Research Article
Relative Gottlieb Groups of Embeddings between
Complex Grassmannians
J. B. Gatsinzi
Department of Mathematics and Statistical Sciences, Botswana International University of Science and Technology,
Palapye, Botswana
CorrespondenceshouldbeaddressedtoJ.B.Gatsinzi;jeangatsinzi@yahoo.fr
Received 15 October 2021; Accepted 27 October 2021; Published 15 November 2021
AcademicEditor:LucaVitagliano
Copyright©2021J.B.Gatsinzi.isisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
LetGr(k, n) bethecomplexGrassmannmanifoldof k-linearsubspacesin C
n
.WecomputerationalrelativeGottliebgroupsofthe
embedding i: Gr(k, n) ⟶ Gr(k, n + r) and show that the G-sequence is exact if r ≥ k(n − k).
1. Introduction
Weworkinthecategoryofspaceshavingthehomotopytype
ofsimplyconnectedCWcomplexesoffinitetype.Wedenote
by h: X ⟶ X
Q
the rationalization of X [1, 2]. Let
f: (X, x
0
) ⟶ (Y, y
0
) be a pointed continuous mapping
andmap(X, Y; f) bethecomponentof f inthespaceofall
continuousmaps g: X ⟶ Y.Considertheevaluationmap
ev: map(X, Y; f) ⟶ Y at the base point x
0
, that is,
ev(g)� g(x
0
). e nth evaluation subgroup of f,
G
n
(Y, X; f), is the image of π
n
(ev) in π
n
(Y) [3]. In the
special case where X � Y and f � 1
X
, one obtains the
Gottlieb group G
n
(X) of X [4]. Gottlieb groups play an
importantroleintopology.Forinstance,if G
n
(X)� 0,then
any fibration X ⟶ E ⟶ S
n+1
admits a section (Corollary
2–7 in [4]).
In[2],LeeandWoointroducerelativeevaluationgroups
G
rel
n
(Y, X; f) and obtain a long sequence,
··· ⟶ G
rel
n+1
(Y, X; f) ⟶ G
n
(X) ⟶ G
n
(Y, X; f)
⟶ G
rel
n
(Y, X; f) ⟶ ··· ,
(1)
calledG-sequence[5].issequenceisexactinsomecases,
for instance, if f is a homotopy monomorphism [6].
2. Rational Relative Gottlieb Groups
e rationalization h: Y ⟶ Y
Q
induces a rationalization
h
∗
: map(X, Y; f) ⟶ map(X, Y; h ∘ f) [7]. erefore,
ev
∗
π
∗
(map(X,Y; f)) ⊗ Q ( � ev
∗
π
∗
map X,Y
Q
; h ∘ f ( ( ( .
(2)
In this paper, we study the G-sequence of the natural
inclusion Gr(k, n) ⟶ Gr(k, n + r) using models of func-
tion spaces in rational homotopy [8, 9]. In particular, we
show that the G-sequence is exact if r ≥ k(n − k).Wework
with algebraic models in rational homotopy theory intro-
duced by Sullivan and Quillen [10, 11]. In this section, we
give relevant definitions and fix notation. Details can be
foundin[1].Allvectorspacesandalgebrasareoverthefield
of rational numbers Q.
Let (A, d) be a cochain algebra. e degree of an ho-
mogeneous element a ∈ A
p
is written |a|. We assume that
Hindawi
International Journal of Mathematics and Mathematical Sciences
Volume 2021, Article ID 1417769, 7 pages
https://doi.org/10.1155/2021/1417769