The Sectional Category of a Map
M. Arkowitz and J. Strom
Abstract
We study a generalization of the Svarc genus of a fiber map. For an
arbitrary collection E of spaces and a map f : X ! Y , we define a
numerical invariant, the E-sectional category of f, in terms of open
covers of Y . We obtain several basic properties of E-sectional cate-
gory, including those dealing with homotopy domination and homo-
topy pushouts. We then give three simple axioms which characterize
the E-sectional category. In the final section we obtain inequalities for
the E-sectional category of a composition and inequalities relating the
E-sectional category to the Fadell-Hussein category of a map and the
Clapp-Puppe category of a map.
MSC Classification Primary: 55M30, Secondary: 55P99
Keywords genus, sectional category, Lusternik-Schnirelmann cate-
gory, category of a map, homotopy pushout
1 Introduction
The sectional category of a fiber map f : X ! Y , denoted secat(f), is
one less than the number of sets in the smallest open cover of Y such
that f admits a cross-section over each member of the cover. This
simple and natural numerical invariant of fiber maps was developed
and studied extensively by Svarc [21] who called it the genus of f.
Subsequently Fet [11] and Berstein-Ganea [2] extended the definition
to arbitrary maps and related it to the Lusternik-Schnirelmann cat-
egory. There have been many applications of sectional category to
questions of classification of bundles, embeddings of spaces and exis-
tence of regular maps [21] as well as applications outside of algebraic
and differential topology [7, 10]. However, since Svarc's papers, the
1
actual study of sectional category has been sporadic and has often
appeared as subsidiary results within a larger work [2, x2], [15, x8], [3,
x4], [6, 9.3]. (An exception to this is the paper of Stanley [20] which
deals with sectional category in the context of rational homotopy the-
ory.)
Recently there has been considerable interest in the study of nu-
merical invariants of the homotopy type of spaces and of the homotopy
class of maps. Classically the Lusternik-Schnirelmann category and
cone length of a space have been studied in [16, 12, 13, 4] and, more
recently, the cone length [17], Clapp-Puppe category [3] and Fadell-
Husseini category of a map [9, 5] have been investigated.
In the paper [1], we gave a unified axiomatic development of many
of these invariants. For the category of spaces and maps and a fixed
collection E of objects, these axioms were of two basic types, namely,
those dealing with numerical functions relative to E defined on the
objects of the category and those dealing with numerical functions
relative to E defined on the morphisms of the category. By special-
izing the collection E, we obtained some of the previously studied
invariants as well as several new invariants. For instance, by setting
E = {all spaces}, our axioms for morphisms yield the Fadell-Husseini
category of a map. One invariant that was not dealt with in [1] was
the sectional category. Whereas the various versions of category and
cone length have been defined in numerous homotopy-invariant ways,
this is not the case for sectional category. The fact that sectional
category does not fit nicely into the general framework that so neatly
encapsulates the category and cone length of spaces and maps may
account for its marginal status in the decades since its introduction.
We begin in Section 3 by defining a generalization of the sectional
category of a map with respect to a collection E of spaces using open
covers of the target of the map. This is a straightforward extension
of the classical definition. We derive several simple basic properties
of this invariant. In particular, when E = {all spaces}, we see that
E_ secat(f) = secat(f). This treatment of sectional category leads to
new invariants obtained by varying the collection E of spaces.
In Section 4 we bring sectional category of maps in line with the
other invariants studied in [1]. This is done by considering maps as
objects in the category whose objects are maps of spaces and whose
morphisms are given by commutative squares. In this category, we
apply the axiomatic approach for invariants of objects, relative to the
collection of all maps with sections which factor through a space in
2
E. The unique numerical invariant obtained from the axioms is then
proved to be E_ secat. This is the content of Theorem 4.7.
Another case of this axiomatic approach is obtained by working in
the category of maps, relative to the the collection of all maps which
factor through some space in E. The unique invariant obtained from
the axioms can then be shown to be the E_ Clapp-Puppe category of
a map (see Remark 4.8). We hope to return to this in the future.
All invariants that fall into our axiomatic scheme will of course
share certain basic properties that follow formally from the axioms. A
major interest, however, is in the new questions which arise regarding
E_ sectional category. Many of these are considered in Section 5 and
take the form of inequalities. There we concentrate on the following:
(1) How is E_ secat(f) related to the domain and the target of f? (2)
How does E_ sectional category behave with respect to composition of
maps? (3) What is the relation between the E_ sectional category, the
Clapp-Puppe category and the Fadell-Husseini category?
2 Preliminaries
In this section we establish our notation and recall some basic def-
initions and results which we shall use. All spaces are to have the
homotopy type of connected CW-complexes. We do not assume that
spaces have base points and hence maps are not base point preserving.
For spaces X and Y , X Y denotes that X and Y have the same
homotopy type. We let * denote a space with one point and we also
write * : X ! Y for any constant map from X to Y . We denote by
idX or id the identity map of X. If f, g : X ! Y are two maps, then
f ' g signifies that f and g are homotopic. Given maps f : X ! Y
and g : Y ! X, if fg ' idY, we say that g is a section of f or that
Y is a retract of X. For maps f : X ! Y and f0 : X0 ! Y 0, if there
is a homotopy commutative diagram
X _____i____//_X0___r____//_X
|f| f0|| f||
fflffl|j fflffl|s fflffl|
Y __________//_Y_0________//Y
such that ri ' idand sj ' id, then we say that f0 dominates f or
that f is a retract of f0. If, in addition, ir ' idand js ' id, then f
and f0 are called equivalent, and we write f f0.
3
We will also use certain basic constructions in homotopy theory.
The pushout of a diagram of the form
g f
C oo_______A__________//_B
is the quotient space of B [ C by the equivalence relation which sets
f(a) equivalent to g(a), for every a 2 A. The homotopy pushout
H of the diagram is the quotient space of B [ (A x [0, 1]) [ C by
the equivalence relation which sets (a, 0) equivalent to f(a) and (a, 1)
equivalent to g(a), for every a 2 A. The pushout and homotopy
pushout constuctions are clearly functors from the category of given
diagrams to the category of spaces.
There are two homotopy pushouts of special interest. The first is
the homotopy pushout of
f
*oo________A _________//_B
which is the mapping cone of f and is denoted B [ CA. The second
is the homotopy pushout of
idA f
A oo_______A__________//_B
which is the mapping cylinder of f and is denoted Mf. We note for
later use that the homotopy pushout of
g f
C oo_______A__________//_B
is homeomorphic to the pushout of the associated mapping cylinder
diagram
Mg oo________A__________//Mf.
We next consider two versions of the category of a map. Given f :
X ! Y , we first define the (reduced) Fadell-Husseini category of
f, denoted catFH(f) [9]. We note that f is equivalent to the inclusion
of X into Mf, and so we regard f as an inclusion. Then catFH(f) is
the smallest n such that there is an open cover {U0, U1, . .,.Un} of Y
with the following properties: (1) if ji: Ui! Y is the inclusion, then
ji' * for i = 0, 1, ..., n, (2) X U0 and (3) there is a map r : U0 ! X
and a homotopy of pairs j0 ' r : (U0, X) ! (Y, X). It follows that
catFH(* ! Y ) is just the Lusternik-Schnirelmann category cat(Y ) of
Y .
4
By a collection E we mean any collection of spaces which con-
tains the one point space *. The second notion of category of a
map f : X ! Y that we consider is the (reduced) E_ Clapp-Puppe
category of f, denoted E_ catCP(f) [3]. This is the smallest non-
negative integer n with the following properties: (1) there exists an
open cover {U0, U1, . .,.Un} of X, (2) there exists spaces Ei 2 E and
maps ui : Ui ! Ei and vi : Ei ! Y , for i = 0, 1, . .,.n and (3)
f|Ui' viui for each i. For a space X, the E_ Clapp-Puppe category of
X is defined by E_ catCP(X) = E_ catCP(idX). If E = {*}, the collec-
tion consisting of a one point space, then E_ catCP(f) is the category
of the map f, as discussed by Berstein-Ganea [2] and others, which
we will denote by catBG(f). We note that catBG(idX) = cat(X).
3 Definition and Basic Properties
In this section we define the sectional category of a map relative to
a collection E of spaces. We then establish basic properties of this
invariant. Of particular importance are the Domination Proposition
(Prop. 3.6) and the Homotopy Pushout Theorem (Thm. 3.9).
Definition 3.1 1. If E is a collection, f : X ! Y a map and
i : U ,! Y the inclusion map, then U is E-section-categorical
(with respect to f) if there is space E 2 E and maps u : U ! E
and v : E ! X such that the following diagram is homotopy-
commutative:
EO_____v____//_XO
u || f||
| i fflffl|
U __________//_Y.
The map vu : U ! X is called an E_ section of f over U.
2. For a map f : X ! Y , the (reduced) E-sectional category
of f, written E_ secat(f), is the smallest integer n such that there
exists an open cover of Y by n + 1 subsets, each of which is E-
section-categorical. If no such integer exists, then E_ secat(f) =
1.
If E is the collection of all spaces, then we write secat(f) for E_ secat(f)
and note that secat(f) is just the sectional category of f as defined in
[2, Def. 2.1].
5
The following result lists a number of useful properties of E_ secat.
The proofs are all straighforward, and we omit them.
Proposition 3.2 Let E be a collection.
1. If f ' f0, then E_ secat(f) = E_ secat(f0).
2. If E = {*}, then E_ secat(f) = cat(Y ), the category of Y .
3. If E and F are two collections and F E, then E_ secat(f)
F_ secat(f).
4. If * : X ! Y is a constant map, then E_ secat(*) = cat(Y ).
Assertions (2) and (4) show that the Lusternik-Schnirelmann category
of Y can be obtained from the E-sectional category of f in two ways:
either by making E trivial or by making f trivial. Since {*} E
{all spaces}, we obtain from (3) the following.
Corollary 3.3 For any map f : X ! Y and collection E,
secat(f) E_ secat(f) cat(Y ).
Next we consider some basic properties of the identity map.
Proposition 3.4 Let E be a collection and X be a space.
1. E_ secat(idX) = E_ catCP(X).
2. X is a retract of a space in E if and only if E_ secat(idX) = 0.
In particular, secat(idX) = 0 for every space X.
The following examples show that the inequalities in Proposition
3.2 and Corollary 3.3 can be strict.
Example 3.5 1. For the inequality in 3.2, let E be the collection
of all spaces and F a collection which does not contain a space
having X as a retract. Then
E_ secat(idX) = 0 < F_ secat(idX).
For a specific example, let F = {*} and take any X 6 *.
2. For the inequality in 3.3, consider f : X ! Y and the inclusion
j : Y ! Y [ CX. Then secat(j) 1 by Lemma 5.3 (below). If
cat(Y [ CX) > 1, then
secat(j) 1 < cat(Y [ CX).
This is the case, for example, when f : S2n+1 ! CPn is the Hopf
map.
6
We next establish a basic property of E-sectional category. In the
next section this will be shown to be one of three properties which
characterize E_ secat.
Proposition 3.6 (Domination) If f : X ! Y is dominated by f0 :
X0 ! Y 0, then
E_ secat(f) E_ secat(f0).
Proof We are given j : Y ! Y 0and r : X0 ! X as in the definition
of domination in x2. Let E_ secat(f0) = n with E-section-categorical
cover {U00, U01, . .,.U0n} of Y 0and maps
ui vi 0
U0i_________//Ei_______//_X
such that f0viui' ji: U0i! Y 0, where Ei2 E and ji is the inclusion.
If Ui= j-1(U0i), then {U0, U1, . .,.Un} is an open cover of Y and
j|Ui 0ui vi 0r
Ui_____//Ui___//Ei___//X____//X
is the desired E_ section of f over Ui. 2
An immediate corollary of Proposition 3.6 is that E_ secatis an
invariant of homotopy equivalence of maps.
Corollary 3.7 If f f0, then E_ secat(f) = E_ secat(f0).
It is well-known that every map is homotopy equivalent to a fiber
map [18, p. 48]. Corollary 3.7 implies that the E-sectional category of
an arbitrary map is equal to the E-sectional category of the equiva-
lent fiber map. Therefore Svarc's definition of sectional category [21],
which applies only to fiber maps, is equivalent to Definition 3.1 in the
special case E = {all spaces}.
We next prove a result about the E_ sectional category of the maps
of one homotopy pushout into another. To establish notation, let
g f
C oo_________A ___________//B
|c| a|| |b|
|fflfflg0 fflffl|f0 fflffl|
C0 oo_________A0__________//_B0
be a commutative diagram, let D and D0 be the homotopy pushouts
of the top and bottom rows, respectively, and let d : D ! D0 be the
induced map. We begin with a lemma.
7
Lemma 3.8 If E_ secat(b) = n, then there exists E-section-categoricalS
open sets N0, N1, . .N.nof D0with respect to d such that Mf0 Ni.
Proof Let ed: Mf ! Mf0 be the map d with restricted domain and
target. Then ed b, so E_ secat(de) = E_ secat(b) = n by Corollary 3.7.
To complete the proof, observe that if U Mf0is E-section-categorical
with respect to ed, then it is also E-section-categorical with respect to
d. 2
Now we can prove the Homotopy Pushout Theorem, which is an-
other basic property which we use in x4 to characterize E_ sectional
category.
Theorem 3.9 (Homotopy Pushout) With the notation above,
E_ secat(d) E_ secat(b) + E_ secat(c) + 1.
Proof Let E_ secat(b) = n and E_ secat(c) = m. Let {U0, U1, . .,.Un}
be a minimal open E-section-categorical cover of B0 with respect to
b. Then there are E-section-categorical open sets N0, N1, . .,.Nn of
D0 with respect to d which cover Mf0 by Lemma 3.8. Similarly,
if {V0, V1, . .,.Vm } is a minimal open E-section-categorical cover of
C0 with respect to c, then there are E-section-categorical open sets
M0, M1, . .,.Mm of D0 with respect to d which cover Mg0. It follows
that {N0, . .,.Nn, M0, . .,.Mm } is an E-section-categorical cover of D0
with respect to d. Thus E_ secat(d) E_ secat(b) + E_ secat(c) + 1. 2
We conclude this section with a simple application of Theorem 3.9
to the maps in a homotopy pushout square.
Corollary 3.10 If
f
A __________//_B
|g| |r|
fflffl|s fflffl|
C __________//_D
is a homotopy pushout square, then
E_ secat(r) E_ secat(g) + E_ catCP(B) + 1.
8
Proof The map of homotopy pushouts obtained from the commuta-
tive diagram
f
A ___________A___________//_B
|g| |||| ||||
fflffl|g || f ||
C oo_________A __________//_B
is equivalent to r : B ! D. Now apply Theorem 3.9, using Proposition
3.4(1). 2
4 Axioms
In this section we characterize the E_ sectional category by simple ax-
ioms. For reasons given in Remark 4.8 we state our axioms in greater
generality than is needed for sectional category.
Definition 4.1 We denote by S a non-empty collection of maps. An
S-length function is a function fl = flS which assigns to every map
f an integer 0 fl(f) 1 such that
1. If f 2 S, then fl(f) = 0.
2. Let
g f
C oo_________A __________//_B
c|| |a| |b|
fflffl|g0 fflffl|f0 |fflffl
C0 oo________A0___________//_B0
be a commutative diagram with induced map d : D ! D0of the
homotopy pushout of the first row into the homotopy pushout
of the second row. If c 2 S, then fl(d) fl(b) + 1.
3. If f is dominated by f0, then fl(f) fl(f0).
We call (1)-(3) the axioms for S-length functions.
It is an immediate consequence of Axiom (3) that if f f0, then
fl(f) = fl(f0). Then, since homotopic maps are equivalent (x2), it
follows that f ' f0 implies that fl(f) = fl(f0).
Remark 4.2 These axioms are analogous to the axioms for the A-
category of a space given in [1, Prop. 5.6(2)]. The following comments
are made to elucidate the analogy. In Definition 4.1 we define the S-
length function on the objects in the category of maps of spaces. In [1,
9
p. 26] an S-length function was defined on the objects in the category
of spaces in the case where S is the collection of contractible spaces,
though we could have taken an arbitrary collection S in the definition.
This is why the analog of Axiom (3) in [1] deals with mapping cone
sequences instead of homotopy pushouts. In addition, the axioms in [1,
p. 26] were made with respect to an arbitrary collection A of spaces.
It is possible to incorporate an arbitrary collection A of maps into
Definition 4.1, but we have not given the definition in this generality.
We are led to the following definition.
Definition 4.3 The maximal S-length function S is defined by
S(f) = max {flS(f) | for everyS_ length functionflS}.
Note that S satisfies the axioms for S-length functions.
We wish to show that E_ secatequals S for some S. For this we
need to choose a collection S of maps that is closely related to E.
Definition 4.4 For a collection of spaces E, let
S(E) = {f : X ! Y | there is anE_ section off over Y }.
The main result of this section is that S(E)= E_ secat. In order to
prove this, we need to show that we can replace open covers with covers
by simplicial subcomplexes in the definition of E_ sectional category.
This is crucial in dealing with maps that are defined by homotopy
pushouts. We require the following lemma.
Lemma 4.5 Let f : X ! Y be a map and B Y an E-section-
categorical subset. If A B, then A is E-section-categorical. If B C
is a deformation retract, then C is E-section-categorical.
The proof is elementary, and we omit it.
Proposition 4.6 Let Y be a simplicial complex and f : X ! Y a
map. Then E_ secat(f) n () there is a simplicial structure on Y
such that Y can be covered by n+1 E-section-categorical subcomplexes
L0, L1, . .,.Ln.
10
Proof (() If Ni is the second regular neighborhood_of Li in Y , __
then Li is a deformation retract of the closure N i[8, p. 72]. Thus N i
is E-section-categorical, and hence so is the open set Ni by Lemma
4.5.
()) Consider an E-section-categorical cover {U0, U1, . .,.Un} of
Y . It is known that there is a simplicial structure on Y with sub-
complexes L0, . .,.Ln which cover Y such that Li Ui for each i [14,
Lem. 2.3]. The result follows from Lemma 4.5. 2
Theorem 4.7 If E is a collection of spaces and f : X ! Y is a map,
then
S(E)(f) = E_ secat(f).
Proof In the proof we write for S(E). To show E_ secat(f) (f),
it suffices to show that E_ secatsatisfies the axioms of Definition 4.1
for an S-length function. But this follows from x3.
Next we show (f) E_ secat(f). We suppose that E_ secat(f) <
1 and prove the result by induction on E_ secat(f). If E_ secat(f) = 0,
then f 2 S, and so (f) = 0. Suppose next that the inequality
holds for all maps g with E_ secat(g) < n and let f : X ! Y with
E_ secat(f) = n. Now X and Y have the homotopy type of simplicial
complexes [19, Thm. 2], so we may assume that X and Y are simplicial
complexes. By the Simplicial Approximation Theorem, we can take
f to be a simplicial map. By Proposition 4.6, there is an E-section-
categorical cover {L0, L1, . .,.Ln} of Y by subcomplexes with respect
to f. We set Ki = f-1 (Li) so that {K0, K1, . .,.Kn} is a cover of X
by subcomplexes.
Thus the following diagram is commutative
Kn oo_________Kn \ (K0 [ . .[.Kn-1)__________//_K0 [ . .[.Kn-1
|f1| |f2| |f3|
fflffl| fflffl| fflffl|
Ln oo_________Ln \ (L0 [ . .[.Ln-1)__________//_L0 [ . .[.Ln-1,
where each fi is induced by f. Clearly f1 2 S(E), and so (f1) = 0.
Furthermore, E_ secat(f3) n - 1, and hence (f3) n - 1 by our
inductive hypothesis. But the pushout of the rows of the diagram
yields the map f : X ! Y . However this is homotopy equivalent to the
homotopy pushout since all horizontal maps are simplicial inclusions
and hence cofibrations [18, p. 78]. By Definition 4.1, (f) (f3) +
1 n. This completes the proof. 2
11
Remark 4.8 1. For a collection E of spaces, the E_ Clapp-Puppe
category of f fits nicely into our axiomatic framework. Precisely,
let T (E) be the collection of all maps g : X ! Y such that
g ' vu, where u : X ! E and v : E ! Y , for some E 2 E. It
can be shown that E_ catCP(f) = T (E)(f) for all maps f.
2. The axioms of Definition 4.1 can be dualized in the sense of
Eckmann-Hilton. This just consists of replacing Axiom 3 with
an appropriate homotopy pull-back axiom. As in Definition 4.3,
we set S equal to the maximum all functions which satisfy the
dual axioms. If E is any collection of spaces, we define
S*(E) = {g : X ! Y | there existsh : Y ! X with hg ' idX}.
Then S*(E)(f), which we can denote by E_ cosecat(f), is the
dual of the E_ sectional category of f. It would be interesting to
investigate this invariant.
5 Inequalities
In this section we consider the questions raised in the introduction
dealing with the E-sectional category of a composition and the rela-
tionship between the various invariants. The main theorem of this
section is Theorem 5.4.
5.1 Composition of Maps
The composition results in this subsection are proved by elementary
covering arguments.
Proposition 5.1 For any maps f : X ! Y and g : Y ! Z,
E_ secat(g) E_ secat(gf).
The proof is obvious, and hence omitted.
The next result deals with maps that have sections.
Proposition 5.2 If f : X ! Y and g : Y ! Z, then
1. If f has a section, then E_ secat(gf) = E_ secat(g).
2. If g has a section, then E_ secat(gf) E_ secat(f).
12
Proof First assume that f has a section s. Let U Z be E-section-
categorical with respect to g. We claim that U is also E-section-
categorical with respect to gf. This follows easily from the following
homotopy-commutative diagram
ppX88
ppp
svpppp |f|
ppppv fflffl|
EO__________//_YO
u|| |g|
| fflffl|
U __________//_Z.
This proves E_ secat(gf) E_ secat(g), and Proposition 5.1 completes
the proof of (1). When g has a section t, one proves (2) by showing
that if U Y is E-section-categorical with respect to f, then t-1(U)
is E-section-categorical with respect to gf. 2
Next we derive additional results on the sectional category of a
composition. We begin with a lemma.
Lemma 5.3 If A-! Y -j!Y [ CA is a mapping cone sequence and
f : X- ! Y is a map, then
E_ secat(jf) E_ secat(f) + 1.
Proof Consider the diagram
* oo_________*___________//X
| | |
| | |f
fflffl| fflffl| fflffl|
* oo________A_ __________//_Y
which induces the map jf : X ! Y [ CA of homotopy pushouts. The
result now follows from Theorem 3.9. 2
Our main result gives an upper bound for the E-sectional category
of a composition.
Theorem 5.4 If f : X- ! Y and g : Y -! Z, then
E_ secat(gf) catFH(g) + E_ secat(f).
13
Proof Suppose catFH(g) = n. Then we can choose a decomposition
of g [1, x3]
j0 j1 jn-2 jn-1
Y0 _____//Y1___//_._._.//_Yn-1___//_YnOO
|| g ||s
|| n|| n
|| g fflffl||
Y ______________________________//_Z
where (1) g ' gnjn-1 . .j.1j0, (2) sng ' jn-1 . .j.1j0, (3) gnsn ' id
ji
and (4) thre is a mapping cone sequence Ai ____//_Yi__//Yi+1. Since
g is dominated by jn-1 . .j.1j0, it follows that gf is dominated by
jn-1 . .j.1j0f. By Lemma 5.3,
E-seccat(gf) E_ secat(jn-1 . .j.1j0f)
E_ secat(jn-2 . .j.1j0f) + 1
..
.
E_ secat(f) + n.
2
We conclude this subsection by showing that the inequality in The-
orem 5.4 can be strict.
Example 5.5 Let f : X ! Y be any map with E_ secat(f) > 0, let
Z = * and let g : Y ! Z be the constant map. Then
E_ secat(gf) = cat(*) = 0 < catFH(g) + E_ secat(f).
5.2 Relations Between Invariants
We begin this subsection with an inequality which provides a lower
bound for the Clapp-Puppe category of a composition.
Proposition 5.6 If f : X ! Y and g : Y ! Z, then
E_ catCP(g) + 1 (secat(f) + 1)(E_ catCP(gf) + 1).
Proof Write secat(f) = n and E_ catCP(gf) = m. Then we have a
cover {U0, U1, . .,.Un} of Y and sections si: Ui! X for each i. Now
E_ catCP(g|Ui) = E_ catCP(gfsi) E_ catCP(gf) = m,
14
so Ui= Vi0[ . .[.Vim with each g|Vijfactoring through some member
of E. Then Y = {Vij| i = 0, 1, 2, . .,.n; j = 0, 1, 2, . .,.m} is desired
cover of Y by (n + 1)(m + 1) E-categorical subsets. 2
If * is a constant map, then clearly E_ catCP(*) = 0. Thus if E =
{*} and gf ' *, then Proposition 5.6 yields the inequality
catBG(g) secat(f)
of Berstein-Ganea [2, Prop 2.6]. More generally, if gf ' *, then
E_ catCP(g) secat(f)
for any collection E by Proposition 5.6. This gives E_ catCP(g)
E_ secat(f) if gf ' *. However this inequality holds without the latter
assumption.
Proposition 5.7 Let f : X ! Y and g : Y ! Z. Then
E_ catCP(g) E_ catCP(Y ) E_ secat(f).
Proof The left inequality follows immediately. The right inequality
follows from the observation that if {U0, U1, . .,.Un} is an open cover
of Y which is E-section-categorical for f, then it is an open cover for
the E_ Clapp-Puppe category of idY . 2
Next we turn to a corollary to Theorem 5.4
Corollary 5.8 For any map f : X ! Y ,
E_ secat(f) catFH(f) + E_ catCP(X).
In particular, secat(f) catFH(f).
f
Proof Apply Theorem 5.4 to the composition X __id_//X___//_Yand
use Proposition 3.4(1). 2
This corollary is used to prove the following relationships between
the invariants E_ secat, E_ catCPand E_ catFH.
Proposition 5.9 Let E be a collection and f : X ! Y and g : Y ! Z
be maps. Then
1. E_ catCP(g) catFH(f) + E_ catCP(X).
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2. If X 2 E, then E_ catCP(g) catFH(f).
3. If gf ' *, then catBG(g) catFH(f).
Proof By Proposition 5.7 and Corollary 5.8, we have
E_ catCP(g) E_ secat(f)
catFH(f) + E_ catCP(X).
Also (2) follows from (1) since E_ catCP(X) = 0.
For (3) we specialize to E = {all spaces} in Corollary 5.8 and use
the Berstein-Ganea inequality catBG(g) secat(f) mentioned above.
2
In the special case E = {*}, Proposition 5.9(1) reduces to
catBG(g) catFH(f) + cat(X).
Next we relate sectional category and the Clapp-Puppe category.
Proposition 5.10 For any collection E and any map f : X ! Y ,
E_ catCP(f) E_ secat(f).
Proof Let E_ secat(f) = n and let {U0, U1, . .,.Un} be a E-section-
categorical cover of Y . Then there are spaces Ei 2 E and maps ui :
Ui ! Ei and vi : Ei ! X such that fviui ' ji, where ji : Ui ,! Y is
the inclusion map. We set Vi = f-1 (Ui). Then {V0, V1, . .,.Vn} is an
open cover of X and the maps
f|Vi ui vi f
Vi_____//Ui___//_Ei___//_X____//Y
show that E_ catCP(f) n. 2
Remark 5.11 This proposition does not show that catBG(f)
secat(f). This is because E_ catCP(f) = catBG(f) when E = {*}
and E_ secat(f) = secat(f) when E = {all spaces}. In a sense, E_ catCP
and E_ secatare dual to each other.
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Dartmouth College
Hanover, NH 03755
Martin.Arkowitz@Dartmouth.edu
Western Michigan University
Kalamazoo, MI 49008
Jeffrey.Strom@wmich.edu
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