# Rational Cohomology Algebra of Mapping Spaces between Complex Grassmannians

Authors: Paul Antony Otieno; Jean Baptiste Gatsinzi; Vitalis Onyango-Otieno
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

We consider the complex GrassmannianGrk,n of k-dimensional subspaces of ℂn. There is a natural inclusion in,r:Grk,n↪Grk,n+r. Here, we use Sullivan models to compute the rational cohomology algebra of the component of the inclusion in,r in the space of mappings from Grk,n to Grk,n+r for r≥1 and in particular to show that the cohomology of mapGrn,k,Grn,k+r;in,r contains a truncated algebra ℚx/xr+n+k2−nk, where x=2, for k≥2 and n≥4.


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## Body

## 1. Introduction

The complex GrassmannGrk,n is the set of k-planes through the origin in ℂn. Moreover,(1)Grk,n≅UnUk×Un−k,where Un is the unitary group ([1], chap. 18). There is a canonical inclusion in,r:Grk,n↪Grk,n+r which is induced by ir:ℂn⟶ℂn+r defined by ix1,…,xn=x1,…,xn,0,…,0.The study of the rational homotopy type of function spaces started with Thom in the case where the codomain is an Eilennerg–Maclane space [2]. The first description of a Sullivan model of function spaces is due to Haefliger [3]. Moreover, a model of function spaces between complex projective spaces was given by Moller and Raussen using Postkinov tower [4]. However, there is no explicit and complete description of the homotopy type of the component of the inclusion Grk,n↪Grk,n+r in the space of mappings from Grk,n to Grk,n+r, r≥1 and n≥4. We begin our work with a model of the inclusion.In our previous work, we studied the rational homotopy ofmapGr2,n,Gr2,n+r;in,1 where in,r:Gr2,n⟶Gr2,n+r is the canonical inclusion. We showed among other things that mapGr2,n,Gr2,n+1;in,r has the rational homotopy type of a product of odd spheres. Under some assumptions on r, the cohomology algebra of mapGr2,n,Gr2,n+r;in,r contains either a polynomial algebra or a truncated algebra over a generator of degree 2.In this paper, we consider the inclusionGrk,n⟶Grk,n+r, k≥3 [5].


## 2. Sullivan Models

Henceforth, we work on the field of rational numbers,ℚ.Definition 1.
A differential graded algebradga is a graded algebra A=⊕n≥0An endowed with a derivation d, of degree +1, such that d2=0. The pair A,d is called a cochain algebra. A graded algebra A is commutative if a⋅b=−1dega⋅degbb⋅a for a,b∈A ([6], chap. 3).Definition 2.
A Sullivan algebra is a commutative cochain algebra of the form∧V,d where V=⊕k≥1Vk and V=∪k=0∞Vk, V0⊂V1⊂⋯ such that dVk⊂∧Vk−1. A Sullivan model for a commutative cochain algebra A,d is a quasi-isomoprphism m:∧V,d⟶A,d from a Sullivan algebra ∧V,d. A Sullivan algebra (or model) is said to be minimal if the differential is decomposable, that is, Imd⊂∧+V⋅∧+V.
Moreover, ifH0A=ℚ, then A,d has a minimal model which is unique up to isomorphism. If X is a nilpotent space and APLX the commutative differential graded algebra (cdga) of piecewise linear forms on X, then a Sullivan model of X is a Sullivan model of APLX ([6], chap.12). A space X is formal if there is a quasi-isomorphism ∧V,d⟶H∗X,ℚ. Moreover, complex Grassmann manifolds are formal [7].
The cohomology ringH∗Grk,n,ℚ of Grk,n has a presentation(2)H∗Grk,n,ℚ=∧c1,c2,…,ckhn−k+1,…,hn,a quotient of the polynomial ring generated by c1,c2,…,ck, ci=2i, modulo the ideal generated by the elements hj,n−k+1≤j≤n. Here, hj is defined as the 2jth degree term in Taylor’s expansion of 1+c1+c2+c3+⋯+ck−1, where 1+c1+c2+c3+⋯+ck is the total Chern class and ci∈H2iGrk,n,ℚ for 1≤i≤k [8].
For instance,H∗Gr2,4,ℚ=∧x2,x4/h3,h4, where h3=2x2x4−x23 and h4=x42−3x22x4+x24. As 2x2x4−x23,x42−3x22x4+x24 forms a regular sequence, the Sullivan model of Gr2,4 is hence given by ∧x2,x4,x5,x7,d with dx2=dx4=0, dx5=2x2x4−x23 and dx7=x42−3x22x4+x24.


## 3. L∞ Models of Function Spaces

Definition 3.
On a graded vector spaceL, an L∞ structure, usually denoted by L,lkk≥1, is a collection of linear maps, lkk≥1 called brackets, where lk:⊗kL⟶L such that the following conditions are satisfied:(1)
lk are graded skew symmetric, that is, for any k permutation σ,(3)lkxσ1,…,xσk=sgnσϵσlkx1,…,xk,whereεσ is the sign given by the Koszul convention.(2)
The generalised Jacobi identity holds, that is,(4)∑i+j=n+1∑σ∈Si,n−isgnσεσ−1ij−1ljlixσ1,…,xσi,xσi+1,…,xσn=0,where Si,n−i denotes the i,n−i shuffles which are permutations σ∈Sn such that σ1<⋯<σi and σi+1<⋯<σn.
AnL∞ algebra L,lkk≥1 is minimal if l1=0. L∞ structures are in one-to-one correspondence with codifferentials on the non unital, free commutative coalgebra Λ+sL in which s denotes suspension, that is sLk=Lk−1 [9].
Note that ifL is a graded vector space of finite type, an L∞ structure on L induces a commutative differential graded algebra structure C∞L=∧sL♯,d, d=∑j≥1dj, where dj:V⟶∧jV is defined by djv;sx1∧⋯∧sxj=−1εv;sljx1,…,xj and ε=−1v+∑​k=1j−1j−kxk [10]. This is a generalisation of the Quillen cochain functor to L∞ algebras ([6], Chap.23).
Further, anL∞ algebra L is an L∞ model of a simply connected space X, if C∞L is a Sullivan model of X [10].Definition 4.
Letϕ:A,dA⟶B,dB be a morphism of cochain algebras. A ϕ-derivation θ of degree n is a linear map A∗⟶A∗−n such that θab=θaϕb+−1naϕaθb. We denote by DernA,B,ϕ the vector space of all ϕ-derivations of degree n. Define a differential of chain complexes δ:DernA,B,ϕ⟶Dern−1A,B,ϕ, by δθ=dBθ−−θθdA.
Ifϕ:∧V,d⟶B,d is a morphism of commutative differential graded algebras, then there is an isomorphism of vector spaces:(5)Der∧V,B,ϕ⟶≃HomV,B.
Moreover, ifv1,…,vn is a basis for V, we will denote by vi,bi the ϕ-derivation θ such that(6)θvi=bi,bi∈B,θvj=0,i≠j.
We now follow [10] for the definition below.
Consider positive derivations defined by(7)Der˜i∧V,B,ϕ=Deri∧V,B,ϕ,ifi≥2,ZDeri∧V,B,ϕ,ifi=1.
IfA,d=∧V,d is a Sullivan algebra, then given j≥2 and φ1,…,φj∈Der˜∧V,B,ϕ, of degrees p1,…,pj, we define their brackets of length j, φ1,…,φj=λjφ1,…,φj∈Der˜∧V,B,ϕ by(8)φ1,…,φjv=−1p1+⋯+pj−1∑∑i1,…,ijεϕv1⋯v^i1⋯v^ij⋯vkφvi1⋯φjvij,where dv=∑v1⋯vk and ϵ is the sign given by the Koszul convention. These operations may be desuspended to define a set of linear maps ljj≥1 each of degree j−2 on s−1Der˜∧V,B,ϕ as follows:
Forj=1,(9)l1:s−1Der˜∧V,B,ϕ⟶s−1Der˜∧V,B,ϕ,l1s−1θ=−s−1δθ.
Forj=2, define(10)lj:s−1Der˜∧V,B,ϕ⊗⋯⊗s−1Der˜∧V,B,ϕ⟶s−1Der˜∧V,B,ϕ,by ljs−1φ1,…,s−1φj=−1αs−1λjφ1,…,φj and α=∑n=1j−1j−n+φn.
Letf:X⟶Y be a continuous function between 1-connected spaces and mapX,Y;f the component of f in the space of continuous maps from X to Y. Let ϕ:∧V,d⟶B be a cdga model of f. Lemma3.3 of [10] proves that s−1Der˜∧V,B,ϕ,ljj≥1 is an L∞ model of(11)mapX,Y;f.


## 4. Inclusion ofGr3,n in Gr3,n+r

Theorem 1.
Letin,r:Gr3,n⟶Gr3,n+r be the canonical inclusion. If r>2n−4, then the rational cohomology algebra of mapGr3,n,Gr3,n+r;in,r contains a truncated algebra ℚx/xr−2n+9, where x=2.Proof.
The cohomology algebras ofGr3,n and Gr3,n+r are given by H∗Gr3,n,ℚ=∧y2,y4,y6/hn−2,hn−1,hn and H∗Gr3,n+r,ℚ=∧x2,x4,x6/hn+r−2,hn+r−1,hn+r, respectively. Recall that Gr3,n is a smooth manifold of dimension 6n−18; hence, HmGr3,n,ℚ=0 for m≥6n−18.
The minimal model ofGr3,n+r is(12)∧V,d=∧x2,x4,x6,x2n+2r−5,x2n+2r−3,x2n+2r−1,d,with dx2=dx4=dx6=0, dx2n+2r−5=hn+r−2, dx2n+2r−3=hn+r−1, and dx2n+2r−1=hn+r. A model of the inclusion is given by(13)ϕ:∧V,d⟶B=∧y2,y4,y6hn−2,hn−1,hn,where ϕx2=y2, ϕx4=y4, ϕx6=y6, and ϕx2n+2r−5=ϕx2n+2r−3=ϕx2n+2r−1=0. Let L=Der∧V,B,ϕ. Consider γ2=x6,y4∈L. Then q1=λkγ2,…,γ2x2n+2r−5, q2=λkγ2,…,γ2x2n+2r−3, and q3=λkγ2,…,γ2x2n+2r−1 are polynomials of degree at least 2n+2r−2k−4. If k<r−2n+7, then q1, q2, and q3 will be of degree >6n−18. Therefore, λkγ2,…,γ2=0 for k=1,…,r−2n+7. In C∞s−1L, let γ2♯=ρ2. Then ρ2i≠0 for i=1,…,r−2n+7.Theorem 2.
Consider the inclusionin,r:Gr3,nGr3,n+r. If r>5n−14, then the cohomology algebra of mapGr3,n,Gr3,n+r;in,r contains a polynomial algebra over a generator of degree 2.Proof.
Let(14)ϕ:∧x2,x4,x6,x2n+2r−5,x2n+2r−3,x2n+2r−1,d⟶∧y2,y4,y6hn−2,hn−1,hn,be a model of the inclusion. Assume n≥6, in particular, if r≥5n−14, then any odd derivation is of degree at least 2n+2r−5−6n−18=2r−4n+13≥13. Hence, L11=L9=L7=⋯=L1=0. As L1=0, then the derivation γ2=x6,y4 is a cycle.
It is sufficient to show thatλmγ2,γ2,…,γ2=0 for m≥2.
We consider the case whenn+r is even, the other case is dealt with in a similar way. For m>n+r/2, λmγ2,…,γ2=0. Assume 2≤m≤n+r/2, and let(15)r=λmγ2,γ2,…,γ2x2n+2r−5,s=λmγ2,γ2,…,γ2x2n+2r−3,t=λmγ2,γ2,…,γ2x2n+2r−1.
The polynomialsr,s, and t are of total degree at least 2n+2r−2m−4. Assume now that r≥5n−14, then 2n+2r−2m−4>6n−18. Therefore, λmγ2,γ2,…,γ2=0 for all m≥1. Hence, H∗C∞s−1L contains a polynomial algebra over a generator of degree 2.


## 5. The General Case

We can generalise the above results.Theorem 3.
Ifr>nk−k2−n+2k−1, then the rational cohomology algebra of mapGrn,k,Grn,k+r;in,r contains a truncated algebra ℚx/xr+n+k2−nk where x=2, for k≥2 and n≥4.Proof.
Considerin,r:Grk,nGrk,n+r. Let(16)∧V,d=∧x2,x4,x6,…,x2k,x2n+r−k+1,…,x2n+r−1,dbe the minimal Sullivan model of Grk,n+r with dx2=dx4=dx6=⋯=dx2k=0, dx2n+r−k+1=hn+r−k+1,…,dx2n+r−1=hn+r. Moreover,(17)B=H∗Grk,n,ℚ=∧y2,y4,y6,…,y2khn−k+1,…,hk.
A Sullivan model ofin,r is given by ϕ:∧V,d⟶B, where ϕx2=y2,ϕx4=y4,ϕx6=y6,…,ϕx2k=y2k, and ϕx2n+2r−2k+1=⋯=ϕx2n+2r−1=0.
Note that the lowest odd degree derivation is of degree2k−1 as k≥2; then, L1=0. Define γ2=x2k,y2k−2. Then γ2 is a cycle. Moreover,(18)b1=λmγ2,γ2,…,γ2x2n+r−k+1,⋮bk=λmγ2,γ2,…,γ2x2n+r−1,are polynomials of degree ≥2n+2r−2k+2−2m. As Grk,n is of dimension 2kn−k, if m<r+n+k2+1−nk−k, then b1,b2,…,bk are of degree >2kn−k.
Therefore,λmγ2,γ2,…,γ2=0 for m=2,…,r+n+k2+1−nk−k. In C∞s−1L, let γ2∗=φ2. Then φ2i≠0 for i=1,…,r+n+k2+1−nk−k.Theorem 4.
Ifr>n−kk−1+2, then the cohomology algebra of mapGrk,n,Grk,n+r;in,r contains a polynomial algebra over a generator of degree 2.Proof.
Let(19)ϕ:∧V=∧x2,x4,x6,…,x2k,x2n+r−k+1,…,x2n+r−1,d⟶B=∧y2,y4,y6,…,y2khn−k+1,…,hk,be a model of the inclusion. By hypothesis, r>n−kk−1+2 and n≥4. Then any odd derivation is of degree at least 2n+2r−2k−2nk+2k2+1. Hence, L1=L3=⋯=Lf=0 where f=2n+2r−2k−2nk+2k2−1. As defined in the proof of Theorem 3, consider γ2=x2k,y2k−2. It suffices to show that λmγ2,γ2,…,γ2=0 for m≥1. Consider n+r even, the other case is dealt with in a similar way.
Form>n+r/2, λmγ2,…,γ2=0. Assume 2≤m≤n+r/2,(20)c1=λmγ2,γ2,…,γ2x2n+2r−2k+1,⋮ck=λmγ2,γ2,…,γ2x2n+2r−1,are polynomials of total degree ≥2n+2r−2k+2−2m. If r>n−kk−1+2, then 2n+2r−2k+2−2m>−k+2−k2−1+nk. Hence, c1=⋯=ck=0.
This implies that−k+2−k2−1+nk>2kn−k. Therefore, λmγ2,…,γ2=0 for all m≥0. This shows that H∗C∞L contains a polynomial algebra over a generator of degree 2.



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