ r ) =( ξ + c D, , · · · , ξ r + c D,r ) for ( ξ , · · · , ξ r ) ∈ E Σ , R D . Let E Σ , R denote the colimit E Σ , R = lim k →∞ E Σ , R D + k e from the stabilizationmaps { f D + k e } k ≥ . Since [ E Σ , R ( B ∗ , σB ∗ ) is Z r × E Σ , R , the restriction q ∞ =ˆ q | E Σ , R : E Σ , R → E Σ , R ( B, σB ) is also a quasifibration. Since the fiber of q D is E Σ , R ( B ) and q ∞ | E Σ , R = q D , we see that the fiber of q ∞ is E Σ , R ( B ) and wemay regard the map q ∞ as the stabilized map of { q D + k e } k ≥ .We also obtain the stabilized scanning map˜ S = lim k →∞ ˜ S D + k e : E Σ , R = lim k →∞ E Σ , R D + k e → Map([0 , , E Σ , R ( B, σB )) . Since [0 ,
1] is contractible and there is a homotopy equivalence E Σ , R =lim k →∞ E Σ , R D + k e ≃ E Σ , R ( B, σB ), we see that ˜ S is a homotopy equivalence.Since B ⊂ B ∗ \ σB ∗ , we may regard E Σ , R ( B ) as a subspace of E Σ , R ( B ∗ , σB ∗ )and we see that ˜ S D | E Σ , R ( B ) = sc HD . Thus, we can identify ˜ S | E Σ , R ( B ) = S ′ and by the diagram (7.18) we have the homotopy commutative diagram E Σ , R ( B ) −−−→ E Σ , R q ∞ −−−→ E Σ , R ( B, σB ) S ′ y ˜ S y ≃ k E Σ , R ( B, σB ) −−−→ Map([0 , , E Σ , R ( B, σB )) res −−−→ E Σ , R ( B, σB ) Since the upper horizontal sequence is a quasifibration sequence and the lowerhorizontal one is a fibration sequence, S ′ is a homotopy equivalence. In this section we prove the stability result (Theorem 8.2).
Definition 8.1.
Let D = ( d , · · · , d r ) ∈ N r be an r -tuple of positive integers.37hen it is easy to see that the following diagram is homotopy commutative:Pol ∗ D ( S , X Σ ) j D −−−→ Ω U ( K Σ ) s D y k Pol ∗ D + e ( S , X Σ ) j D + e −−−→ Ω U ( K Σ )Hence we can stabilize the map(8.1) j D + ∞ = lim t →∞ j D + t e : Pol Σ D + ∞ = lim t →∞ Pol ∗ D + t e ( S , X Σ ) → Ω U ( K Σ ) . The main purpose of this section is to prove the following result.
Theorem 8.2.
The map j D + ∞ : Pol Σ D + ∞ ≃ −→ Ω U ( K Σ ) is a homotopy equiv-alence. Before proving Theorem 8.2 we need the following definition and lemma.
Definition 8.3.
Now we identify C = R in a usual way and let us write U = { w ∈ C : | Re( w ) | < , | Im( w ) | < } = ( − , × ( − ,
1) and I = [ − , X ⊂ C , let F ( X ) denote the space of r -tuples( f ( z ) , · · · , f r ( z )) ∈ C [ z ] r of (not necessarily monic) polynomials satisfyingthe following condition ( ∗ ):( ∗ ) For any σ = { i , · · · , i s } ∈ I ( K Σ ), the polynomials f i ( z ) , · · · , f i s ( z )have no common real roots in X (i.e. no common roots in X ∩ R ).Similarly, let F C ( X ) ⊂ F ( X ) denote the subspace of all ( f ( z ) , · · · , f r ( z )) ∈ F ( X ) satisfying the following condition ( ∗ ) C :( ∗ ) C For any σ = { i , · · · , i s } ∈ I ( K Σ ), the polynomials f i ( z ) , · · · , f i s ( z )have no common roots in X .(ii) Let ev : F ( U ) → U ( K Σ ) denote the map given by evaluation at 0,i.e. ev ( f ( z ) , · · · , f r ( z )) = ( f (0) , · · · , f r (0)) for ( f ( z ) , · · · , f r ( z )) ∈ F ( U ).(iii) Let ˜ F ( U ) ⊂ F ( U ) (resp. ˜ F C ( U ) ⊂ F C ( U )) denote the subspace ofall ( f ( z ) , · · · , f r ( z )) ∈ F ( U ) (resp. ( f ( z ) , · · · , f r ( z )) ∈ F C ( U )) such thatno f i ( z ) is identically zero. Let j F : ˜ F C ( U ) ⊂ −→ ˜ F ( U ) denote the inclusionmap.(iv) Let ev R : ˜ F ( U ) → U ( K Σ ) denote the map given by the restriction ev R = ev | ˜ F ( U ) . (v) Note that the group T r R = ( R ∗ ) r (resp. T r C = ( C ∗ ) r ) acts freely on thespace ˜ F ( U ) (resp. ˜ F C ( U )) in a natural way, and let(8.2) p : ˜ F ( U ) → ˜ F ( U ) / T r R ( q : ˜ F C ( U ) → ˜ F C ( U ) / T r C )38enote the natural projection, where ˜ F ( U ) / T r R (resp. ˜ F C ( U ) / T r C ) denotesthe orbit space. Note that U ∩ R = I × { } , and let(8.3) v : ˜ F ( U ) / T r R → E Σ ( I, ∂I )denote the natural map which assigns to an r -tuple ( f ( z ) , · · · , f r ( z )) ∈ ˜ F ( U )the r -tuple of their configurations represented by their real roots in I =[ − , U = D and let(8.4) u : ˜ F C ( U ) / T r C → E Σ ( D , S )denote the natural map which assigns to an r -tuple ( f ( z ) , · · · , f r ( z )) ∈ ˜ F C ( U ) the r -tuple of their configurations represented by their roots in D .(vi) By using this identification ( I, ∂I ) = ( I × { } , ∂I × { } ) ⊂ ( D , S ),we obtain the inclusion map(8.5) i Σ : E Σ ( I, ∂I ) ⊂ −→ E Σ ( D , S ) . Lemma 8.4. (i)
The space
Pol Σ D + ∞ is simply connected for any D ∈ N r . (ii) The map ev R : ˜ F ( U ) ≃ −→ U ( K Σ ) is a homotopy equivalence. (iii) The inclusion map j F : ˜ F C ( U ) ≃ −→ ˜ F ( U ) is a homotopy equivalence. (iv) The map u : ˜ F C ( U ) / T r C ≃ −→ E Σ ( D , S ) is a homotopy equivalence. (v) The induced homomorphisms ( ( i Σ ) ∗ : π k ( E Σ ( I, ∂I )) ∼ = −→ π k ( E Σ ( D , S )) v ∗ : π k ( ˜ F ( U ) / T r R ) ∼ = −→ π k ( E Σ ( I, ∂I )) are isomorphisms for any k ≥ .Proof. (i) Since Pol ∗ D ( S , X Σ ) is simply connected for any D ∈ N r by Propo-sition 5.9, the space Pol Σ D + ∞ is also simply connected.(ii) Let i : U ( K Σ ) → F ( U ) be the inclusion map given by viewing con-stants as polynomials. Clearly ev ◦ i = id. Let f : F ( U ) × [0 , → F ( U )be the homotopy given by f (( f , · · · , f t ) , t ) = ( f ,t ( z ) , · · · , f r,t ( z )), where f i,t ( z ) = f i ( tz ). This gives a homotopy between i ◦ ev and the identitymap, and this proves that ev is a deformation retraction. Since F ( U ) isan infinite dimensional manifold and ˜ F ( U ) is a closed submanifold of F ( U )of infinite codimension, it follows from [8, Theorem 2] that the inclusion˜ F ( U ) → F ( U ) is a homotopy equivalence. Hence the restriction ev R is alsoa homotopy equivalence.(iii) It follows from [20, Lemma 5.4] that the evaluation map ev = ev R ◦ j F :˜ F C ( U ) ≃ −→ U ( K Σ ) is a homotopy equivalence. Since ev R is a homotopy39quivalence by the assertion (ii), we see that the map j F : ˜ F C ( U ) ≃ −→ ˜ F ( U )is also a homotopy equivalence.(iv) We know that the map u is a homotopy equivalence by [11, page133]. However, for the sake of completeness of this paper we will give theanother proof. Since all the maps and homotopies which appear in the proofof [20, Lemma 5.4] are T r C -equivariant maps, we see that the map ev isindeed a T r C -equivariant homotopy equivalence. Thus the map ev induces ahomotopy equivalence e ev : ˜ F C ( U ) / T r C ≃ −→ EG × G U ( K Σ ), where G = T r C and EG × G U ( K Σ ) denotes the Borel construction.Recall that there is a natural deformation retraction rt : EG × G U ( K Σ ) ≃ −→ DJ ( K Σ ) by [3, Theorem 6.29]. If r Σ : E Σ ( D , S ) ≃ −→ DJ ( K Σ ) denotes thedeformation retraction given in [20, Lemma 4.3], then one can check that thefollowing diagram is commutative (up to homotopy equivalence):(8.6) ˜ F C ( U ) / T r C e ev −−−→ ≃ EG × G U ( K Σ ) u y rt y ≃ E Σ ( D , S ) r Σ −−−→ ≃ DJ ( K Σ )Thus, we see that u is a homotopy equivalence.(v) Let p ′ : ˜ F ( U ) / T r R → ˜ F ( U ) / T r C be the natural projection. Since p ′ is abundle projection with fiber T r C / T r R ∼ = ( S ) r , we see that the map p ′ inducesan isomorphism on homotopy groups π k ( ) for any k ≥
3. Moreover, since theinclusion j F is a T r C -equivariant map and the group T r C acts freely on bothspaces spaces ˜ F C ( U ) and ˜ F ( U ), there is a map ˜ j F : ˜ F C ( U ) / T r C → ˜ F ( U ) / T r C such that the following diagram is commutative:(8.7) T r C −−−→ ˜ F C ( U ) q −−−→ ˜ F C ( U ) / T r C k j F y ≃ ˜ j F y T r C −−−→ ˜ F ( U ) −−−→ ˜ F ( U ) / T r C Since two horizontal sequences of (8.7) are fibration sequences and the map j F is a homotopy equivalence, the map ˜ j F is a homotopy equivalence. Thenif we consider the commutative diagram(8.8) ˜ F ( U ) / T r R v −−−→ E Σ ( I, ∂I ) i Σ −−−→ E Σ ( D , S ) p ′ y k ˜ F ( U ) / T r C ˜ j F ←−−− ≃ ˜ F C ( U ) / T r C u −−−→ ≃ E Σ ( D , S )40e see that the map i Σ induces an epimorphism on homotopy groups π k ( ) forany k ≥
3. Let R : E Σ ( D , S ) → E Σ ( I, ∂I ) denote the restriction map givenby R ( ξ , · · · , ξ r ) = ( ξ ∩ I, · · · , ξ r ∩ I ). Since R ◦ i Σ = id, the map i Σ inducesa monomorphism on homotopy groups π k ( ) for any k ≥
1. Thus the map i Σ induces an isomorphism on homotopy groups π k ( ) for any k ≥
3. By thediagram (8.8), we can easily see that the map v also induces an isomorphismon homotopy groups π k ( ) for any k ≥ Proof of Theorem 8.2.
Since Pol Σ D + ∞ and Ω U ( K Σ ) are simply connected, itsuffices to show that( † ) k ( j D + ∞ ) ∗ : π k (Pol Σ D + ∞ ) ∼ = −→ π k (Ω U ( K Σ ))is an isomorphism for any k ≥
2. Let us identify C = R and let U =( − , × ( − ,
1) as before. Let scan : ˜ F ( C ) → Map( R , ˜ F ( U )) denote themap given by scan ( f ( z ) , · · · , f r ( z ))( w ) = ( f ( z + w ) , · · · , f r ( z + w )) for w ∈ R , and consider the diagram˜ F ( U ) ev R −−−→ ≃ U ( K Σ ) p y ˜ F ( U ) / T r R v −−−→ E Σ ( I, ∂I )This induces the commutative diagram below˜ F ( C ) scan −−−→ Map( R , ˜ F ( U )) ( ev R ) −−−−→ ≃ Map( R , U ( K Σ )) p y p y ˜ F ( C ) / T r R scan −−−→ Map( R , ˜ F ( U ) / T r R ) v −−−→ Map( R , E Σ ( I, ∂I ))Observe that Map( R , · ) can be replaced by Map ∗ ( S , · ) by extending from R to S = R ∪ ∞ (as base-point preserving maps). Thus by setting (c j D : Pol ∗ D ( S , X Σ ) ⊂ −→ ˜ F ( C ) scan −→ Map ∗ ( S , ˜ F ( U )) = Ω ˜ F ( U ) c j ′ D : E Σ , R D ( C ) ⊂ −→ ˜ F ( C ) scan −→ Map ∗ ( S , ˜ F ( U ) / T r R ) = Ω ˜ F ( U ) / T r R we obtain the following commutative diagram(8.9) Pol ∗ D ( S , X Σ ) c j D −−−→ Ω ˜ F ( U ) Ω ev R −−−→ ≃ Ω U ( K Σ ) ∼ = y Ω p y ≃ E Σ , R D ( C ) c j ′ D −−−→ Ω ˜ F ( U ) / T r R Ω v −−−→ Ω E Σ ( I, ∂I )41ince p is a covering projection, Ω p is a homotopy equivalence. If we identifyPol Σ D + ∞ with the colimit lim t →∞ E Σ , R D + t e ( C ), by replacing D by D + t e ( t ∈ N )and letting t → ∞ , we obtain the following homotopy commutative diagram:(8.10) Pol Σ D + ∞ \ j D + ∞ −−−→ Ω ˜ F ( U ) Ω ev R −−−→ ≃ Ω U ( K Σ ) k Ω p y ≃ Pol Σ D + ∞ \ j ′ D + ∞ −−−→ Ω ˜ F ( U ) / T r R Ω v −−−→ Ω E Σ ( I, ∂I )where we set [ j D + ∞ = lim t →∞ [ j D + t e and [ j ′ D + ∞ = lim t →∞ [ j ′ D + t e . Since (Ω ev R ) ◦ [ j D + t e = j D + t e and (Ω v ) ◦ [ j ′ D + t e = sc D + t e (up to homotopyequivalence), we also obtain the following two equalities:(8.11) j D + ∞ = (Ω ev R ) ◦ [ j D + ∞ , S H = (Ω v ) ◦ [ j ′ D + ∞ . Since the map ev R is a homotopy equivalence, it suffices to prove that( †† ) k ( [ j D + ∞ ) ∗ : π k (Pol Σ D + ∞ ) ∼ = −→ π k (Ω ˜ F ( U ))is an isomorphism for any k ≥
2. Since S H = (Ω v ) ◦ [ j ′ D + ∞ is a homotopyequivalence and the map Ω v induces an isomorphism on homotopy groups π k ( ) for any k ≥ [ j D + ∞ induces an isomorphism on homotopy groups π k ( ) for any k ≥
2. This completes the proof of Theorem 8.2.
Corollary 8.5.
Let X Σ be a simply connected non-singular toric variety suchthat the condition (2.14.1) is satisfied. (i) The two-fold loop map Ω v : Ω ˜ F ( U ) / T r R ≃ −→ Ω E Σ ( I, ∂I ) is a homo-topy equivalence. (ii) The loop map Ω i Σ : Ω E Σ ( I, ∂I ) → Ω E Σ ( D , S ) is a universal cover-ing projection with fiber Z r (up to homotopy equivalence).Proof. (i) The assertion (i) follows from (v) of Lemma 8.4.(ii) Since Ω i Σ is a homotopy equivalence (by (v) of Lemma 8.4), theassertion (ii) easily follows from the following two equalities:(8.12) π (Ω E Σ ( I, ∂I )) = 0 , π (Ω E Σ ( D , S )) = Z r . Since Pol Σ D + ∞ is simply connected (by (i) of Lemma 8.4), by Theorem 8.2we have an isomorphism 0 = π (Pol Σ D + ∞ ) ∼ = π (Ω E Σ ( I, ∂I )) and the first42quality of (8.12) holds. It remains the second isomorphism in (8.12). Sincethere is a homotopy equivalence E Σ ( D , S ) ≃ DJ ( K Σ ) ([20, Lemma 4.3]),it suffices to show that there is an isomorphism π ( DJ ( K Σ )) ∼ = Z r . Since X Σ is simply connected and π ( X Σ ) = Z r − n (by (2.26)), it follows fromthe homotopy exact sequence induced from (3.5) that there is a short exactsequence 0 → π ( X Σ ) = Z r − n → π ( DJ ( K Σ )) ∂ −→ π ( T n C ) = Z n → . Thus,we have an isomorphism π ( DJ ( K Σ )) ∼ = Z r . In this section we give the proofs of the main results (Theorem 2.16, Corollary2.17, Theorem 2.20, Corollary 2.21, Corollary 2.18, Corollary 2.22).
Proofs of Theorem 2.16 and Theorem 2.20. If P rk =1 d k n k = n , Theorem2.16 easily follows from Theorem 6.5 and Theorem 8.2. Next assume that P rk =1 d k n k = n . It follows from the assumption (2.14.1) that there is an r -tuple D ∗ = ( d ∗ , · · · , d ∗ r ) ∈ N r such that P rk =1 d ∗ k n k = n . If we choose asufficiently large integer m ∈ N , then the condition d k < m d ∗ k holds foreach 1 ≤ k ≤ r . Then note that j D : Pol ∗ D ( S , X Σ ) → Ω Z K Σ is given by thecomposite of maps j D = j D ◦ s D,D : Pol ∗ D ( S , X Σ ) s D,D −→ Pol ∗ D ( S , X Σ ) j D −→ Ω Z K Σ as in (6.17), where D = m D ∗ = ( m d ∗ , m d ∗ , · · · , m d ∗ r ) . Since the maps s D,D and j D are homotopy equivalences through dimension d ( D, Σ) and d ( D , Σ) (by Theorem 6.6 and Theorem 2.16), by using d ( D, Σ) ≤ d ( D , Σ)the map j D is a homotopy equivalence through dimension d ( D, Σ) and The-orem 2.20 follows.
Proof of Corollary 2.17.
Since Ω q Σ : Ω Z K Σ → Ω X Σ is a universal covering,the assertion follows from (2.23) and Theorem 2.16. Proofs of Corollary 2.21.
Consider the map i D : Pol ∗ D ( S , X Σ ) → Ω X Σ de-fined by the composite of maps(9.1) i D = Ω q Σ ◦ j D , where j D : Pol ∗ D ( S , X Σ ) → Ω Z K Σ denotes the map given by (6.17). SinceΩ q Σ : Ω Z K Σ → Ω X Σ is a universal covering, the assertion follows from The-orem 2.20. 43 roofs of Corollary 2.18 and Corollary 2.22. If X Σ is compact and Σ(1) ⊂ Σ $ Σ, the condition (2.14.1) holds for the fan Σ . Thus, Corollary 2.18follows from Theorem 2.16 and Corollary 2.17. Similarly, Corollary 2.22 alsofollows from Theorem 2.20 and Corollary 2.21. Acknowledgements.
The second author was supported by JSPS KAK-ENHI Grant Number 18K03295. This work was also supported by the Re-search Institute for Mathematical Sciences, a Joint Usage/Research Centerlocated in Kyoto University.
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