Rationalized evaluation subgroups of mapping spaces between complex Grassmannians


<jats:p>We determine evaluation subgroups of the inclusion <jats:inline-formula><jats:alternatives><jats:tex-math>$$Gr(2,n)\hookrightarrow Gr(2,n+1)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mi>r</mml:mi><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo><mml:mo>↪</mml:mo><mml:mi>G</mml:mi><mml:mi>r</mml:mi><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> between complex Grassmannians.</jats:p>


Introduction
Given a pointed topological space (X , x 0 ), the nth Gotttlieb group of X , also called the evaluation subgroup of π n (X ) and denoted by G n (X ), consists of those α ∈ π n (X ) for which there is a map F : X × S n −→ X such that the following diagram commutes: x 0 , then it follows from the definition that G n (X ) = im ev : π n (aut X, 1 X ) −→ π n (X , x 0 ) .
Similarly, if f : X −→ Y is a based map between simply connected CW complexes and map(X , Y ; f ), the space of maps from X to Y which are homotopic to f , then G n (Y , X ; f ) = im(ev : map(X , Y ; f ) −→ π n (Y )) is the nth evaluation subgroup of f [9]. In [10], Woo and Lee defined relative evaluation subgroups G rel n (X , Y ; f ) and showed that they fit in a sequence · · · −→ G rel n+1 (Y , X ; f ) −→ G n (X ) −→ G n (Y , X ; f ) −→ · · · called the G-sequence of f . We use Sullivan models to compute rational relative Gottlieb groups of the inclusion Gr(2, n) → Gr(2, n + 1). We refer to [4] for details and work over a field of characteristic zero in this case Q.
Definition 2 A Sullivan algebra is a commutative cochain algebra of the form (∧V , d) where V = {V p } p≥2 and ∧V denotes the graded free commutative algebra on V . A Sullivan model for a commutative cochain algebra has a minimal model which is unique up to isomorphism. If X is a nilpotent space and A P L (X ) the commutative differential graded algebra (cdga) of piecewise linear forms on X , then a Sullivan model of X is a Sullivan model of A P L (X ) [2, Chap.12].

Derivation spaces and the rationalized G-sequence
Given commutative differential graded algebras (A, d A ) and (B, d B ) and a map φ : A −→ B, define a φ-derivation of degree n to be a linear map θ : A * −→ B * −n which satisfies θ(x y) = θ(x)φ(y) + (−1) n|x| φ(x)θ (y). We only consider derivations of positive degree. Let Der n (A, B; φ) denote the vector space of all φ-derivations of degree n for n > 0. Define a linear map D : is a chain complex. In case A = B and φ = 1 B , the chain complex of derivations Der * (B, B; 1) is just the usual complex of derivations on the commutative differential graded algebra B [6].
Pre-composition with φ, respectively post-composition with the augmentation ε : B −→ Q, gives a map of chain complexes Definition 3 Let φ : V −→ W be a map of differential graded vector spaces. Define a differential graded vector space, Rel * (φ), called the mapping cone as follows. Rel n (φ) = sV n−1 ⊕ W n with differential δ of degree −1 given by [7]. There are chain maps J : W n −→ Rel n (φ) and P : Rel n (φ) −→ V n−1 defined by J (w) = (0, w) and P(sv, w) = v. These give a short exact sequence of chain complexes which leads to a long exact sequence in homology whose connecting homomorphism is H (φ). We refer to this sequence as the long exact homology sequence of φ.

We use Theorem 3.3 of [6]
Theorem Let f : X −→ Y be a map between simply connected C W -complexes and φ : Here, ε denotes the augmentation of either A or B. On passing to homology and using the naturality of the mapping cone construction, we obtain the following homology ladder (n ≥ 2), H n (Der(B, B; 1)) H n (Der(A, B; φ) is a map of commutative differential graded algebras, we define the evaluation subgroups of φ by The nth relative evaluation subgroup of φ is defined by; Then the image of the upper long sequence in the lower, of the ladder above, gives a sequence that terminates in G 2 (A, B; φ).
We refer to this sequence as the G-sequence of the map φ :

The inclusion Gr(2, n) → Gr(2, n + 1)
Let Gr(k, n) be the Grassmann manifold of k-dimensional subspaces of C n . The cohomology ring H * (Gr(k, n), Q) is generated by the Chern classes c i ∈ H 2i (Gr(k, n), Q), for 1 ≤ i ≤ k. Further, the cohomology ring has a presentation H * (Gr(k, n), as the quotient of the polynomial ring generated by c 1 , c 2 , . . . , c k , |c i | = 2i, modulo the ideal generated by the elements h j , n − k + 1 ≤ j ≤ n. Here, h j is defined as the 2 jth degree term in the Taylor's expansion of (1+c 1 +c 2 +c 3 +· · ·+c k ) −1 where (1+c 1 +c 2 +c 3 +· · ·+c k ) is the total Chern class [4].