Rationalized evaluation subgroups of mapping spaces between complex Grassmannians

Abstract

We determine evaluation subgroups of the inclusion $Gr(2,n)\hookrightarrow Gr(2,n+1)$ between complex Grassmannians.

1 Introduction

Given a pointed topological space $(X,x_{0})$, the nth Gotttlieb group of X, also called the evaluation subgroup of $\pi _{n}(X)$ and denoted by $G_{n}(X)$, consists of those $\alpha \in \pi _{n}(X)$ for which there is a map $F : X \times S^{n} \longrightarrow X$ such that the following diagram commutes:

where $f: S^{n}\longrightarrow X$ is a representative of $\alpha$ and $\nabla$ is the folding map. Thus for every $\alpha \in G_{n}(X,x_{0})$, there exists at least one map $F:X \times S^{n}\longrightarrow X$ such that $F(x_{0},s)=f(s)$. We say that F is an affiliated map to $\alpha$ [3]. If X has a base point $x_{0}$ and $aut X$ denotes the monoid of self homotopy equivalences of X with $ev :aut X \longrightarrow X$ the evaluation map at $x_{0}$, then it follows from the definition that

$$\begin{aligned} G_{n}(X)=im \Big ( ev_{\sharp }:\pi _{n}(aut X,1_{X}) \longrightarrow \pi _{n}(X,x_{0}) \Big ) . \end{aligned}$$

Similarly, if $f: X \longrightarrow Y$ 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 $G_{n}(Y,X;f)= im ( ev_{\sharp }:map(X,Y;f)\longrightarrow \pi _{n}(Y) )$ is the nth evaluation subgroup of f [9]. In [10], Woo and Lee defined relative evaluation subgroups $G_{n}^{rel}(X,Y;f)$ and showed that they fit in a sequence

$$\begin{aligned} \cdots \longrightarrow G_{n+1}^{rel}(Y,X;f)\longrightarrow G_{n}(X)\longrightarrow G_{n}(Y,X;f) \longrightarrow \cdots \end{aligned}$$

called the G-sequence of f.

We use Sullivan models to compute rational relative Gottlieb groups of the inclusion $Gr(2,n)\hookrightarrow Gr(2,n+1)$. We refer to [4] for details and work over a field of characteristic zero in this case ${\mathbb {Q}}$.

Definition 1

A differential graded algebra (dga) is a graded algebra $A=\bigoplus _{n\ge 0}A^{n}$ together with a derivation d, $d=d_{n}:A^{n} \longrightarrow A^{n+1}$ such that $d \circ d=0$. Then (A, d) is called a cochain algebra. A graded algebra A is commutative if $a\cdot b=(-1)^{deg a \cdot deg b } b \cdot a$ for $a, b\in A$ [2, Chap. 3].

Definition 2

A Sullivan algebra is a commutative cochain algebra of the form $(\wedge V, d)$ where $V = \left\{ V^{p} \right\} _{p\ge 2}$ and $\wedge V$ denotes the graded free commutative algebra on V. A Sullivan model for a commutative cochain algebra (A, d) is a quasi-isomorphism $m:(\wedge V,d) \longrightarrow (A,d)$ from a Sullivan algebra $(\wedge V,d)$. A Sullivan algebra is said to be minimal if the differential is decomposable, that is, $Im d \subset \wedge ^{+}V \cdot \wedge ^{+}V$. Moreover, if $H^{0}(A)={\mathbb {Q}}$ then (A, d) has a minimal model which is unique up to isomorphism. If X is a nilpotent space and $A_{PL}(X)$ the commutative differential graded algebra (cdga) of piecewise linear forms on X, then a Sullivan model of X is a Sullivan model of $A_{PL}(X)$ [2, Chap.12].

2 Derivation spaces and the rationalized G-sequence

Given commutative differential graded algebras $(A, d_{A})$ and $(B, d_{B})$ and a map $\phi : A \longrightarrow B$, define a $\phi$-derivation of degree n to be a linear map $\theta :A^{*} \longrightarrow B^{*-n}$ which satisfies $\theta (xy)=\theta (x)\phi (y)+ (-1)^{n|x|}\phi (x)\theta (y)$. We only consider derivations of positive degree. Let $Der_{n}(A,B;\phi )$ denote the vector space of all $\phi$-derivations of degree n for $n>0$. Define a linear map $D:Der_{n}(A,B;\phi )\longrightarrow Der_{n-1}(A,B;\phi )$ by $D(\theta )=d_{B}\circ \theta - (-1)^{|\theta |}\theta \circ d_{A}$. Then, $(Der_{*}(A,B;\phi ),D)$ is a chain complex. In case $A=B$ and $\phi =1_{B}$, the chain complex of derivations $Der_{*}(B,B;1)$ is just the usual complex of derivations on the commutative differential graded algebra B [6]. If $\phi :(\wedge V,d)\longrightarrow (\wedge W,d)$ is a Sullivan minimal model of $f:X \longrightarrow Y$, then $H_{n}(Der(\wedge V,\wedge W;\phi ),D ) \cong \pi _{n}( map(X,Y;f))\otimes {\mathbb {Q}}; n\ge 2$ [6, 1].

We note that $Der(\wedge V,B;\phi )\cong Hom(V,B)$. If $\lbrace v_{i} \rbrace$ is a basis of V, we denote by $(v_{i},b)$, the unique $\phi$-derivation $\theta$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \theta (v_{i})=b_{i}&{} b_{i}\in B, \\ \theta (v_{j})=0 &{} i \ne j. \end{array}\right. } \end{aligned}$$

Pre-composition with $\phi$, respectively post-composition with the augmentation $\varepsilon : B \longrightarrow {\mathbb {Q}}$, gives a map of chain complexes

$$\begin{aligned} \phi ^{*}:Der_{*}(B,B;1)\longrightarrow Der_{*}(A,B;\phi ), \end{aligned}$$

respectively

$$\begin{aligned} \varepsilon _{*}:Der_{*}(A,B;\phi )\longrightarrow Der_{*}(A,{\mathbb {Q}};\varepsilon ). \end{aligned}$$

Definition 3

Let $\phi : V \longrightarrow W$ be a map of differential graded vector spaces. Define a differential graded vector space, $Rel_{*}(\phi )$, called the mapping cone as follows. $Rel_{n}(\phi )=sV_{n-1} \oplus W_{n}$ with differential $\delta$ of degree $-1$ given by $\delta (sv, w)=(-sd_{V}(v),\phi (v)+d_{W}(w) )$ [7]. There are chain maps $J: W_{n}\longrightarrow Rel_{n}(\phi )$ and $P: Rel_{n}(\phi )\longrightarrow V_{n-1}$ 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 $H(\phi )$. We refer to this sequence as the long exact homology sequence of $\phi$.

We use Theorem 3.3 of [6]

Theorem

Let $f: X \longrightarrow Y$ be a map between simply connected CW-complexes and $\phi :(\wedge V,d)\longrightarrow (\wedge W,d)$ its Sullivan model. The long exact sequence induced by $f_{*}:map(X,X;1)\longrightarrow map(X,Y;f)$ on rational homotopy groups is equivalent to the long exact homology sequence of the map

$$\begin{aligned} \phi ^{*}:Der_{*}(\wedge W,\wedge W;1)\longrightarrow Der_{*}(\wedge V,\wedge W;\phi ) \end{aligned}$$

induced by the minimal model $\phi :(\wedge V,d)\longrightarrow (\wedge W,d)$ of the map $f:X \longrightarrow Y$.

Definition 4

Given a commutative differential graded algebra map $\phi :A \longrightarrow B$, we have the following commutative diagram of differential graded vector spaces;

Here, $\varepsilon$ 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 $(n\ge 2)$,

Definition 5

Suppose $\phi : A \longrightarrow B$ is a map of commutative differential graded algebras, we define the evaluation subgroups of $\phi$ by

$$\begin{aligned} G_{n}(A,B;\phi )= im\lbrace H(\varepsilon _{*}) :H_{n}(Der(A,B;\phi )) \longrightarrow H_{n}(Der(A,{\mathbb {Q}};\varepsilon )) \rbrace . \end{aligned}$$

In the special case $A=B$ and $\phi =1_{B}$, we refer to the Gottlieb group of B, and use the notation $G_{n}(B)$. If $B=(\wedge V,d)$ is a model of a simply connected space X, then an element $\alpha \in G_{n}(B)$ is represented by a linear mapping $f: V^{n}\longrightarrow {\mathbb {Q}}$ that extends into a derivation $\theta$ of $\wedge V$ such that $\delta \theta =0$. Moreover, $G_{n}(B)\cong G_{n}(X_{{\mathbb {Q}}})$ [2, Proposition 29.8].

The nth relative evaluation subgroup of $\phi$ is defined by;

$$\begin{aligned} G_{n}^{rel}(A,B;\phi )=im\lbrace H(\varepsilon _{*},\varepsilon _{*}):H_{n}(Rel(\phi ^{*}))\longrightarrow H_{n}(Rel(\widehat{\phi ^{*}})) \rbrace . \end{aligned}$$

Then the image of the upper long sequence in the lower, of the ladder above, gives a sequence

(1)

that terminates in $G_{2}(A,B;\phi )$.

We refer to this sequence as the G-sequence of the map $\phi : A \longrightarrow B$. This can be applied to the minimal model $\phi :(\wedge V,d) \longrightarrow (\wedge W,d)$ of the map $f :X \longrightarrow Y$ as stated and proved in [6, Theorem 3.5].

3 The inclusion $Gr(2,n) \hookrightarrow Gr(2,n+1)$

Let Gr(k, n) be the Grassmann manifold of k-dimensional subspaces of ${\mathbb {C}}^{n}$. The cohomology ring $H^{*}(Gr(k,n), {\mathbb {Q}})$ is generated by the Chern classes $c_{i}\in H^{2i}(Gr(k,n),{\mathbb {Q}})$, for $1\le i \le k$. Further, the cohomology ring has a presentation

$$\begin{aligned} H^{*}(Gr(k,n),{\mathbb {Q}})=\wedge (c_{1},c_{2},\ldots , c_{k}) / ( h_{n-k+1},\ldots ,h_{n} ), \end{aligned}$$

as the quotient of the polynomial ring generated by $c_{1},c_{2}, \ldots , c_{k}$, $\vert c_{i} \vert =2i$, modulo the ideal generated by the elements $h_{j},n-k+1\le j\le n$. Here, $h_{j}$ is defined as the 2jth degree term in the Taylor’s expansion of $(1+c_{1}+c_{2}+c_{3}+ \cdots + c_{k})^{-1}$ where $(1+c_{1}+c_{2}+c_{3}+ \cdots +c_{k})$ is the total Chern class [4].

In particular, the cohomology rings of Gr(2, n) and $Gr(2, n+1)$ are:

• $H^{*} (Gr(2,n),{\mathbb {Q}})=\wedge (y_{2},y_{4})/ (h_{n-1},h_{n})$ and

• $H^{*} (Gr(2,n+1),{\mathbb {Q}})=\wedge (x_{2},x_{4})/ (h_{n},h_{n+1})$ respectively.

The minimal model of Gr(2, n) is $(\wedge (y_{2},y_{4},y_{2n-3},y_{2n-1}),d )$ with $d(y_{2})=d(y_{4})=0$, $d(y_{2n-3})=h_{n-1}$, $d(y_{2n-1})=h_{n}$. In the same way, a model of $Gr(2,n+1)$ is given by $( \wedge (x_{2},x_{4},x_{2n-1},x_{2n+1}),d )$ with $dx_{2}=dx_{4}=0$, $dx_{2n-1}=h_{n}$ and $dx_{2n+1}=h_{n+1}$.

Lemma 1

$h_{n+1}=-x_{2}h_{n}-x_{4}h_{n-1}$.

Proof

Write the Taylor series $(1+x_{2}+x_{4})^{-1}=1+h_{1}+h_{2}+ \cdots$ where, $|h_{i}|=2i$. From $(1+x_{2}+x_{4})(1+x_{2}+x_{4})^{-1}=1$, one gets the relation $h_{n+1}=-x_{2}h_{n}-x_{4}h_{n-1}$$\square$

In particular, $h_{n+1}$ is co-boundary in $( \wedge (x_{2},x_{4},x_{2n-3},x_{2n-1}),d )$, that is, there exists $\alpha$ of degree $2n+1$ such that $d\alpha =h_{n+1}$.

Theorem 1

Let $B= (\wedge (y_{2},y_{4},y_{2n-3},y_{2n-1}),d )$. Then $G_{n}(B)=\langle [ y_{2n-3}^{*} ], [ y_{2n-1}^{*} ] \rangle$.

Proof

Let $\alpha _{2n-1}=(y_{2n-1},1)$ and $\alpha _{2n-3}=(y_{2n-3},1)$. Then $\delta \alpha _{n-1}=\delta \alpha _{2n-3}=0$. Moreover, $\alpha _{2n-3}$ and $\alpha _{2n-1}$ can not be boundaries for degree reason. Therefore, $[ \alpha _{2n-3} ]$ and $\left[ \alpha _{2n-1} \right]$ are non zero homology classes in $H_{*}( Der(B,B;1))$. Further, $\varepsilon _{*}(\alpha _{2n-3})=y_{2n-3}^{*}$ and $\varepsilon _{*}(\alpha _{2n-1})=y_{2n-1}^{*}$.

As Gr(2, n) is a finite CW-complex then $G_{even}(B)=0$ [2, Pg.379] . Hence, $G_{n}(B)= \left\langle \left[ y_{2n-3}^{*} \right] ,\left[ y_{2n-1}^{*} \right] \right\rangle$. $\square$

The inclusion $Gr(2,n)\hookrightarrow Gr(2,n+1)$ has a model of the form

$$\begin{aligned} \phi :\wedge V= \Big ( \wedge (x_{2},x_{4},x_{2n-1},x_{2n+1}),d \Big ) \longrightarrow \Big ( \wedge (y_{2},y_{4},y_{2n-3},y_{2n-1}),d \Big )=B \end{aligned}$$

where $\phi (x_{2})=y_{2}$, $\phi (x_{4})=y_{4}$, $\phi (x_{2n-1})=y_{2n-1}$ and $\phi (x_{2n+1})=\alpha$ where $d\alpha =h_{n+1}$ by Lemma 1.

Theorem 2

Consider the inclusion $Gr(2,n)\hookrightarrow Gr(2,n+1)$ and $\phi :(\wedge V,d)\longrightarrow (B,d)$ its Sullivan model, then $G_{*}(\wedge V, B;\phi ) \cong \langle [ x_{2n-1}^{*}] ,[ x_{2n+1}^{*}] \rangle$.

Proof

As Gr(2, n) is formal, $Der(\wedge V,B;\phi ) \overset{\simeq }{\longrightarrow } Der (\wedge V, H(B);f \circ \phi )$ where $f:B\xrightarrow {\simeq }H(B)$ is a quasi isomorphism. Similarly, since B is formal $Der(B,B;1)\simeq Der(B,H(B);f)$. Define $\theta _{2n-1}=(x_{2n-1},1)$, $\theta _{2n+1}=(x_{2n+1},1)$ in $Der(\wedge V,H(B);f \circ \phi )$. Then $\delta \theta _{2n-1}=\delta \theta _{2n+1}=0$. Moreover, $[\theta _{2n-1} ]$ and $[\theta _{2n+1} ]$ are nonzero cohomology classes in $H_{*}( Der(\wedge V,H^{*}(B);f \circ \phi ))$.

We note that, $\theta _{2}=(x_{2},1)$ and $\theta _{4}=(x_{4},1)$ are not cycles in $Der (\wedge V, H(B);f \circ \phi )$ [8].

Further, $H(\varepsilon _{*})([\theta _{2n-1} ]) = [x^{*}_{2n-1} ] \in G_{2n-1}(\wedge V,B;f \circ \phi )$. In a similar way, $H(\varepsilon _{*})([ \theta _{2n+1} ]) = [x^{*}_{2n+1} ]\in G_{2n+1}(\wedge V,B;f \circ \phi )$. It then follows that $G_{*}(\wedge V,B;f \circ \phi )= \langle [x^{*}_{2n-1}], [x^{*}_{2n+1} ] \rangle$. $\square$

Theorem 3

Consider the inclusion $Gr(2,n)\hookrightarrow Gr(2,n+1)$ and

$$\begin{aligned} \phi : (\wedge V,d)&=\Big (\wedge (x_{2},x_{4},x_{2n-1},x_{2n+1}),d \Big ) \\&\longrightarrow \Big (\wedge V(y_{2},y_{4},y_{2n-3},y_{2n-1}),d \Big )=B \end{aligned}$$

its Sullivan model, then $G_{*}^{rel}(\wedge V,B;\phi )= \langle [ ( x_{2n-3}^{*},0)] , [ ( 0, y_{2n+1}^{*})] \rangle$.

Proof

Consider the diagram below [6].

Let $\alpha _{2n-1}=(y_{2n-1},1)$, $\alpha _{2n-3}=(y_{2n-3},1) \in Der(B,H(B);f)$ and $\theta _{2n-1}$, $\theta _{2n+1} \in Der(\wedge V,H^{*}(B);\phi )$ as defined above. Then $\phi ^{*}(\alpha _{2n-1})=\theta _{2n-1}$ and $\phi ^{*}(\alpha _{2n-3})=0$.

Further, $D(\alpha _{2n-1},0)=(0,\theta _{2n-1})$, $D(\alpha _{2n-3},0)=(0,0)$ and $D(0,\theta _{2n-1},0)=0=D(0,\theta _{2n+1})$. Therefore, $[(\alpha _{2n-3},0) ]$ and $[(0,\theta _{2n+1}) ]$ are non zero homology classes in $H_{*} (Rel(\phi ^{*}) )$. Moreover, $H(\varepsilon _{*},\varepsilon _{*})([\alpha _{2n-3},0 ] )= [(x^{*}_{2n-3},0) ]$ and $H(\varepsilon _{*},\varepsilon _{*})([(0,\theta _{2n+1}) ] )= [(0,y^{*}_{2n+1}) ]$. A straightforward computation shows that $[(x^{*}_{2n-3},0) ]$ and $[(0,y^{*}_{2n+1}) ]$ span $H(\varepsilon _{*},\varepsilon _{*})$. $\square$

The G-sequence reduces to

and is exact.

Example 1

Consider $Gr(2,4)\hookrightarrow Gr(2,5)$. A model of the inclusion is given by

$$\begin{aligned} \phi :\wedge V= \Big ( \wedge (x_{2},x_{4},x_{7},x_{9}),d \Big ) \longrightarrow \Big (\wedge (y_{2},y_{4},y_{5},y_{7}),d \Big )=B, \end{aligned}$$

where $dx_{2}=dx_{4}=0, dx_{7}=x_{4}^{2}-3x^{2}x_{4}+x_{2}^{4}$, $dx_{9}=4x_{2}^{3}x_{4}-3x_{2}x_{4}^{2}-x_{2}^{5}$$dy_{2}=dy_{4}=0$, $dy_{5}=2y_{2}y_{4}-y_{2}^{3}$ and $dy_{7}=y_{4}^{2}-3y_{2}^{2}y_{4}+y_{2}^{4}$.

Moreover, $\phi (x_{2})=y_{2}$, $\phi (x_{4})=y_{4}$, $\phi (x_{7})=y_{7}$ and $\phi (x_{9})=-y_{2}y_{7}-y_{4}y_{4}$.

We compute $G_{*}^{rel}(\wedge V,B;\phi )$. Let $\alpha _{7}=(y_{7},1), \alpha _{5}=(y_{5},1) \in Der_{7}(B,H(B);f)$ where $f:B \overset{\cong }{\rightarrow } H(B)$ and $\theta _{7}=(x_{7},1)$, $\theta _{9}=(x_{9},1)\in Der(\wedge V,H^{*}(B);f\circ \phi )$ then $\phi ^{*}(\alpha _{7})=\theta _{7}$ and $\phi ^{*}(\alpha _{5})=0$. Moreover, $D(s\alpha _{7},0)=(0,\theta _{7})$, $D(s\alpha _{5},0)=(0,0)$ and $D(0,\theta _{7})=D(0,\theta _{9})=(0,0)$. Hence $\left[ (s\alpha _{5},0) \right]$ and $\left[ (0,\theta _{9}) \right]$ are non zero homology classes. Moreover, $(\epsilon _{*},\epsilon _{*})(s\alpha _{5},0)=(sy_{5}^{*},0)$, ($\epsilon _{*},\epsilon _{*}) (0,\theta _{9})=(0,y_{9}^{*})$. Therefore

$$\begin{aligned} G_{*}^{rel}(\wedge V,B ;\phi )= \left\langle \left[ (0,x_{9}^{*}) \right] ,\left[ ( sy_{5}^{*},0) \right] \right\rangle . \end{aligned}$$