# Derivations of a Sullivan Model and the RationalizedG-Sequence

Authors: Oteng Maphane
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

LetGk,nℂ for 2≤k<n denote the Grassmann manifold of k-dimensional vector subspaces of ℂn. In this paper, we compute, in terms of the Sullivan models, the rational evaluation subgroups and, more generally, the G-sequence of the inclusion G2,nℂ↣G2,n+rℂ for r≥1.


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## Body

## 1. Introduction

Throughout this paper, we rely on the theory of minimal Sullivan models in rational homotopy theory for which [1] is our standard reference. Let X be a based CW-complex; the nth Gottlieb group of X (or the nth evaluation subgroup [2] of πnX), denoted by GnX, consists of those elements a∈πnX for which there is a continuous map H:X×Sn⟶X such that the following diagram commutes:(1)where h:Sn⟶X is a representative of a and ▽ is the folding map. Let f:X⟶Y be a based map of simply connected finite CW-complexes. In [3], the evaluation at the base point of X gives the evaluation mapω:MapX,Y;f⟶Y, where MapX,Y;f is the component of f in the space of mappings from X to Y. The image of the homomorphism induced in homotopy groups(2)ω♯:π∗MapX,Y;f⟶π∗Yis called the nth evaluation subgroup of p, and it is denoted by GnY,X;f. Moreover, if f=idX, the space MapX,Y;f is the monoid aut1X of self-equivalences of X homotopic to the identity of X; then, ev:aut1X⟶X is the evaluation map, and the image of the induced homomorphism(3)ev♯:π∗aut1X⟶π∗Xis GnX, i.e., the nth Gottlieb group. Moreover, in [4], Woo and Lee studied the relative evaluation subgroups GnrelX,Y;p and proved that they fit in a sequence(4)⋯⟶Gn+1relX,Y;f⟶GnX⟶GnX,Y;f⟶⋯called the G-sequence of f. Finally, in [3], Smith and Lupton identify the homomorphism induced on rational homotopy groups by the evaluation map ω:MapX,Y;f⟶Y, in terms of a map of complexes of derivations constructed directly from the Sullivan minimal model of f. In [5], the authors use a map of complexes of derivations of minimal Sullivan models of mapping spaces to compute rational relative Gottlieb groups of the inclusion G2,nℂ↣G2,n+1ℂ between complex Grassmannians. In this paper, we generalize their work to compute rational relative Gottlieb groups of the inclusion Gk,nℂ↣Gk,n+rℂ,r≥1 between complex Grassmannians for 2≤k<n.


## 2. Preliminaries

Here, we fix terminology and recall some standard facts on differential graded algebras. All vector spaces and algebras are taken over a fieldℚ of rational numbers.Definition 1.
A graded algebraA is a sum A=⊕n≥0An,where An is a vector space, together with an associative multiplication Ai⊗Aj⟶Ai+j,x⊗y↦xy, and has 1∈A0. It is graded commutative if, for any homogeneous elements x and y,(5)xy=−1xyyx,where x=i for x∈Ai. If A is a graded algebra equipped with a linear differential map d:An⟶An+1 such that d°d=0 and(6)dxy=dxy+−1xxdy,then A,d is called a differential-graded algebra and d is called a differential. Moreover, if A is also a graded commutative algebra, then A,d is a commutative differential graded algebra (cdga). It is said to be connected if A0≅ℚ.Definition 2.
LetV=⊕i≥0Vi with Veven:=⊕i≥0V2i and Vodd:=⊕i≥1V2i−1. A commutative-graded algebra A is called free commutative if A=∧V=SVeven⊗EVodd, where SVeven is the symmetric algebra on Veven and EVodd is the exterior algebra on Vodd.Definition 3.
A Sullivan algebra is a commutative differential-graded algebra∧V,d, where V=∪k≥0Vk and V0⊂V1⋯ such that dV0=0 and dVk⊂∧Vk−1. It is called minimal if dV⊂∧≥2V.IfA,d is a cdga whose cohomology is connected and finite dimensional in each degree, then there always exists a quasi-isomorphism from a Sullivan algebra ∧V,d to A,d [1]. To each simply connected space, Sullivan associates a cdga APLX of rational polynomial differential forms on X that uniquely determines the rational homotopy type of X [6]. A minimal Sullivan model of X is a minimal Sullivan model of APLX. More precisely, H∗∧V,d≅H∗X;ℚ as graded algebras and V≅π∗X⊗ℚ as graded vector spaces.


## 3. Derivations of a Sullivan Model and theG-Sequence

LetA,d be a commutative differential-graded algebra. A derivation θ of degree k is a linear mapping θ:An⟶An−k such that θab=θab+−1kaaθb. Denote by DerkA the vector space of all derivation of degree k and DerA=⊕kDerkA. The commutator bracket induces a graded Lie algebra structure on DerA. Moreover, DerA,δ is a differential graded Lie algebra [6], with the differential δ defined in the usual way by(7)δθ=d∘θ+−1k+1θ∘d.Let∧V,d be a Sullivan algebra, where V is spanned by v1,…,vk. Then, Der∧V is spanned by θ1,…,θk, where θi is the unique derivation of ∧V defined by θivj=δij. The derivation θi will be denoted by vi,1. Moreover, an element v∈V≅π∗X⊗ℚ is a Gottlieb element of π∗X⊗ℚ if and only if there is a derivation θ of ∧V satisfying θv=1 and such that δθ=0,see page 392 in [1].Letϕ:A,d⟶B,d be a morphism of cdga’s. A ϕ-derivation of degree k is a linear mapping θ:An⟶Bn−k for which θab=θaϕb+−1kaϕaθb.We consider only derivations of positive degree. Denote byDernA,B;ϕ the vector space of ϕ-derivations of degree n for n>0 and by DerA,B;ϕ=⊕nDernA,B;ϕ the ℤ-graded vector space of all ϕ-derivations. The differential-graded vector space of ϕ-derivations is denoted by DerA,B;ϕ,∂, where the differential ∂ is defined by ∂θ=dB°θ+−1k+1θ°dA. In case A=B and ϕ=1B, then DerB,B;1,∂ is just the usual differential-graded Lie algebra of derivations on the cdga B [3]. Whenever A=∧V,d is a Sullivan algebra, we note that there is an isomorphism of graded vector spaces:(8)DerA,B;ϕ≅HomV,B.Ifvi is a basis of V, then the vector space DerA,B;ϕ is spanned by the unique ϕ-derivation θ denoted by vi,bi such that θivi=bi, where bi∈B and θivj=0 for i≠j. Moreover, in [3], precomposition with ϕ gives a chain complex map ϕ∗:DerB,B;1⟶DerA,B;ϕ and postcomposition with the augmentation ε:B⟶ℚ gives a chain complex map ε∗:DerA,B;ϕ⟶DerA,ℚ;ε. The evaluation subgroup of ϕ is defined as follows:(9)GnA,B;ϕ=ImHε∗:HnDerA,B;ϕ⟶HnDerA,ℚ;ε.In caseA=B and ϕ=1B, we get the Gottlieb group of B,d defined as follows:(10)GnB=ImHε∗:HnDerB,B;1⟶HnDerB,ℚ;ε.In particular,GnB≅GnXℚ, if B is the minimal Sullivan model of a simply connected space X (see Proposition 29.8 in [1]).Definition 4.
(see [3, 7]). Let ϕ:A⟶B be a map of differential-graded vector spaces. A differential-graded vector space, Rel∗ϕ, called the mapping cone of ϕ is defined as follows. Relnϕ=An−1⊕Bn with the differential δa,b=−dAa,ϕa+dBb. There are inclusion and projection chain maps J:Bn⟶Relnϕ and P:Relnϕ⟶An−1 defined by Jw=0,w and Pa,b=a. These yields a short exact sequence of chain complexes(11)0⟶B∗⟶JRel∗ϕ⟶PA∗−1⟶0,and a long exact homology sequence of ϕ(12)⋯⟶Hn+1Relϕ⟶HPHnA⟶HϕHnB⟶HJHnRelϕ⟶⋯whose connecting homomorphism is Hϕ.Following [3], we consider a commutative diagram of differential-graded vector spaces:(13)where ε is the augmentation of either A or B, which leads to the following homology ladder for n≥2:(14)Thenth relative evaluation subgroup of ϕ is defined as follows:(15)Gnrel=ImHε∗,ε∗:HnRelϕ∗⟶HnRelϕ¯∗.TheG-sequence of the map ϕ:A⟶B is given by(16)⋯⟶HJ¯Gn+1relA,B;ϕ⟶HP¯GnB⟶Hϕ¯∗GnA,B;ϕ⟶HJ¯⋯which ends in G2A,B;ϕ. Moreover, Theorem 3.5 in [3] can be applied to the Sullivan model ϕ:∧V,d⟶B,d of the map f:X⟶Y.


## 4. The InclusionGk,nℂ↣Gk,n+1ℂ

The complex GrassmannianGk,nℂ is a simply connected homogeneous space as Gk,nℂ≅Un/Uk×Un−k for 1≤k<n, where Un is the unitary group. It is a symplectic manifold of dimension 2m, where m=kn−k. As the complex Grassmannian Gk,nℂ is simply connected, so we may associate a minimal Sullivan model.The method to compute a Sullivan model of the homogeneous spaceGk,nℂ is given in detail in [8, 9].Following [9], a Sullivan model of Gk,nℂ for 1≤k<n is given by(17)∧b2,b4,…,b2k,x2,x4,…,x2n−k,y1,y3,…,y2n−1,d,with(18)dbi=0=dxj,dy2p−1=∑p1+p2=pb2p1x2p2,1≤p≤n.Lemma 1.
The minimal Sullivan model ofGk,nℂ for 2≤k<n is given by(19)∧b2,…,b2k,y2n−k+1,…,y2n−1,d,where dbi=0 and dy2n−k+1∈∧b2,…,b2k.Proof.
Consider the Sullivan model(20)∧b2,b4,…,b2k,x2,x4,…,x2n−k,y1,y3,…,y2n−1,d,of Gk,nℂ for 2≤k<n:(21)dy1=b2+x2,dy3=b4+x4+b2x2,⋮dy2n−1=b2kx2n−k.
The model is not minimal as the linear part is not zero. To find its minimal Sullivan model, we make a change of variablet2=b2+x2 and replace x2 by t2−b2 wherever it appears in the differential. This gives an isomorphic Sullivan algebra(22)∧b2,t2,b4,…,b2k,x4,…,x2n−k,y1,y3,…,y2n−1,d,where(23)dy1=t2,dy3=b4+x4+b2t2−b2,⋮dy2n−1=b2kx2n−k.
As the ideal generated byy1 and t2 is acyclic, the above Sullivan algebra is quasi-isomorphic to(24)∧b2,b4,…,b2k,x4,…,x2n−k,y3,…,y2n−1,d,where(25)dy3=b4+x4−b22,⋮dy2n−1=b2kx2n−k.
One continues in this fashion and makes another change of variablet4=b4+x4−b22 and replaces x4 by t4−b4+b22 wherever it appears in the differential and does so until they reach a change of variable of the form(26)t2n−k=b2n−k+x2n−k+αforn=2kort2n−k=x2n−k+βforn>2k,where α∈∧b2,…,b2k−1,β∈∧b2,…,b2k and replace(27)x2n−k=t2n−k−b2k+α,forn=2k,t2n−k+β,forn>2k,wherever it appears in the differential. This gives an isomorphic Sullivan algebra:(28)∧b2,…,b2k,y2n−k−1,y2n−k+1,…,y2n−1,d,where(29)dy2n−k−1=t2n−k,⋮dy2n−1=b2kx2n−k.
As the ideal generated byt2n−k and y2n−k−1 is acyclic, we get the minimal Sullivan model:(30)∧b2,…,b2k,y2n−k+1,…,y2n−1,d,with dbi=0 and dy2n−k+1∈∧b2,…,b2k.
In the same way, by Lemma 1, the minimal Sullivan model ofGk,n+rℂ for 2≤k<n+r and r≥1 is given by(31)∧a2,…,a2k,z2n+r−k+1,…,z2n+r−1,d,where dai=0 and dz2n+r−k+1∈∧a2,…,a2k. We establish the following results.Theorem 1.
LetB=∧b2,…,b2k,y2n−k+1,…,y2n−1,d. Then, GnB=y2n−k+1∗,…,y2n−1∗.Proof.
Lety2n−t+1,1 denote the derivation θ2n−t+1 for t∈1,…,k such that θ2n−t+1y2n−t+1=1 and zero on other generators. Then,(32)δθ2n−t+1y2n−t+1=0.
Moreover, the generatorsθ2n−t+1 cannot be boundaries for degree reasons. Therefore, θ2n−t+1 are nonzero homology classes in H∗DerB,B;1. Furthermore, ε∗θ2n−t+1=y2n−t+1∗.
AsGk,nℂ is a simply connected finite CW-complex, then GevenB=0 (see Proposition 28.8 in [1]. Thus, GnB=y2n−k+1∗,…,y2n−1∗.Theorem 2.
Given the inclusionGk,nℂ↣Gk,n+rℂ,r≥1 for 2≤k<n and ϕ:∧V,d⟶B,d for its Sullivan model, then G∗∧V,B;ϕ=z2n+r−k+1∗,…,z2n+r−1∗.Proof.
The vector spaceDer∧V,B;ϕ is generated by the derivations θ2n+r−t+1=z2n+r−t+1,1 for t∈1,…,k. The differential is given by(33)δθ2n+r−t+1z2n+r−t+1=0.
Hence,θ2n+r−t+1 are the nonzero homology classes in H∗Der∧V,B;ϕ. Moreover, Hε∗θ2n+r−t+1=z2n+r−t+1∗, where z2n+r−t+1∗∈Der∧V,\\ℚ,ε. Thus, G∗∧V,B;ϕ=z2n+r−k+1∗…,z2n+r−1∗.Theorem 3.
Given the inclusionGk,nℂ↣Gk,n+rℂ,r≥1 for 2≤k<n and ϕ:∧V,d⟶B,d for its Sullivan model, then G∗rel∧V,B;ϕ=y2n−k+1∗,0,0,z2n+r−1∗.Proof.
Define the derivationsα2n−t+1=y2n−t+1,1 for t∈1,…,k in DerB,B;1 and θ2n+r−t+1=z2n+r−t+1,1 in Der∧V,B;ϕ. Then,(34)ϕ∗α2n−t+1=θ2n+r−t+1,if2n−t+1=2n+r−t+1,0,if2n−t+1≠2n+r−t+1,such that(35)Dα2n−t+1,0=0,θ2n+r−t+1,if2n−t+1=2n+r−t+1,0,0,if2n−t+1≠2n+r−t+1,and D0,θ2n+r−t+1=0,0. Thus, α2n−t+1,0 if 2n−t+1≠2n+r−t+1 and 0,θ2n+r−t+1 are the nonzero homology classes in H∗Relϕ∗. Moreover, Hε∗,ε∗α2n−t+1,0=y2n−t+1∗,0, for 2n−t+1≠2n+r−t+1, and in the same way, Hε∗,ε∗0,θ2n+r−t+1=0,z2n+r−t+1∗. A straightforward calculation shows that y2n−t+1∗,0 for 2n−t+1≠2n+r−t+1 and 0,z2n+r−t+1∗ span Hε∗,ε∗.
TheG-sequence reduces to(36)0⟶G2n+r−1∧V,B;ϕ⟶≃HJG2n+r−1rel∧V,B;ϕ⟶0,0⟶G2n−t+1=2n+r−t+1B⟶≃Hϕ∗G2n−t+1=2n+r−t+1∧V,B;ϕ⟶0,0⟶G2n−k+1rel∧V,B;ϕ⟶≃HPG2n−k+1B⟶0,and it is exact.



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