Rational homotopy type of projectivization of the tangent bundle of certain spaces

Abstract

Purpose

The paper aims to determine the rational homotopy type of the total space of projectivized bundles over complex projective spaces using Sullivan minimal models, providing insights into the algebraic structure of these spaces.

Design/methodology/approach

The paper utilises techniques from Sullivan’s theory of minimal models to analyse the differential graded algebraic structure of projectivized bundles. It employs algebraic methods to compute the Sullivan minimal model of P(E) and establish relationships with the base space.

Findings

The paper determines the rational homotopy type of projectivized bundles over complex projective spaces. Of great interest is how the Chern classes of the fibre space and base space, play a critical role in determining the Sullivan model of P(E). We also provide the homogeneous space of P(E) when n = 2. Finally, we prove the formality of P(E) over a homogeneous space of equal rank.

Research limitations/implications

Limitations may include the complexity of computing minimal models for higher-dimensional bundles.

Practical implications

Understanding the rational homotopy type of projectivized bundles facilitates computations in algebraic topology and differential geometry, potentially aiding in applications such as topological data analysis and geometric modelling.

Social implications

While the direct social impact may be indirect, advancements in algebraic topology contribute to broader mathematical knowledge, which can underpin developments in science, engineering, and technology with societal benefits.

Originality/value

The paper’s originality lies in its application of Sullivan minimal models to determine the rational homotopy type of projectivized bundles over complex projective spaces, offering valuable insights into the algebraic structure of these spaces and their associated complex vector bundles.

Citation

Gastinzi, J.B. and Ndlovu, M. (2024), "Rational homotopy type of projectivization of the tangent bundle of certain spaces", Arab Journal of Mathematical Sciences, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/AJMS-02-2024-0029

1. Introduction

This section will outline the fundamental concepts and definitions of differential graded algebras. We consider a setting where all algebras and vector spaces are taken over the field Q of rational numbers. The primary reference for the definitions in this paper is [1].

Definition 1.1.

A graded algebra is a graded vector space A = ⊕_p ≥ 0A^p together with an associative multiplication of degree zero:

Ap⊗Aq→Ap+q,x⊗y→xy,

such that there is an identity 1 ∈ A⁰. A graded algebra is called commutative (cga) if

xy=(−1)|x||y|yx,

where, x, y are homogeneous elements of degree |x| and |y| respectively. A differential in a graded algebra A is a linear map d: A^p → A^p+1 satisfying d◦d = 0,

d(xy)=(dx)y+(−1)|x|x(dy).

The pair (A, d) is a commutative differential graded algebra (cdga), if A is commutative.

If V is a graded vector space, then the free commutative graded algebra ΛV is defined by ΛV = S(V^even) ⊗ E(V^odd), where S(V^even) is the symmetric algebra and E(V^odd) is the exterior algebra. If {v₁, v₂, ⋯ } is a basis of V, then ΛV is often written as Λ(v₁, v₂, ⋯).

Definition 1.2.

Let C*(X,Q) be the cochain algebra of the normalized singular cochains on a topological space X. Sullivan defines a cdga APL(X,Q) of polynomial forms on X, with natural cochain algebra quasi-isomorphisms

C*(X,Q)→≅D(X)←≅APL(X,Q),

where D(X) is a third natural cochain algebra [2, §10]. Moreover, APL(X,Q) is a contravariant functor from the category of topological spaces to a category of cdgas.

Definition 1.3.

A commutative cochain algebra (ΛV, d) is called a Sullivan algebra if

V=⋃k=0∞V(k)

such that, d = 0 in V(0) and d: V(k) = ΛV(k − 1), k ≥ 1. Moreover, a Sullivan algebra (ΛV, d) is called minimal if dV ⊂ Λ^≥2V. Let (A, d) be a cdga with H0(A)=Q, there always exists a quasi-isomorphism m: (ΛV, d) → (A, d), where (ΛV, d) is a Sullivan algebra. A cdga map φ: (A, d) → (B, d) is a quasi-isomorphism if H*(φ) is an isomorphism.

Definition 1.4.

Consider the cdga Λ(t, dt), where |t| = 0, |dt| = 1 and d(t) = dt. There are augmentation maps εi:Λ(t,dt)→Q, where ɛ₀(t) = 0 and ɛ₁(t) = 1. Two cdga maps ϕ₀, ϕ₁: (ΛV, d) → (B, d) are homotopic (i.e. ϕ₀ ≃ ϕ₁) if there exists a cdga map Φ: (ΛV, d) → B ⊗Λ(t, dt) such that (1 ⊗ ɛ_i)◦Φ = ϕ_i [2, §12].

Definition 1.5.

[1, §2] Let X be a path-connected space. The Sullivan minimal model of X is the Sullivan minimal model of A_PL(X). If f: X → Y is a map between path-connected spaces, the minimal model of A_PL(f) is called the Sullivan minimal model of f.

Let φ: (A, d) → (B, d) be a map of cdga’s, and m_A: (ΛV, d) → (A, d) and m_B: (ΛW, d) → (B, d) be the Sullivan models. Then there exists a morphism of cdga’s g: (ΛV, d) → (ΛW, d) that is unique up to homotopy, such that m_B◦g ≃ φ◦m_A is called the Sullivan minimal model of φ.

Definition 1.6.

A Sullivan minimal algebra (ΛV, d) is said to be formal if there exists a homomorphism ϕ: (ΛV, d) → H *(ΛV, d) inducing an isomorphism on cohomology. A space X is said to be formal if its minimal model is formal.

Definition 1.7.

A relative Sullivan model of a morphism of commutative differential graded algebras φ: (A, d) → (B, d) is a morphism (A,d)→i(A⊗ΛV,d) and V = ∪_k ≥ 0V(k), where V(0) ⊂ V(1) ⊂ ⋯ is an increasing sequence with d(V(0)) ⊆ A and dV(k) ⊆ A ⊗ V(k − 1), k ≥ 1 and there is a quasi isomorphism ψ: (A ⊗ΛV, d) → (B, d) such that ψ◦i = φ.

Let F→E→ξB be a fibration between simply connected spaces, then there exists a relative Sullivan model (ΛV, d) → (ΛV ⊗ΛW, D) → (ΛW, d), where (ΛV, d) and (ΛW, d) are respective Sullivan models of B and F. Moreover, (ΛV ⊗ΛW, D) is a Sullivan model of E, not necessarily minimal [3, §12].

Definition 1.8.

Let Q be a finite-dimensional graded vector space concentrated in even degrees. A regular sequence is defined as an ordered set of elements u₁, …, u_m belonging to Λ⁺Q such that u₁ is not a zero divisor in ΛQ, and for i ≥ 2, then the image of u_i is likewise not a zero divisor in the quotient graded algebra ΛQ/(u₁, …, u_i−1). In particular, any given sequence of the form u₁, …, u_m can be used to define a pure Sullivan algebra denoted as (ΛQ ⊗ΛP, d), for ΛP = Λ(x₁, …, x_m), where the differential operator is defined by dx_i = u_i [2, p. 437, 4 p. 157].

Definition 1.9.

[1, p. 188] A closed manifold (M²ⁿ, ω) is cohomologically symplectic (or c-symplectic) if there is ω∈H2(M;Q) such that ωⁿ ≠ 0.

2. Model of the projectivization of a complex bundle

A projectivized bundle is constructed by replacing each fibre of a complex vecto bundle with the corresponding projective space. Specifically, let

π:ℂn→E→B,

be a complex vector bundle over a complex smooth manifold B, the projectivized bundle P(E) is constructed by replacing each fibre Cn of E with the corresponding projective space CPn−1 [5, §20]. The operation of projectivization applied to the bundle π yields a fibre bundle,

(1)P(π):CPn−1→P(E)→B.

The cohomology algebra of the total space P(E) is given by

H*(P(E))=H*(B)[x]/xn+c1(E)xn−1+⋯+cn−1(E)x+cn(E),

where ci∈H(B,Q) are the Chern classes on the complex bundle π and x is a generator of H*(CPn−1;Q) [1, p. 333, 4, §4, 6, p. 1963]. According to the Leray-Hirsch theorem, H*(P(E))≅H*(B)⊗H*(CPn−1) as vector spaces [5].

Theorem 2.1.

Let π:Cn→E→B be a complex fibre bundle and P(π) its projectivization. If (A, d) is a cdga model of B, then a model of the total space of P(π) is given by (A ⊗ Λ(x_2n, x_2n−1), D), Dx2n−1=x2n+∑i=1ncix2n−i and ci∈H2i(A,Q) represent the Chern classes of π.

Proof

If B is a complex manifold and π corresponds to the tangent bundle, the structure group of the complex vector bundle can be reduced to U(n). Moreover, the structural group of P(π) reduces to U(n)/S¹ ≅ PU(n), where S¹ is considered as a subgroup of U(n) under the identification

λ↦λ⋯0⋮⋱⋮0⋯λ,λ∈S1.

Therefore, P(π) is classified by a map

f:B→BPU(n).

As BPU(n) has the rational homotopy type SU(n), a Sullivan model of BPU(n) is given by (Λ(y₄, ⋯y_2n), 0), and a model of f is ϕ: (Λ(y₄, ⋯y_2n), 0) → (A, d) with Chern classes [c_i] = ϕ(y_2i) ∈ H²ⁱ(A, d), for i = {1, 2, …, n}. A relative model of projectivization is then given by,

Φ:(A,d)→(A⊗(Λ(x2,x2n−1),D)→(Λ(x2n,x2n−1),d),

with

Dx2n−1=x2n+∑i=1ncix2n−i.

This agrees with the rational homotopy type of the rational universal fibration of fibre CPn−1 [7, 8, §11].

3. Projectivization and tangent sphere bundles over complex manifolds

For a complex vector bundle π: E → B²ⁿ let the unit tangent sphere bundle be denoted by S²ⁿ⁻¹ → S(E) → B²ⁿ. Also, the complex structure on E implies that the circle S¹ acts on the sphere bundle. Therefore, there exists a bundle map ζ: S(E) → P(E).

A model for the unit sphere bundle is given by (A ⊗ Λv_2n−1, D), where D_|A = d, Dv_2n−1 = w, where w is a cocycle that represents the fundamental class of B [9]. The following diagram of cdga’s commutes:

where, q(x₂) = 0 and q(x_2n−1) = v_2n−1.

Theorem 3.1.

Consider the fibration of the unit tangent sphere bundle over a complex projective space CPn and the projectivization fibration over CPn,

S(E)→ζP(E),

then πi(ζ)⊗Q yields πi(P(E))⊗Q≃πi(S(E))⊗Q, for i ≥ 3.

Proof: If Cn→E→CPn is a tanget bundle and y₂ a generater of H2(CPn,Q), then the Chern classes are given by y2i∈H2i(CPn,Q), i = 1, 2, …, n up to a non-zero rational factor (see [4, §21]). Given that the Sullivan model of P(E) is (Λ(y₂, y_2n+1) ⊗ Λ(x₂, x_2n−1), d) with dy₂ = dx₂ = 0, dy2n+1=y2n+1, and dx2n−1=x2n+⋯+x22y2n−2+y2n. Also a model of the sphere bundle S(E) is given by the Sullivan model (Λ(y₂, y_2n+1) ⊗ Λv_2n−1, d), with dy₂ = 0, dy2n+1=y2n+1 and dv2n−1=y2n. The Sullivan model of ζ is

q:(Λ(y2,y2n+1)⊗Λ(x2,x2n−1),d)→(Λ(y2,y2n+1)⊗Λv2n−1,D),

where q(x₂) = 0, q(x_2n−1) = v_2n−1 (see Ref. [10]).

The dual homotopy groups generated by π*(P(E))⊗Q#≅〈y2,y2n+1,x2,x2n−1〉 and π*(S(E))⊗Q#≅〈y2,y2n+1,v2n−1〉. The restriction of q to the algebra generators yields

q¯:〈y2,y2n+1,x2,x2n−1〉→〈y2,y2n+1,v2n−1〉

where q¯(y2)=y2, q¯(y2n+1)=y2n+1, q¯(x2)=0, and q¯(x2n−1)=v2n−1. Hence, q¯ is surjective and hence π*(ζ)⊗Q is injective. Moreover, q¯ is an isomorphism in the degree greater than 2. Hence, for i ≥ 3, πi(ζ)⊗Q is an isomorphism.

Theorem 3.2.

The rational homotopy type of the total space P(E) of the projectivized bundle

CP1→P(E)→CP2,

is that of the homogeneous space U(3)/U(1) × U(1) × U(1).

Proof: Consider the projectivization fibration CP1→P(E)→CP2. The Sullivan model of the total space P(E) is given by

Λy2,y5⊗Λx2,x3,d,

with dy₂ = dx₂ = 0, dx3=x22+x2y2+y22 and dy5=y23 which is quasi isomorphic to ((Λy2/y23)⊗Λ(x2,x3);d¯),d¯y2=d¯x2=0andd¯x3=x22+x2y2+y22. The cohomology ring is

H*(P(E),Q)=(Λy2/y23)⊗Λx2/(x22+x2y2+y22).

Now, consider the homogeneous space G/H where G = U(3) and H = U(1) × U(1) × U(1).

Let j: H = U(1) × U(1) × U(1) ↪ G = U(3) the inclusion, and B_j: BH → BG the classifying map. Then G/H is the homotopy pullback of the following diagram:

A Sullivan model for G/H is

Λa2,b2,c2⊗Λw1,w3,w5,d,

where

dw3=a2b2+b2c2+a2c2,dw5=a2b2c2,dw1=a2+b2+c2.

A change of variable t₂ = a₂ + b₂ + c₂ yields an isomorphism (Λ(a₂, b₂, t₂) ⊗ Λ(w₁, w₃, w₅), d) with dw₁ = t₂, dw₃ = a₂b₂ + (a₂ + b₂)(t₂ − a₂ − b₂) and dw₅ = a₂b₂(t₂ − a₂ − b₂) which is quasi-isomorphic to

(Λ(a2,b2,w3,w5),d),

where

da2=db2=0,dw3=−a22−a2b2−b22,dw5=−a22b2−a2b22.

The map f: (Λ(a₂, b₂, w₃, w₅), d) → (Λ(x₂, y₂, x₃, y₅), d), with f(a₂) = x₂, f(w₃) = x₃,

f(b₂) = y₂, f(w₅) = y₂x₃ − y₅ is a quasi-isomorphism. Therefore, P(E) has the rational homotopy type of U(3)/U(1) × U(1) × U(1).

Proposition 3.3.

Consider a non-trivial complex vector bundle δ:Cn→E→S2n and the projectivization of vector bundle

P(δ):CPn−1→P(E)→πS2n.

Then P(E) has a rational homotopy type of CP2n−1, for n ≥ 2.

Proof:

The Sullivan minimal model of S²ⁿ is (Λ(a_2n, b_4n−1), d) with da₂ = 0, and db4n−1=a2n2. Again, the Sullivan minimal model of CPn−1 is Λ(x₂, x_2n−1) with dx₂ = 0, and dx2n−1=x2n. Therefore, the KS model of π;

Λa2n,b4n-1,d→Λa2n,b4n-1⊗Λx2,x2n-1,D→Λx2,x2n-1,d,

is classified by a mapping,

f:(Λ(y4,…,y2n),0)→(Λ(a2n,b4n−1),d).

The Chern classes given by c₁ = [f(y₂)] = 0, c₂ = [f(y₄)] = 0, c₃ = [f(y₆)] = 0, ⋯ , c_n−1 = [f(y_2(n−1))] = 0, c_n = [f(y_2n)] = [a_2n] ∈ H*(S²ⁿ).

Then, the total space P(E) of the projectivized bundle has a Sullivan model,

(Λ(a2n,b4n−1,x2,x2n−1),D),

Da2n=0,Dx2=0,Db4n−1=a2n2,Dx2n−1=x2n+a2n.

Making the change of variables to eliminate the linear part, let t2n=x2n+a2n. Then we get an isomorphic cdga

Λt2n,b4n-1,x2,x2n-1,D,

Da2n=0,Dx2=0,Db4n−1=(t2n−x2n)2,Dx2n−1=t2n.

As the ideal (x_2n−1, t_2n) is acyclic, the minimal Sullivan model of P(E) is,

(Λ(x2,b4n−1),D),

Dx2=0,Db4n−1=x22n,

Hence P(E), has a rational homotopy type of CP2n−1.

Remark: As P(E)≅QCP2n−1, then P(E) satisfies the hard Lefschetz property (See Theorem 3.1 in Ref. [6]).

Theorem 3.4.

([4, p. 149, 6], Theorem 4.1). Let M be a simply connected smooth manifold of dimension 2n. Given a fibration

CPn−1→E→M,

then E is formal if and only if M is formal.

The proof of this Theorem 3.4 is given in Refs. [4, 11]. We give here a simple proof of a particular case of this theorem.

Theorem 3.5.

If B is a homogenous space of equal rank with a complex structure of dimension n, and Cn→E→B is the complex vector bundle, then in the projectivization bundle

CPn−1→P(E)→B,

have a formal total space P(E).

Proof:

Let B be a homogeneous space of equal rank, implying the existence of a pure model

(Λ(y1,…,ym,v1,…,vm),d),

where y_i and v_i are even and odd generators respectively, dy_i = 0, and dv_i ⊆ Λ(y₁, …, y_m). Moreover, (dv₁, …, dv_m) is a regular sequence in Λ(y₁, …, y_m). Hence B is formal. A model for the total space of the projectivized bundle is given by

(Λ(y1,…,ym,x2,v1,…,vm,x2n−1),d),

where

dy1=⋯=dym=dx2=0anddx2n−1=x2n+∑i=1ncix2n−i,ci=H2iB.

Let u_i = dv_i we show that (u₁, …, u_m, dx_2n−1) forms a regular sequence in Λ(y₁, …, y_m, x₂). It suffices to show that

dx2n−1=x2n+∑i=1ncix2n−i

is not a zero divisor in Λ(y1,…,ym)/(u1,…,um)⊗Λx2). By the contrary, assume that

x2n+∑i=1ncix2n−i

is a zero divisor. Then there exists

β0+∑k=1mβkx2k,

such that

(2)β0+∑k=1mβkx2kx2n+∑i=1ncix2n−i=0,

where β_i ∈ Λ(y₁, …, y_m)/(u₁, …, u_m). The expansion of equation (2) yields

βmx2m+n+polynomial of degree<m+ninx2.

This cannot be a zero divisor in Λ(y1,…,ym)/(u1,…,um).