Rational hom*otopy type of projectivization of the tangent bundle of certain spaces (2024)

Abstract

Purpose

The paper aims to determine the rational hom*otopy 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 hom*otopy 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 hom*ogeneous space of P(E) when n=2. Finally, we prove the formality of P(E) over a hom*ogeneous 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 hom*otopy 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 hom*otopy type of projectivized bundles over complex projective spaces, offering valuable insights into the algebraic structure of these spaces and their associated complex vector bundles.

Keywords

Projectivization bundle
Sullivan minimal models
Formality
Complex manifolds
Primary 55P62

Citation

Gastinzi, J.B. and Ndlovu, M. (2024), "Rational hom*otopy 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

Publisher

Emerald Publishing Limited

License

Published in the Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode

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:

$A^{p} \otimes A^{q} \to A^{p + q}, x \otimes y \to x y,$

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

$x y = {(- 1)}^{| x | | y |} y x,$

where, x, y are hom*ogeneous 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,

$d (x y) = (d x) y + {(- 1)}^{| x |} x (d y) .$

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 $A_{P L} (X, Q)$ of polynomial forms on X, with natural cochain algebra quasi-isomorphisms

$C^{*} (X, Q) \overset{≅}{\to} D (X) \overset{≅}{\leftarrow} A_{P L} (X, Q),$

where D(X) is a third natural cochain algebra [2, §10]. Moreover, $A_{P L} (X, Q)$ 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

$V = ⋃_{k = 0}^{\infty} V (k)$

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 $H^{0} (A) = Q$ , 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 $ε_{i} : Λ (t, d t) \to Q$ , where ɛ₀(t)=0 and ɛ₁(t)=1. Two cdga maps ϕ₀, ϕ₁: (ΛV, d) → (B, d) are hom*otopic (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 hom*otopy, such that m_B◦g≃φ◦m_A is called the Sullivan minimal model of φ.

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

$π : ℂ^{n} \to E \to B,$

be a complex vector bundle over a complex smooth manifold B, the projectivized bundle P(E) is constructed by replacing each fibre

C^{n}

of E with the corresponding projective space

C P^{n - 1}

[5, §20]. The operation of projectivization applied to the bundle π yields a fibre bundle,

(1) $P (π) : C P^{n - 1} \to P (E) \to B .$

The cohom*ology algebra of the total space P(E) is given by

$H^{*} (P (E)) = H^{*} (B) [x] / (x^{n} + c_{1} (E) x^{n - 1} + \dots + c_{n - 1} (E) x + c_{n} (E)),$

where

c_{i} \in H (B, Q)

are the Chern classes on the complex bundle π and x is a generator of

H^{*} (C P^{n - 1}; Q)

[1, p. 333, 4, §4, 6, p. 1963]. According to the Leray-Hirsch theorem,

H^{*} (P (E)) ≅ H^{*} (B) \otimes H^{*} (C P^{n - 1})

as vector spaces [5].

Theorem 2.1.

Let $π : C^{n} \to E \to B$ 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), $D x_{2 n - 1} = x_{2}^{n} + \sum_{i = 1}^{n} c_{i} x_{2}^{n - i}$ and $c_{i} \in H^{2 i} (A, Q)$ 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

$λ \mapsto (\begin{matrix} λ & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & \dots & λ \end{matrix}), λ \in S^{1} .$

Therefore, P(π) is classified by a map

$f : B \to B P U (n) .$

As BPU(n) has the rational hom*otopy 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,

$Φ : (A, d) \to (A \otimes (Λ (x_{2}, x_{2 n - 1}), D) \to (Λ (x_{2 n}, x_{2 n - 1}), d),$

with

$D x_{2 n - 1} = x_{2}^{n} + \sum_{i = 1}^{n} c_{i} x_{2}^{n - i} .$

This agrees with the rational hom*otopy type of the rational universal fibration of fibre

C P^{n - 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 $C P^{n}$ and the projectivization fibration over $C P^{n}$ ,

$S (E) \overset{ζ}{\to} P (E),$

then $π_{i} (ζ) \otimes Q$ yields $π_{i} (P (E)) \otimes Q ≃ π_{i} (S (E)) \otimes Q$ , for i≥3.

Proof: If $C^{n} \to E \to C P^{n}$ is a tanget bundle and y₂ a generater of $H^{2} (C P^{n}, Q)$ , then the Chern classes are given by $y_{2}^{i} \in H^{2 i} (C P^{n}, Q)$ , 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, $d y_{2 n + 1} = y_{2}^{n + 1}$ , and $d x_{2 n - 1} = x_{2}^{n} + \dots + x_{2}^{2} y_{2}^{n - 2} + y_{2}^{n}$ . 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, $d y_{2 n + 1} = y_{2}^{n + 1}$ and $d v_{2 n - 1} = y_{2}^{n}$ . The Sullivan model of ζ is

$q : (Λ (y_{2}, y_{2 n + 1}) \otimes Λ (x_{2}, x_{2 n - 1}), d) \to (Λ (y_{2}, y_{2 n + 1}) \otimes Λ v_{2 n - 1}, D),$

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

The dual hom*otopy groups generated by ${[π_{*} (P (E)) \otimes Q]}^{#} ≅ 〈 y_{2}, y_{2 n + 1}, x_{2}, x_{2 n - 1} 〉$ and ${[π_{*} (S (E)) \otimes Q]}^{#} ≅ 〈 y_{2}, y_{2 n + 1}, v_{2 n - 1} 〉$ . The restriction of q to the algebra generators yields

$\bar{q} : 〈 y_{2}, y_{2 n + 1}, x_{2}, x_{2 n - 1} 〉 \to 〈 y_{2}, y_{2 n + 1}, v_{2 n - 1} 〉$

where

\bar{q} (y_{2}) = y_{2}

\bar{q} (y_{2 n + 1}) = y_{2 n + 1}

\bar{q} (x_{2}) = 0

, and

\bar{q} (x_{2 n - 1}) = v_{2 n - 1}

. Hence,

\bar{q}

is surjective and hence

π_{*} (ζ) \otimes Q

is injective. Moreover,

\bar{q}

is an isomorphism in the degree greater than 2. Hence, for i≥3,

π_{i} (ζ) \otimes Q

is an isomorphism.

Theorem 3.2.

The rational hom*otopy type of the total space P(E) of the projectivized bundle

$C P^{1} \to P (E) \to C P^{2},$

is that of the hom*ogeneous space U(3)/U(1) × U(1) × U(1).

Proof: Consider the projectivization fibration $C P^{1} \to P (E) \to C P^{2}$ . The Sullivan model of the total space P(E) is given by

$(Λ (y_{2}, y_{5}) \otimes Λ (x_{2}, x_{3}), d),$

with dy₂=dx₂=0,

d x_{3} = x_{2}^{2} + x_{2} y_{2} + y_{2}^{2}

and

d y_{5} = y_{2}^{3}

which is quasi isomorphic to

((Λ y_{2} / y_{2}^{3}) \otimes Λ (x_{2}, x_{3}); \bar{d}), \bar{d} y_{2} = \bar{d} x_{2} = 0 and \bar{d} x_{3} = x_{2}^{2} + x_{2} y_{2} + y_{2}^{2}

. The cohom*ology ring is

$H^{*} (P (E), Q) = (Λ y_{2} / y_{2}^{3}) \otimes (Λ x_{2} / (x_{2}^{2} + x_{2} y_{2} + y_{2}^{2})) .$

Now, consider the hom*ogeneous 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 hom*otopy pullback of the following diagram:

A Sullivan model for G/H is

$(Λ (a_{2}, b_{2}, c_{2}) \otimes Λ (w_{1}, w_{3}, w_{5}), d),$

where

$\begin{matrix} d w_{3} & = & a_{2} b_{2} + b_{2} c_{2} + a_{2} c_{2}, \\ d w_{5} & = & a_{2} b_{2} c_{2}, \\ d w_{1} & = & a_{2} + b_{2} + c_{2} . \end{matrix}$

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

$(Λ (a_{2}, b_{2}, w_{3}, w_{5}), d),$

where

$\begin{matrix} d a_{2} & = & d b_{2} = 0, \\ d w_{3} & = & - a_{2}^{2} - a_{2} b_{2} - b_{2}^{2}, \\ d w_{5} & = & - a_{2}^{2} b_{2} - a_{2} b_{2}^{2} . \end{matrix}$

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 hom*otopy type of U(3)/U(1) × U(1) × U(1).

Proposition 3.3.

Consider a non-trivial complex vector bundle $δ : C^{n} \to E \to S^{2 n}$ and the projectivization of vector bundle

$P (δ) : C P^{n - 1} \to P (E) \overset{π}{\to} S^{2 n} .$

Then P(E) has a rational hom*otopy type of $C P^{2 n - 1}$ , for n≥2.

Proof:

The Sullivan minimal model of S²ⁿ is (Λ(a_2n, b_4n−1), d) with da₂=0, and $d b_{4 n - 1} = a_{2 n}^{2}$ . Again, the Sullivan minimal model of $C P^{n - 1}$ is Λ(x₂, x_2n−1) with dx₂=0, and $d x_{2 n - 1} = x_{2}^{n}$ . Therefore, the KS model of π;

$(Λ (a_{2 n}, b_{4 n - 1}), d) \to (Λ (a_{2 n}, b_{4 n - 1}) \otimes Λ (x_{2}, x_{2 n - 1}), D) \to (Λ (x_{2}, x_{2 n - 1}), d),$

is classified by a mapping,

$f : (Λ (y_{4}, \dots, y_{2 n}), 0) \to (Λ (a_{2 n}, b_{4 n - 1}), d) .$

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,

$(Λ (a_{2 n}, b_{4 n - 1}, x_{2}, x_{2 n - 1}), D),$

$D a_{2 n} = 0, D x_{2} = 0, D b_{4 n - 1} = a_{2 n}^{2}, D x_{2 n - 1} = x_{2}^{n} + a_{2 n} .$

Making the change of variables to eliminate the linear part, let $t_{2 n} = x_{2}^{n} + a_{2 n}$ . Then we get an isomorphic cdga

$(Λ (t_{2 n}, b_{4 n - 1}, x_{2}, x_{2 n - 1}), D),$

$D a_{2 n} = 0, D x_{2} = 0, D b_{4 n - 1} = {(t_{2 n} - x_{2}^{n})}^{2}, D x_{2 n - 1} = t_{2 n} .$

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

$(Λ (x_{2}, b_{4 n - 1}), D),$

$D x_{2} = 0, D b_{4 n - 1} = x_{2}^{2 n},$

Hence P(E), has a rational hom*otopy type of $C P^{2 n - 1}$ .

Remark: As $P (E) ≅_{Q} C P^{2 n - 1}$ , 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

$C P^{n - 1} \to E \to M,$

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 hom*ogenous space of equal rank with a complex structure of dimension n, and $C^{n} \to E \to B$ is the complex vector bundle, then in the projectivization bundle

$C P^{n - 1} \to P (E) \to B,$

have a formal total space P(E).

Proof:

Let B be a hom*ogeneous space of equal rank, implying the existence of a pure model

$(Λ (y_{1}, \dots, y_{m}, v_{1}, \dots, v_{m}), d),$

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

$(Λ (y_{1}, \dots, y_{m}, x_{2}, v_{1}, \dots, v_{m}, x_{2 n - 1}), d),$

where

$d y_{1} = \dots = d y_{m} = d x_{2} = 0 and d x_{2 n - 1} = x_{2}^{n} + \sum_{i = 1}^{n} c_{i} x_{2}^{n - i}, [c_{i}] = H^{2 i} (B) .$

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

$d x_{2 n - 1} = x_{2}^{n} + \sum_{i = 1}^{n} c_{i} x_{2}^{n - i}$

is not a zero divisor in

(Λ (y_{1}, \dots, y_{m}) / (u_{1}, \dots, u_{m})) \otimes Λ x_{2})

. By the contrary, assume that

$x_{2}^{n} + \sum_{i = 1}^{n} c_{i} x_{2}^{n - i}$

is a zero divisor. Then there exists

$β_{0} + \sum_{k = 1}^{m} β_{k} x_{2}^{k},$

such that

(2) $(β_{0} + \sum_{k = 1}^{m} β_{k} x_{2}^{k}) (x_{2}^{n} + \sum_{i = 1}^{n} c_{i} x_{2}^{n - i}) = 0,$

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

$β_{m} x_{2}^{m + n} + polynomial of degree < m + n in x_{2} .$

This cannot be a zero divisor in

(Λ (y_{1}, \dots, y_{m}) / (u_{1}, \dots, u_{m}))

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Acknowledgements

This paper is based on a part of Ndlovu’s Ph.D. thesis carried out under Gatsinzi’s supervision. We leveraged language editing by utilizing ChatGPT.

Corresponding author

Meshach Ndlovu can be contacted at: nm21100072@studentmail.biust.ac.bw