## 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

### 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.

## Keywords

## 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

## Publisher

:Emerald Publishing Limited

Copyright © 2024, Jean Baptiste Gastinzi and Meshach Ndlovu

## 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

**Definition 1.1.**

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

*A*

^{0}. A graded algebra is called commutative (cga) if

*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,

*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 Λ*V* is defined by Λ*V* = *S*(*V*^{even}) ⊗ *E*(*V*^{odd}), where *S*(*V*^{even}) is the symmetric algebra and *E*(*V*^{odd}) is the exterior algebra. If {*v*_{1}, *v*_{2}, ⋯ } is a basis of *V*, then Λ*V* is often written as Λ(*v*_{1}, *v*_{2}, ⋯).

**Definition 1.2.**

Let *X*. Sullivan defines a cdga *X*, with natural cochain algebra quasi-isomorphisms

*D*(

*X*) is a third natural cochain algebra [2, §10]. Moreover,

**Definition 1.3.**

A commutative cochain algebra (Λ*V*, *d*) is called a Sullivan algebra if

*d*= 0 in

*V*(0) and

*d*:

*V*(

*k*) = Λ

*V*(

*k*− 1),

*k*≥ 1. Moreover, a Sullivan algebra (Λ

*V*,

*d*) is called minimal if

*dV*⊂ Λ

^{≥2}

*V*. Let (

*A*,

*d*) be a cdga with

*m*: (Λ

*V*,

*d*) → (

*A*,

*d*), where (Λ

*V*,

*d*) is a Sullivan algebra. A cdga map

*φ*: (

*A*,

*d*) → (

*B*,

*d*) is a quasi-isomorphism if

**Definition 1.4.**

Consider the cdga Λ(*t*, *dt*), where |*t*| = 0, |*dt*| = 1 and *d*(*t*) = *dt*. There are augmentation maps *ɛ*_{0}(*t*) = 0 and *ɛ*_{1}(*t*) = 1. Two cdga maps *ϕ*_{0}, *ϕ*_{1}: (Λ*V*, *d*) → (*B*, *d*) are homotopic (i.e. *ϕ*_{0} ≃ *ϕ*_{1}) if there exists a cdga map Φ: (Λ*V*, *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}: (Λ*V*, *d*) → (*A*, *d*) and *m*_{B}: (Λ*W*, *d*) → (*B*, *d*) be the Sullivan models. Then there exists a morphism of cdga’s *g*: (Λ*V*, *d*) → (Λ*W*, *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 (Λ*V*, *d*) is said to be formal if there exists a homomorphism *ϕ*: (Λ*V*, *d*) → *H* *(Λ*V*, *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 *V* = ∪_{k ≥ 0}*V*(*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* ⊗Λ*V*, *d*) → (*B*, *d*) such that *ψ*◦*i* = *φ*.

Let *V*, *d*) → (Λ*V* ⊗Λ*W*, *D*) → (Λ*W*, *d*), where (Λ*V*, *d*) and (Λ*W*, *d*) are respective Sullivan models of *B* and *F*. Moreover, (Λ*V* ⊗Λ*W*, *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 *u*_{1}, …, *u*_{m} belonging to Λ^{+}*Q* such that *u*_{1} is not a zero divisor in Λ*Q*, and for *i* ≥ 2, then the image of *u*_{i} is likewise not a zero divisor in the quotient graded algebra Λ*Q*/(*u*_{1}, …, *u*_{i−1}). In particular, any given sequence of the form *u*_{1}, …, *u*_{m} can be used to define a pure Sullivan algebra denoted as (Λ*Q* ⊗Λ*P*, *d*), for Λ*P* = Λ(*x*_{1}, …, *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 (*M*^{2n}, *ω*) is cohomologically symplectic (or c-symplectic) if there is *ω*^{n} ≠ 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

*B*, the projectivized bundle

*P*(

*E*) is constructed by replacing each fibre

*E*with the corresponding projective space

*π*yields a fibre bundle,

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

*π*and

*x*is a generator of

*Let *

*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*)/*S*^{1} *≅ P*U(*n*), where *S*^{1} is considered as a subgroup of U(*n*) under the identification

*P*(

*π*) is classified by a map

As *BP*U(*n*) has the rational homotopy type *S*U(*n*), a Sullivan model of *BP*U(*n*) is given by (Λ(*y*_{4}, ⋯*y*_{2n}), 0), and a model of *f* is *ϕ*: (Λ(*y*_{4}, ⋯*y*_{2n}), 0) → (*A*, *d*) with Chern classes [*c*_{i}] = *ϕ*(*y*_{2i}) ∈ *H*^{2i}(*A*, *d*), for *i* = {1, 2, …, *n*}. A relative model of projectivization is then given by,

with

## 3. Projectivization and tangent sphere bundles over complex manifolds

For a complex vector bundle *π*: *E* → *B*^{2n} let the unit tangent sphere bundle be denoted by *S*^{2n−1} → *S*(*E*) → *B*^{2n}. Also, the complex structure on *E* implies that the circle *S*^{1} 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* ⊗ Λ*v*_{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*(*x*_{2}) = 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 *,

*Proof:* If *y*_{2} a generater of *i* = 1, 2, …, *n* up to a non-zero rational factor (see [4, §21]). Given that the Sullivan model of *P*(*E*) is (Λ(*y*_{2}, *y*_{2n+1}) ⊗ Λ(*x*_{2}, *x*_{2n−1}), *d*) with *dy*_{2} = *dx*_{2} = 0, *S*(*E*) is given by the Sullivan model (Λ(*y*_{2}, *y*_{2n+1}) ⊗ Λ*v*_{2n−1}, *d*), with *dy*_{2} = 0, *ζ* is

*q*(

*x*

_{2}) = 0,

*q*(

*x*

_{2n−1}) =

*v*

_{2n−1}(see Ref. [10]).

The dual homotopy groups generated by *q* to the algebra generators yields

*i*≥ 3,

**Theorem 3.2.**

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

*Proof:* Consider the projectivization fibration *P*(*E*) is given by

*dy*

_{2}=

*dx*

_{2}= 0,

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

*t*

_{2}=

*a*

_{2}+

*b*

_{2}+

*c*

_{2}yields an isomorphism (Λ(

*a*

_{2},

*b*

_{2},

*t*

_{2}) ⊗ Λ(

*w*

_{1},

*w*

_{3},

*w*

_{5}),

*d*) with

*dw*

_{1}=

*t*

_{2},

*dw*

_{3}=

*a*

_{2}

*b*

_{2}+ (

*a*

_{2}+

*b*

_{2})(

*t*

_{2}−

*a*

_{2}−

*b*

_{2}) and

*dw*

_{5}=

*a*

_{2}

*b*

_{2}(

*t*

_{2}−

*a*

_{2}−

*b*

_{2}) which is quasi-isomorphic to

where

The map *f*: (Λ(*a*_{2}, *b*_{2}, *w*_{3}, *w*_{5}), *d*) → (Λ(*x*_{2}, *y*_{2}, *x*_{3}, *y*_{5}), *d*), with *f*(*a*_{2}) = *x*_{2}, *f*(*w*_{3}) = *x*_{3},

*f*(*b*_{2}) = *y*_{2}, *f*(*w*_{5}) = *y*_{2}*x*_{3} − *y*_{5} 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

*P*(

*E*) has a rational homotopy type of

*n*≥ 2.

*Proof:*

The Sullivan minimal model of *S*^{2n} is (Λ(*a*_{2n}, *b*_{4n−1}), *d*) with *da*_{2} = 0, and *x*_{2}, *x*_{2n−1}) with *dx*_{2} = 0, and *π*;

*c*

_{1}= [

*f*(

*y*

_{2})] = 0,

*c*

_{2}= [

*f*(

*y*

_{4})] = 0,

*c*

_{3}= [

*f*(

*y*

_{6})] = 0, ⋯ ,

*c*

_{n−1}= [

*f*(

*y*

_{2(n−1)})] = 0,

*c*

_{n}= [

*f*(

*y*

_{2n})] = [

*a*

_{2n}] ∈

*H**(

*S*

^{2n}).

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

Making the change of variables to eliminate the linear part, let

As the ideal (*x*_{2n−1}, *t*_{2n}) is acyclic, the minimal Sullivan model of *P*(*E*) is,

Hence *P*(*E*), has a rational homotopy type of

**Remark:** As *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* 2*n*. *Given a fibration*

*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* *is the complex vector bundle, then in the projectivization bundle*

*have a formal total space P(E)*.

*Proof:*

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

*y*

_{i}and

*v*

_{i}are even and odd generators respectively,

*dy*

_{i}= 0, and

*dv*

_{i}⊆ Λ(

*y*

_{1}, …,

*y*

_{m}). Moreover, (

*dv*

_{1}, …,

*dv*

_{m}) is a regular sequence in Λ(

*y*

_{1}, …,

*y*

_{m}). Hence

*B*is formal. A model for the total space of the projectivized bundle is given by

Let *u*_{i} = *dv*_{i} we show that (*u*_{1}, …, *u*_{m}, *dx*_{2n−1}) forms a regular sequence in Λ(*y*_{1}, …, *y*_{m}, *x*_{2}). It suffices to show that

*β*

_{i}∈ Λ(

*y*

_{1}, …,

*y*

_{m})/(

*u*

_{1}, …,

*u*

_{m}). The expansion of equation (2) yields

## References

1Félix Y, Oprea J, Tanré D. Algebraic models in geometry. In: Graduate texts in Mathematics. New York: Oxford University Press; 2008; 17.

2Félix Y, Halperin S, Thomas J-C. Rational homotopy theory. In: Graduate texts in Mathematics. Springer Science + Business Media; 2001; 205.

3Félix Y, Halperin S. Rational homotopy theory via sullivan models: a survey. arXiv preprint arXiv:1708.05245. 2017; 5(2): 14-36. doi: 10.4310/iccm.2017.v5.n2.a3.

4Tralle A, Oprea J. Symplectic manifolds with no Kähler structure. In: Lecture notes in Mathematics. New York: Springer-Verlag Berlin Heidelberg; 1997; 1661.

5Bott R, Tu LW. Differential forms in algebraic topology. In: Graduate texts in Mathematics. New York: Springer Science + Business Media; 1982; 82.

6Nishinobu H, Yamaguchi T. The Lefschetz condition on projectivizations of complex vector bundles. Commun Korean Math Soc. 2014; 29(4): 569-79. doi: 10.4134/ckms.2014.29.4.569.

7Gatsinzi JB. On the genus of elliptic fibrations. Proc Am Math Soc. 2004; 132(2): 597-606. doi: 10.1090/s0002-9939-03-07203-4.

8Sullivan D. Infinitesimal computations in topology. Publications Mathématiques de l’IHÉS. 1977; 47(1): 269-331. doi: 10.1007/bf02684341.

9Banyaga A, Gatsinzi JB, Massamba F. A note on the formality of some contact manifolds. J Geometry. 2018; 109: 1-10. doi: 10.1007/s00022-018-0409-3.

10Lambrechts P, Stanley D. Examples of rational homotopy types of blow-ups. Proc Am Math Soc. 2005; 133(12): 3713-19. doi: 10.1090/s0002-9939-05-07750-6.

11Lupton G, Oprea J. Symplectic manifolds and formality. J Pure Appl Algebra. 1994; 91(1-3): 193-207. doi: 10.1016/0022-4049(94)90142-2.

## 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.