INFINITENESS OF A∞-TYPES OF GAUGE GROUPS
DAISUKE KISHIMOTO
Let G be a topological group, and let P be a principal G-bundle over a base K. The gauge group ofP, denoted by G(P), is the topological group of automorphisms of P. We pose:
Problem 0.1. Consider all principal G-bundles overK for fixed Gand K: how many homotopy types ofG(P) are there?
Precise counting has been done in several special cases, and Crabb and Sutherland gave a rough estimate such that if G is a compact Lie group and K is a finite complex, then the number of the homotopy types of G(P) is finite even when there are infinitely many P. Recently Tsutaya generalized this finiteness result to the homotopy types as An-spaces (An-type) for n < ∞. So one can naturally think of A∞-types, and I will explain the following infiniteness result.
Theorem 0.2. Let G be a compact connected simple Lie group. If K is a sphere and there are infinitely many P, then the number of the A∞-types of G(P) is infinite.
This talk is based on joint work with Mitsunobu Tsutaya.
Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan E-mail address: [email protected]
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INDEPENDENCE COMPLEXES AND INCIDENCE GRAPHS
SHUICHI TSUKUDA
We show that the independence complex of the incidence graph of a hypergraph is homotopy equivalent to the combinatorial Alexander dual of the independence complex of the hypergraph, generalizing a result of Csorba. We will discuss some applications.
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Certain homotopy invariants related to self-maps
Kee Young Lee
Abstract. For a connected space X, let E(X) be the group of self-homotopy equivalences of X. We define Aut]k(X) by the set of homotopy classes of self maps of X that induce an automor- phism on πi(X) for i = 0,1,· · · , k, that is, [f] ∈ Aut]k(X) if and only if πi(f) : πi(X) → πi(X) is an isomorphism for i = 0,1,· · · , k. Then Aut]0(X) = [X, X]. Moreover, if k < n, then Aut]n(X) ⊂ Aut]k(X). For any connected CW-complex X, We have Aut]∞(X) =E(X). Thus, for a connected CW-complex X, we have
E(X) =Aut]∞(X)⊆...⊆Aut]1(X)⊆Aut]0(X) = [X, X].
The self closeness number NE(X) is the minimum number k such that E(X) =Aut]k(X). In this work, we study the properties of Aut]k(X) and the self closeness numberNE(X). We will discuss the computation of self closeness numbers and give some examples.
Department of Mathematics, Korea University, Sejong 339-700, Korea
E-mail address: [email protected]
2010 Mathematics Subject Classification. 55p10, 55Q10, 55Q45.
Key words and phrases. self-homotopy equivalence, self closeness number, ho- motopy group.
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A FINITE POSET IN THE INFINITE ORDER SUBGROUP OF GOTTLIEB GROUP
Let ξ : X →j E →p Y be a fibration of simply connected CW com- plexes. Let autX the space of unpointed homotopy self-equivalences of X and aut(p) denote the space of unpointed fibre-homotopy self- equivalences of p, which is the subspace of autE with g : E →E sat- isfying p◦g = p. We define the n-th fibre-restricted Gottlieb group of X with respect to ξ, denoted by Gξn(X) as Im πn(evX ◦ R) for evX ◦ R : aut1(p) → aut1X → X with R restriction map. It is a natural subgroup of the n-th Gottlieb group Gn(X) of X. We ap- proach
Realization Problem. For which subgroup G of Gn(X) does there exist a fibrationξ such thatGξn(X) =G ?
via Sullivan minimal models in rational homotopy theory.
Toshihiro Yamaguchi
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LIFTING SOME GENERALIZED H-STRUCTURES AND THEIR DUALS
YEON SOO YOON
The concepts ofCk-spaces is introduced in [IMOY] which is situated at an in- termediate stage between H-spaces and T-spaces. It is known that a spaceX is a Ck-space if and only if G(Z, X) = [Z, X] for any space Z with cat Z ≤k. Ck- spaces are generalized to the Ckf=spaces for a map f : A → X. On the other hand, the dual spaces, DCkp-spaces for a map p:X →A, ofCkf-spaces are intro- duced and studied [KY]. It is known that a space X is a DCkp-space if and only ifDGp(X, Z) = [X, Z] for any space Z with cocat Z ≤k. In this talk, we study about liftingCkf-structures and extendingDCkp-structures.
Department of Mathematics Education, Hannam University, Daejeon 306-791, Korea E-mail address:[email protected]
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MAPPING SPACES FROM PROJECTIVE SPACES
MITSUNOBU TSUTAYA
We denote the n-th projective space of a topological monoid G by BnGand the classifying space byBG. LetGbe a topological monoid of pointed homotopy type of a CW complex andG0 a grouplike topological monoid. We prove the weak equivalence between the pointed mapping space Map0(BnG, BG) and the space of all An-maps from G to G0. This fact has several applications. As the first application, we show that the connecting map G→Map0(BnG, BG) of the evaluation fiber sequence Map0(BnG, BG) → Map(BnG, BG) → BG is delooped. As other applications, we consider higher homotopy commutativity, An- types of gauge groups and Tkf-spaces by Iwase–Mimura–Oda–Yoon.
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The homotopy type of the space of rational curves on a toric variety
Andrzej Kozlowski
∗and Kohhei Yamaguchi
†Polyhedral products. LetKbe a simplicial complex on the index set [r] ={1,2,· · · , r},1 and let (X, A) be a collection of spaces of pairs {(Xk, Ak)}rk=1. We denote by ZK(X, A) the polyhedral product of (X, A) with respect to K given by ZK(X, A) = ∪
σ∈K(X, A)σ, where we set (X, A)σ ={(x1,· · · , xr)∈X1×· · ·×Xr :xj ∈Aj if j /∈σ}forσ ∈K. When (Xj, Aj) = (X, A) for each j, we writeZK(X, A) = ZK(X, A). For each subset σ ⊂[r], let Lσ ⊂ Cr denote the coordinate subspace of type σ defined by Lσ = {(x1,· · · , xr) ∈ Cr : xj = 0 if j ∈ σ}, and U(K) denotethe complement of the coordinate subspaces of type K defined by U(K) =Cr\∪
σ /∈K,σ⊂[r]Lσ. Note that U(K) = ZK(C,C∗).
The homogenous coordinate of the toric variety. Let Σ be the fan inRn,XΣ denote the toric variety associated to the fan Σ, and let Σ(1) ={ρ1,· · · , ρr} be the set of all one dimensional strongly convex rational polyhedral cones in Σ. Then for each 1≤k ≤ r, let nk ∈ Zn denote the primitive generator of ρk such that ρk∩Zn = Z≥0·nk. Then define the simplicial complex KΣ on [r] and the subgroup GΣ ⊂TrC = (C∗)r by
KΣ = {
{i1,· · · , ik} ⊂[r] :ni1,· · · ,nik span a cone in Σ} , GΣ = {(µ1,· · · , µr)∈TrC:
∏r
k=1
µhkm,nki = 1 for all m ∈Zn}.
It is known that that if the set {n1,· · · ,nr} spans Rn, there is an isomorphism (0.1) XΣ ∼=ZKΣ(C,C∗)/GΣ =U(KΣ)/GΣ,
where the groups GΣ acts on ZKΣ(C,C∗) by the coordinate-wise multiplications. It is known that GΣ acts freely on the space ZKΣ(C,C∗) if XΣ is a smooth toric variety.
We say that a set of primitive generators {ni1,· · · ,nik}isprimitiveif they do not span a cone in Σ but every proper subset does. Let rmin(Σ) denote the positive integer
(0.2) rmin(Σ) = min{
k ∈Z≥1 :{ni1,· · · ,nik} is primitive} .
∗Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, Warsaw, Poland.
†Department of Mathematics, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan.
1In this talk, a simplicial complexKalways means an abstract simplicial complex and we assume that it always contains the empty set∅.
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Assumption (∗). From now on, we assume that the following condition is satisfied:
(∗) The set Σ(1) = {n1,· · · ,nr} spans Rn, and let D = (d1,· · · , dr) be an r-tuple of positive integers such that∑r
k=1dknk =0.
Then we can identify XΣ =ZKΣ(C,C∗)/GΣ as in (0.1), and we denote by [a1,· · · , ar] the corresponding element of XΣ for each element (a1,· · · , ar)∈ ZKΣ(C,C∗).
Spaces of based holomorphic maps. Let Hdm ⊂ C[z0, . . . , zm] denote the subspace consisting of all homogeneous polynomials of degree d such that the coefficient of (z0)d is 1. Let Hol∗D(CPm, XΣ) denote the space consisting of r-tuples (f1,· · · , fr) ∈ Hmd1 ×
· · · × Hmdr, such that (f1(x),· · ·, fr(x)) ∈ ZKΣ(C,C∗) for any x ∈ Cm+1 \ {0}. Now we choose x0 = [1,1,· · ·,1] ∈ XΣ and [1 : 0 : · · · : 0] ∈ CPm as the base-points of XΣ and CPm, respectively. Then one can define the the natural map iD : Hol∗D(CPm, XΣ) → Map∗(CPm, XΣ) byiD(f1,· · · , fr)([x]) = [f1(x),· · · , fr(x)] for x ∈Cm+1\ {0}. Since the space Hol∗D(CPm, XΣ) is connected, the image of iD is contained in some path-component of Map∗(CPm, XΣ), which is denoted by Map∗D(CPm, XΣ). Thus we obtain the natural inclusion map
(0.3) iD : Hol∗D(CPm, XΣ)→Map∗D(CPm, XΣ).
Now recall the following recent result due to Mostovoy-Villanueva.
Theorem 0.1 (Mostovoy-Villanueva). Let Σ be a fan in Rn, {dk : 1 ≤ k ≤ r} the set of positive integers satisfying the condition (∗), and XΣ be a smooth compact toric variety associated to Σ. If 1 ≤ m ≤ 2rmin(Σ) − 2 and D = (d1,· · · , dr), the inclusion map iD : Hol∗D(CPm, XΣ) → Map∗D(CPm, XΣ) is a homology equivalence through dimension N(d1,· · · , dr;m,Σ), where N(d1,· · · , dr;m,Σ) := (2rmin(Σ)−2m−1) min{d1,· · ·, dr} − 2.
Conjecture 0.2. Let Σ be a fan in Rn, let {dk: 1≤k≤r} be the set of positive integers satisfying the condition (∗), andXΣ be a smooth non-compact toric variety associated toΣ.
If 1≤m ≤2rmin(Σ)−2 and D = (d1,· · ·, dr), the inclusion map iD : Hol∗D(CPm, XΣ) → Map∗D(CPm, XΣ) is a homology equivalence through dimension N(d1,· · · , dr;m,Σ)?
In this talk, first we shall recall several basic facts concerning the toric topology and the Aitiyah-Jones-Segal type problem. Next, we shall consider the above conjecture 0.2 for certain non-compact toric varietyXI, and show that the conjecture 0.2 is true forXΣ =XI when m= 1.
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