# Relative Gottlieb Groups of Embeddings between Complex Grassmannians

Authors: J. B. Gatsinzi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

LetGrk,n be the complex Grassmann manifold of k-linear subspaces in ℂn. We compute rational relative Gottlieb groups of the embedding i:Grk,n⟶Grk,n+r and show that the G-sequence is exact if r≥kn−k.


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## Body

## 1. Introduction

We work in the category of spaces having the homotopy type of simply connected CW complexes of finite type. We denote byh:X⟶Xℚ the rationalization of X [1, 2]. Let f:X,x0⟶Y,y0 be a pointed continuous mapping and mapX,Y;f be the component of f in the space of all continuous maps g:X⟶Y. Consider the evaluation map ev:mapX,Y;f⟶Y at the base point x0, that is, evg=gx0. The nth evaluation subgroup of f, GnY,X;f, is the image of πnev in πnY [3]. In the special case where X=Y and f=1X, one obtains the Gottlieb group GnX of X [4]. Gottlieb groups play an important role in topology. For instance, if GnX=0, then any fibration X⟶E⟶Sn+1 admits a section (Corollary 2–7 in [4]).In [2], Lee and Woo introduce relative evaluation groups GnrelY,X;f and obtain a long sequence,(1)⋯⟶Gn+1relY,X;f⟶GnX⟶GnY,X;f⟶GnrelY,X;f⟶⋯,called G-sequence [5]. This sequence is exact in some cases, for instance, if f is a homotopy monomorphism [6].


## 2. Rational Relative Gottlieb Groups

The rationalizationh:Y⟶Yℚ induces a rationalization h∗:mapX,Y;f⟶mapX,Y;h∘f [7]. Therefore,(2)ev∗π∗mapX,Y;f⊗ℚ≅ev∗π∗mapX,Yℚ;h∘f.In this paper, we study theG-sequence of the natural inclusion Grk,n⟶Grk,n+r using models of function spaces in rational homotopy [8, 9]. In particular, we show that the G-sequence is exact if r≥kn−k. We work with algebraic models in rational homotopy theory introduced by Sullivan and Quillen [10, 11]. In this section, we give relevant definitions and fix notation. Details can be found in [1]. All vector spaces and algebras are over the field of rational numbers ℚ.LetA,d be a cochain algebra. The degree of an homogeneous element a∈Ap is written a. We assume that A,d is 1-connected, that is, H0A,d=ℚ and H1A,d=0. The algebra A is called commutative if ab=−1|abba for homogeneous elements a,b∈A.Definition 1.
A commutative differential graded algebra (cdga, for short)A,d is called a Sullivan algebra if A=SVeven⊗EVodd, where V=⊕k≥2Vk. It will be denoted by ∧V,d.
Moreover, a Sullivan algebra∧V,d is called minimal if dV⊂∧≥2V. A Sullivan model of A,d is given by a Sullivan algebra ∧V,d together with a quasi-isomorphism f:∧V,d⟶A,d. It is unique up to isomorphism.Definition 2.
IfX is a simply connected space of finite type, then the (minimal) Sullivan model of X is the (minimal) Sullivan model of cdga APLX of polynomial differential forms on X [1, 5]. A simply connected topological space X is called formal if there exists a quasi-isomorphism ∧V,d⟶H∗X,ℚ, where ∧V,d is a Sullivan model of X. Formal spaces include homogeneous spaces G/H, where G and H have the same rank.The complex Grassmann manifoldGrk,n is the space of k-dimensional subspaces of ℂn. Moreover, Gk,n≃Un/Uk×Un−k, where Un is the unitary group. Hence, Gk,n is formal (see also [11, 12]). As Gk,n≅Gn−k,n, we will assume that k≤n/2. As Gk,n is a formal, its Sullivan model can be computed from its cohomology algebra. Precisely,(3)H∗Grk,n=∧x2,x4,…,x2khn−k+1,…,hn−1,hn,where hj is the polynomial of degree 2j in the Taylor expansion of the expression 1/1+x2+⋯+x2k [13]. A Sullivan model is given by(4)∧x2,…,x2k,x2n−2k+1,…,x2n−1,d,where dx2i=0 and dx2n−2k+2i−1=hn−k+i, i=1,…,k. Moreover, this model is minimal.Let(5)∧V,d=∧x2,…,x2k,x2n+2r−2k+1,…,x2n+2r−1,d,∧W,d=∧y2,…,y2k,y2n−2k+1,…,y2n−1,d,be respective minimal Sullivan models of Grk,n+r and Grk,n. A Sullivan model of the inclusion i:Grk,n⟶Grk,n+r is then(6)ϕ:∧V,d⟶∧W,d,which is defined by(7)ϕx2=y2,…,ϕx2k=y2k,ϕx2n+2r−2k+2i+1=∑j=0k−1pijy2n−2k+2j+1,where pij is a polynomial of degree 2r+i−j in y2,…,y2k, for i,j=0,1,2,…,k−1, provided that r+i−j≥0.The polynomialspij encode the relationships between hi’s. They can be explicitly expressed from the equality:(8)1+x2+⋯+x2k1+h1+h2+⋯=1.For instance, fork=2,(9)h1=−x2,h2=x22−x4,h3=−x23+2x2x4,h4=x24−3x22x4+x42,h5=−x2h4−x4h3,h6=x22−x4h4+x2x4h3,h7=h4h3+−x22x4+x42h3.Example 1.
The inclusionGr2,4⟶Gr2,7 has a Sullivan model:(10)ϕ:∧x2,x4,x11,x13,d⟶∧y2,y4,y5,y7,d,where(11)dx2=dx4=0,dx11=x22−x4h4+x2x4h3,dx13=h4h3+−x22x4+x42h3,dy2=dy4=0,dy5=−y23+2y2y4,dy7=y24−3y22y4+y42,ϕx2=y2,ϕx4=y4,ϕx11=y2y4y5+y22−y4y7,ϕx13=−y22y4+y42y5+−y23+2y2y4y7.
We note that−y22y4+y42=dy2y5+y7; therefore,(12)ϕx13=dy2y5+y7y5+dy5y7.Recall that ifϕ:A,dA⟶B,dB is a map of chain complexes; the mapping cone of ϕ, denoted by Relϕ, is defined by(13)Relϕ∗=sA∗−1⊕B∗,D,where the differential is defined by Dsa,b=−sdAa,ϕa+dBb [9] or p. 46 in [14]. Define chain maps J:Bn⟶Relnϕ and P:Relnϕ⟶An−1 by Jb=0,b and Psa,b=a. There is an exact sequence of chain complexes:(14)0⟶B∗⟶JRel∗ϕ⟶PA∗−1⟶0,which induces a long exact sequence:(15)⟶HnB⟶HnJHnRelϕ⟶HnPHn−1A⟶Hn−1ϕHn−1B⟶,(see Proposition 4.3 in [14]).Definition 3.
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 such that θab=θaϕb+−1kaϕaθb. We denote by DernA,B;ϕ the vector space of ϕ-derivations of degree n and by DerA,B;ϕ=⊕nDernA,B;ϕ the ℤ-graded vector space of all ϕ-derivations. The differential on DerA,B;ϕ is defined by δθ=dθ−−1kθd. We will restrict to derivations of positive degree; however, in degree one, we only consider those derivations which are cycles.Ifϕ:A⟶A is the identity mapping, we simply write DerA for DerA,A;1A. Moreover, if A=∧V, where v1,v2,… is a basis of V and ϕ:∧V,d⟶B,d is a morphism of cdga’s, we denote by vi,b the unique ϕ-derivation θ such that θvi=b and zero on other elements of the basis.Define the Gottlieb group of∧V,d:(16)Gn∧V=θ∈HnDer∧V,δ:θv=1,v∈Vn.Hence,G∗∧V≅imH∗ϵ∗, where ϵ∗:Der∧V⟶Der∧V,ℚ;ϵ is the postcomposition with the augmentation map ϵ:∧V⟶ℚ. If X is simply connected and ∧V,d is the minimal Sullivan model of X, then Gn∧V≅GnXℚ, where h:X⟶Xℚ is the rationalization (Propostion 29.8 in[1]).Similarly, ifϕ:∧V,d⟶∧W,d is a map between Sullivan algebras, then the Gottlieb group G∗∧V,∧W;ϕ is defined as H∗ϵ∗, where ϵ∗:Der∧V,∧W;ϕ⟶Der∧V,ℚ;ϵ is the postcomposition with ϵ:∧W,d⟶ℚ. Moreover, if ϕ is a Sullivan model of a map f:X⟶Y, where Y is finite, then G∗∧V,∧W;ϕ≅GnYℚ,X;h∘f, where h:Y⟶Yℚ is the rationalization map.Letϕ:∧V,d⟶∧W,d be a Sullivan model of a map f:X⟶Y between simply connected spaces. It induces a chain map ϕ∗:Der∧W⟶Der∧V,∧W;ϕ by precomposition by ϕ. We get the following commutative diagram:(17)Der∧W⟶ϕ∗Der∧V,∧W;ϕ⟶JRelϕ∗↓ϵ∗↓ϵ∗↓ϵ∗;ϵ∗Der∧W,ℚ;ϵ⟶ϕ∗^Der∧V,ℚ;ϵ⟶J^Relϕ∗^.Then, rational evaluation subgroups are corresponding images in the lower ladder induced in homology by vertical maps. Therefore, there is a long sequence:(18)⋯⟶Gn∧W⟶Hϕ^∗Gn∧V,∧W;ϕ⟶HJ^Gnrel∧V,∧W;ϕ⟶HP^⋯.We will use the following result for our computations (Theorem 2.1 in [9] or Corollary 1 in [15]).Theorem 1 (see [9]).
Letf:X⟶Y be a map between simply connected CW complexes, where X is of finite type and ϕ:∧V,d⟶∧W,d its Sullivan model. The long exact sequence induced by the map f∗:mapX,X;1X⟶mapX,Y;f on rational homotopy groups is equivalent to the long exact sequence of(19)ϕ∗:Der∧W,d⟶Der∧V,∧W;ϕ.We consider the particular case, wheref is the inclusion i:Grk,n⟶Grk,n+r, where r≥1 and its Sullivan model ϕ:∧V,d⟶∧W,d as given in equation (6).Theorem 2.
Letϕ:∧V,d⟶∧W,d be a Sullivan model of the inclusion i:Grk,n⟶Grk,n+r, where r≥kn−k:(1)
G∗∧V,∧W;ϕ≅V#, the dual ofV(2)
G∗rel∧V,∧W;ϕ≅sG∗∧W⊕G∗∧V,∧W;ϕ≅sG∗∧W⊕V#Proof.
(1)
Recall that∧V=∧x2,…,x2k,x2n+2r−2k+1,…,x2n+2r−1, ∧W=∧y2,…,y2k,y2n−2k+1,…,y2n−1, and ϕ:∧V,d⟶∧W,d are defined by ϕx2=y2,…,ϕx2k=y2k, ϕx2n+2r−2k+2i−1=∑j=1kpijy2n−2k+2j−1, and pij is a polynomial of degree 2r+i−j in y2,…,y2k and i∈1,…,k.We consider the compositionφ:∧V,d⟶ϕ∧W,d⟶pH∗∧W,d. As p is a quasi-isomorphism, then the G-sequence of the inclusion is computed from the long exact sequence induced by the cone of the map:(20)ϕ∗:Der∧W,H∗∧W;p⟶Der∧V,H∗∧W;φ.Each of the derivationsx2n+2r−2k+2i+1∗=x2n+2r−2k+2i−1,1∈Der∧V,H∗∧W;φ is a cycle of degree at least 2k+2r+2i−1>2k+2r and cannot be boundary as all even degree derivations in Der∧V,H∗∧W,φ are of degree at most 2k. Hence, x2n+2r−2k+2i+1∗ is nonzero in G∗∧V,H∗∧W;φConsider the derivationsx2i∗=x2i,1∈Der∧V,H∗∧W,d,φ, for i=1,…,k. Then,(21)δx2i∗x2n+2r−2k+2j−1∈H2n+r−k+j−i∧W,d.Moreover, as1≤1,j≤k, then j−i≥−k+1. Therefore,(22)2n+r−k+j−i≥2n+r−k−k+1≥2r+1,asn≥2k.Therefore,δx2i∗x2n+2r−2k+2j−1∈H≥2kn−k+2=0. Hence, x2i∗ is a cycle for i=1,…,k. Moreover, x2i∗ cannot be a boundary as all odd degree derivations are of degree at least 2n+2r−2k+1−2kn−k>2n−k+1≥2k+1. Therefore, x2n+2r−2k+2i−1∗ are cycles which cannot be boundaries for degree reasons. Hence, G∗∧V,H∗∧W,d,φ≅V#.(2)
First, we note thatHevenDer∧W,∧H∗∧W,d;p=0, and consequently, Geven∧W,∧H∗∧W,d;p=0 [1, 16]. Moreover, a straightforward calculation shows that(23)Godd∧W,∧H∗∧W,d,p≅y2n−2k+1∗,…,y2n−1∗.
We consider the vector space:(24)Relϕ∗=sDer∧W,H∗∧W;p⊕Der∧V,∧W;φ,where the differential is defined by Dsα,β=−sδα,ϕ∗α+δβ. Consider W1#=y2n−2k+1∗,…,y2n−1∗ in Der∧W,H∗∧W;p. For degree reasons, ϕ∗W1#=0. Therefore, Dsy∗,0=0, for y∗∈W1#. Hence, sy2n−2k+1∗,…,sy2n−1∗ represent nonzero homology classes in G∗rel∧V,H∗∧W;φ. We conclude that G∗rel∧V,H∗∧W;φ=sG∗∧W,H∗∧W;p⊕G∗∧V,H∗∧W;φ.Corollary 1.
Ifr≥kn−k, then the rational G-sequence of the inclusion i:Grk,n⟶Grk,n+r is exact.Proof.
It comes from the previous lemma that theG-sequence is(25)0⟶G∗∧V,H∗∧W;φ⟶G∗∧V,H∗∧W;φ⊕sG∗∧W,H∗∧W;p⟶G∗∧W,H∗∧W;p⟶0,which is exact.


## 3. InclusionGrk,n⟶Grk,n+1

In the range1≤r<kn−k, the G-sequence of the inclusion Grk,n⟶Grk,n+r is more challenging to characterize, as shown in the following example.Example 2.
Consider the inclusionGr2,4⟶Gr2,7 of which a Sullivan model is given by(26)ϕ:A=∧x2,x4,x11,x13,d⟶∧y2,y4,y5,y7,d=B,where ϕ is defined in Example 1. We compose with the quasi-isomorphism p:B,d⟶H∗B,0 and consider ϕ∗:DerB,H∗B;p⟶DerA,H∗B;φ, where φ=p∘ϕ. Moreover, G∗B,H∗B;p=y5∗,y7∗, where y5∗=y5,1 and similarly y7∗=y7,1. Furthermore, δx2∗=0; hence, x2∗ represents a nonzero homology class in DerA,H∗B;φ. A simple calculation shows that δx4∗=x11,ω/2, where ω=x24. Hence,(27)G∗A,H∗B;φ≅x2∗,x11∗,x13∗.
Consider(28)Rel∗ϕ∗=sDerB,H∗B;p⊕DerA,H∗B;φ,D.
Then,(29)Dsy5∗,0=0,α5,Dsy7∗,0=0,α7,where α5=x11,y2y4 and α7=x11,y22−y4. Therefore, the image of(30)H∗P:G∗relA,H∗B,φ⟶G∗−1B,H∗B;p,is zero. Hence, the sequence(31)G6relA,H∗B;φ⟶H∗PG5B,H∗B;p⟶H5ϕ∗G5A,H∗B;φ,reduces to(32)0⟶y5∗⟶0,which is not exact.
In the same way,(33)G8relA,H∗B;φ⟶H∗PG7B,H∗B;p⟶H7ϕ∗G5A,H∗B;φis not exact. Moreover, H∗J:G∗A,H∗B;φ⟶G∗relA,H∗B;φ is an isomorphism.Although theG-sequence of the inclusion Grk,n⟶Grk,n+r might not be exact for some values of 1≤r<kn−k, we have the following result for r=1.Theorem 3.
Letϕ:∧V,d⟶∧W,d be a Sullivan model of the inclusion Grk,n⟶Grk,n+1:(1)
G∗rel∧V,∧W;ϕ has dimension 1(2)
TheG-sequence of the inclusion Grk,n⟶Grk,n+1 is not exactProof.
Recall from Section2 that the minimal Sullivan model of Grk,n is ∧W,d, where(34)W=y2,y4,…,y2k,y2n−k+1,…,y2n−1,dy2=⋯=dy2k=0,dy2n−k+i−1=hn−k+i,fori=1,…,k.
Similarly, a model ofGk,n+1 is ∧V,d, where(35)V=x2,…,x2k,x2n−k+3,…,x2n+1,dx2=⋯=dx2k=0,dx2n−k+i+1=hn−k+i+1,fori=1,…,k.
Moreover, a model of the inclusioni:Grk,n⟶Grk,n+1 is given by ϕ:∧V,d⟶∧W,d and defined by(36)ϕx2i=y2i,ϕx2n−k+3=y2n−k+3,…,ϕx2n−1=y2n−1,ϕx2n+1=−y2y2n−1−y4y2n−3−⋯−y2ky2n−k+1.
We consider the quasi-isomorphism(37)p:∧W,d⟶H∗∧W,d=∧y2,…,y2kdy2n−k+1,…,dy2n−1,and set φ=p∘ϕ. Consider(38)ϕ∗:Der∧W,H∗∧W,d;p⟶Der∧V,H∗∧W;φ.
We have the following relations:(39)ϕ∗y2i∗=x2i∗,fori=1,…,nϕ∗y2n−1∗=x2n−1∗−x2n+1,y2ϕ∗y2n−3∗=x2n−3∗−x2n+1,y4,…ϕ∗y2n−k+3∗=x2n−k+3∗−x2n+1,y2k−2ϕ∗y2n−k+1∗=−x2n+1,y2k.
As a result, in(40)Relϕ∗=sDer∧W,H∗∧W,d;p⊕Der∧V,H∗∧W;φ,we have the following relations:(41)D0,x2n−k+2i+1∗=0,fori=1,…,kDsy2n−1∗,0=0,x2n−1∗−x2n+1,y2Dsy2n−3∗,0=0,x2n−3∗−x2n+1,y4…Dsy2n−2k+3∗,0=0,x2n−2k+3∗−x2n+1,y2k−2Dsy2n−2k+1∗,0=0,−x2n+1,y2k.
We consider the commutative diagram:(42)Der∧W,H∗∧W;p⟶ϕ∗Der∧V,H∗∧W;φ↓ϵ∗↓ϵ∗Der∧W,ℚ;ϵ⟶ϕ^∗Der∧V,ℚ;ϵ.
Letyi∗^=ϵ∗yi∗ and xj∗^=ϵ∗xj∗. Consider(43)Relϕ^∗=sDer∧W,ℚ;ϵ⊕Der∧V,ℚ;ϵ,D^.
Then,(44)D^sy^2n−2k+1∗,0=0,0D^sy^2n−2k+3∗,0=0,x^2n−2k+3∗…D^sy^2n−1∗,0=0,x^2n−1∗.
Hence,(45)H∗Relϕ^∗=sy^2n−2k+1∗,0,0,x^2n+1∗.
Moreover, the image ofH∗ϵ∗,ϵ∗:H∗Relϕ∗⟶H∗Relϕ^∗ is 0,x^2n+1∗. Therefore,(46)G∗rel∧V,H∗∧W;φ=0,x^2n+1∗.
This shows the first part of the theorem and corrects Theorem 3 in [17] and Theorem 3 [18].
Moreover,(47)HP^:G∗rel∧V,H∗∧W,d;φ⟶G∗−1∧W,H∗∧W,d;pis the zero map. The G-sequence then reduces to exact portions(48)G2n−2k+2i+1∧W,H∗∧W;p⟶≃Hϕ^∗G2n−2k+2i+1∧V,H∗∧W;φ,for i=1,…,k−1, and(49)G2n+1∧V,H∗∧W;p⟶≃HJ^G2n+1rel∧V,H∗∧W;φ,and a nonexact part,(50)0⟶G2n−2k+1∧W,H∗∧W;p⟶0.Example 3.
We consider a model of the inclusionGr2,4⟶Gr2,5 which is of the form(51)ϕ:∧V,d=∧x2,x4,x7,x9,d⟶∧y2,y4,y5,y7,d=∧W,d,where dx2=dx4=0, dx7=x24−3x22x4+x42, dx9=−x2x24−3x22x4+y42−x4−x23+2x2x4, dy2=dy4=0, dy5=−y23+2y2y4, dy7=y24−3y22y4+y42, ϕx2=y2, ϕx4=y4, ϕx7=y7, and ϕx9=−y2y7−y4y5.We compose which the quasi-isomorphismp:∧W,d⟶H∗∧W,d to get φ:∧V,d⟶H∗∧W,d. In(52)Relϕ∗=sDer∧W,H∗∧W,d;p⊕Der∧V,H∗∧W,d,φ,we have the following relations:(53)Dsy5∗,0=0,x9,−y4,Dsy7∗,0=0,x7∗+x9,−y2.Consider(54)Relϕ∗^=sDer∧W,ℚ;ϵ⊕Der∧V,ℚ,ϵ,D≅sW#⊕V#,D,where(55)Dsy^5∗,0=0,0,Dsy^7∗,0=0,x^7∗,D0,x^7∗=D0,x^9=0,0.Hence,(56)H∗Relϕ∗^≅sy^5∗,0,0,x^9∗.However,imHϵ∗,ϵ∗=0,x^9∗. Therefore, G∗rel∧V,H∗∧W,d;φ≅0,x^9∗. As G∗∧V,H∗∧W,d;φ≅x^7∗,x^9∗ and G∗∧W,H∗∧W,d,p=y^5∗,y^7∗, then the G-sequence reduces to exact nonzero fragments:(57)0⟶G9∧V,H∗∧W,d;φ⟶≅G9rel∧V,H∗∧W,d;φ⟶0,0⟶G7∧W,H∗∧W,d;p⟶≅G7∧V,H∗∧W,d;φ⟶0,and a nonexact sequence,(58)0⟶G5∧W,H∗∧W,d;p⟶0.



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