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305 A

Obstructions

1150

305 (IX.1 1)

Obstructions

and by @“(jr) the set of thomotopy classes

of mappings in %“(f’) relative to L. The set

&O(f’) consists of a single element ‘because of

the arcwise connectedness of Y, @(f’) is non-

empty, and @‘(f’) (na 2) may be empty. Let ,f”

be an element of @“(f’). If we consider the

restriction off” to the boundary Oni1 of an

oriented (n + l)-cell (r”+’ of K, then f”: $‘+l -t

Y determines an element c(f”, on+‘) of the

thomotopy group 7~,,(Y) (- 202 Homotopy

Theory). This element gives a measure of ob-

struction for extending f” to the interior of

C?+I. We obtain an (n+ 1)-tcocycle c”+‘(f”) of

the tsimplicial pair (K, L) with coefficients in

n,J Y), called the obstruction cocycle off”, by

assigning c(fn,a”+‘) to each (n+ l)-ce11 a”+‘.

This obstruction cocycle c”“(f”) is the mea-

sure of obstruction for extending f” to R”“. A

necessary and sufflcient condition for the

extensibility is given by c”“(f”) ==0. Clearly,

c”+l (j”“) is uniquely determined for each ele-

ment f of @“(f’). The set of a11c”+‘(f”) with

S”E@“(~‘) forms a subset o”+‘(f’) of the group

of cocycles Z”‘l(K, L; n,(Y)). @“‘l(f) is non-

empty if and only if o”+l(,f’) contains the zero

element 0.

LetKn=KxI,Lu=(KxO)L(LxZ)U

(K x 1). Given two mappings SO, fi : K-t Y

satisfying f. 1L =f, 1L, we cari defïne a natural

mapping F’ : La -) Y such that an element

F” of @“(F’) corresponds to a thomotopy

h”-’ relative to L connecting f. ( K”-’ with

fi 1K”-‘. Given an element F”E@“(F’), we

have the element P’(F”) of Z”“(Km, L”;

n,(Y)), which we identify with Z”(K, L; n,(Y))

through the natural isomorphism of chain

groups of the pair (Ko, L”) to those of the

pair (K, L). Thus we cari regard c”+‘(F”) as an

element of Z”(K, L; 7c”(Y)), which is denoted

by d”(f,, h”-‘,f,), and cal1 it the separation (or

difference) cocycle. If ,jo ( R”-l =SI ) i?‘, we

have the canonical mapping F”: L” U (Ko)

* Y, and the separation cocycle is denoted

simply by d”(fo,fi). The set of :separation

cocycles corresponding to elements of @“(Fr) is

considered to be a subset of Z”(K, L; n,( Y))

and is denoted by o”(.fO,fi). A necessary and

sufflcient condition for h”-’ to be extensible

to a homotopy on l?” is d”(f,,h”-‘,f,)=O.

Therefore a necessary and suffccient condition

for f0 1R” =fi ( R” (rel L) (i.e., relative to L) is

O~o”(f,,f,). Givenf;,f;:K”-tYwithfO[L=

.fTIL, then d”(&‘,h”-‘,f;) (~o’(f;,f;)) is

an element of Z”(K”, L; 7c”(Y)): which is also

considered to be a cochain of the pair (K, L).

In this sense, we ca11 @(SO, h” -‘, ,f,“) the sepa-

ration (or deformation) cochain over (K, L).

The coboundary of the separation cochain

d”(fl, h”-‘,,f;) coincides (except possibly for

sign) with c”“(&‘)-c”“(f,“).

For a tïxed fo E a>“( f’), any n-cochain dn

A. History

The theory of obstructions aims at measuring

the extensibility of mappings by means of

algebraic tools. Such classical results as the

+Brouwer mapping theorem and Hopf’s exten-

sion and tclassification theorems in homotopy

theory might be regarded as the origins of this

theory. A systematic study of the theory was

initiated by S. Eilenberg [l] in connection

with the notions of thomotopy and tcoho-

mology groups, which were introduced at the

same time. A. Komatu and P. Olum [L] ex-

tended the theory to mappings into spaces not

necessarily +n-simple. For mappings of poly-

hedra into certain special spaces, the +homo-

topy classification problem, closely related to

the theory of obstructions, was solved in the

following cases (K” denotes an m-dimensional

polyhedron): K”+‘+S” (N. Steenrod [SI), Knt2

4s” (J. Adem), Kntk*Y, where ni( Y)=0 for

i <n and n < i < n + k (M. Nakaoka). There

are similar results by L. S. Pontryagin, M.

Postnikov, and S. Eilenberg and S. MacLane.

Except for the special cases already noted, it is

extremely difflcult to discuss higher obstruc-

tions in general since they involve many com-

plexities. Nevertheless, it is significant that the

idea of obstructions has given rise to various

important notions in modern algebraic topol-

ogy, including cohomology operations (- 64

Cohomology Operations) and characteristic

classes (- 56 Characteristic Classes).

The notion of obstruction is also very useful

in the treatment of cross sections of fiber bun-

dles (- 147 Fiber Bundles), tdiffeomorphisms

of differentiable manifolds, etc.

B. General Theory for an n-Simple Space Y

The question of whether two (continuous)

mappings of a topological space X into an-

other space Y are +homotopic to each other

cari be reduced to the extensibility of the given

mapping: (X x {O))U(X x {l})+ Y to a map-

ping of the product X x 1 of X and the unit

interval I= [0, 11 into Y. Therefore the prob-

lem of classifying mappings cari be treated

in the same way as that of the extension of

mappings.

Let K be a tpolyhedron, L a subpolyhedron

of K, and R” = LU K” the union of L and the

+n-skeleton K” of K. Let Y be an tarcwise

connected n-simple space, and ,f’ be a mapping

of L into Y. Denote by O”(f’) the set of map-

pings of I?” into Y that are extensions off’,

1151

305 c

Obstructions

of the pair (K, L) with coefficients in n,(Y)

is expressible as a separation cochain d”=

d”(fl,f;) wheref/EQ”(f’) is a suitable map-

ping such that fil R”-’ =fr” 1R”-’ (existence

theorem).

Therefore if we take an element f”-’ of

@-I(f’) whose obstruction cocycle c”(f”-‘)

is zero, the set of a11 obstruction cocycles

c”“(J”) of a11 such ~“E@“(S’) that are exten-

sions off”-’ forms a subset of O”“(f’) and

coincides with a coset of Z”+‘(K, L; n,(Y))

factored by B”+‘(K, L; rcL,(Y)). Thus a coho-

mology class ?+r(f”-~)EH”+~(K, L; rrn( Y))

corresponds to an f”-’ E Qn-l (f’) such that

c”(f”-‘) = 0, and ?“+i(f”-‘) = 0 is a necessary

and sufftcient condition for f”-’ to be exten-

sible to I?“” (first extension theorem).

For the separation cocycle, d”(f,, h”-‘,fJ~

H”(K, L;a,(Y)) corresponds to each homotopy

h”-’ on I?n-Z such that d”-‘(f,, h”-‘,f,)=O,

and 6”( f& h”-‘, fi) = 0 is a necessary and

sufficient condition for h”-’ to be extensible to

a homotopy on R” (tirst homotopy theorem).

The subset of H”+‘(K, L, K,( Y)) correspond-

ing to on+’ (f ‘) is denoted by On+’ (f ‘) and is

called the obstruction to an (n + 1)dimensional

extension off ‘. Similarly, the subset On( fo, fi)

of H”(K, L, n,( Y)) corresponding to o”( fo, fi) is

called the obstruction to an ndimensional

homotopy connecting f. with fi. Clearly, a

condition for f' to be extensible to Rn+’ is

given by 0 E O”+l (f ‘), and a necessary and

sufftcient condition for f. 1K” = fi 1K” (rel L) is

given by OeO”(fo, fi).

A continuous mapping <p:(K’, L’)+(K, L)

induces homomorphisms of cohomology

groups ‘p*: H”+‘(K, L; n,( Y))+H”+‘(K’, L’,

n,(Y)), H”(K, L; w,( Y))+H”(K’, L’; n,(Y)). Then

for f’:LtY, O”“(f’ocp)~rp*O”+‘(f’), and

for f,, fi : K + Y such that f. 1L = fi ) L, On( f, o

cp,f, o cp)~ p*O”( fo, fi). Therefore we also lïnd

that the obstruction to an extension and the

obstruction to a homotopy are independent

of the choice of subdivisions of K, L, and con-

sequently are topological invariants.

Let fo, fi, and fi be mappings K-* Y such

that f. 1L = fi 1L = f, 1L. Given homotopies

h~;l:f,)~“-l~fi)Rn-l(relL),h;;l:fl)~-l~

f2 1R”-’ (rel L), then for the composite h”,;’ =

h;;’ o ht;‘, we have

general, if On( fo, fi) is nonempty, it is a coset

of H”(K, L; x,(Y)) factored by the subgroup

O”(f,, fo). Combined with the existence theo-

rem on separation cochains, this cari be uti-

lized to show the following theorem.

Assume that O”(f ‘) is nonempty. The set

of a11 elements @‘(f ‘) that are extensions of

an element of @-l (f ‘) is put in one-to-one

correspondence with the quotient group of

Hn(I?‘, L; A,( Y)) modulo On( fo, fô) by pairing

the obstruction On( fô, f “) with each f” for a

fixed fô. Among such elements of @“( f ‘), the

set off” that are extensible to @+’ is in one-

to-one correspondence with the quotient

group of H”(R”+1 ,L; ~n(Y))=H”(KL; G(Y))

modulo the subgroup On( fô”, f$+‘), assuming

that fo is extended to fo”+’ (fkst classification

theorem).

C. Primary Obstructions

Assume that H’+‘(K, L; ni( Y))= H’(K, L; ni( Y))

=O,whereO<i<p(e.g.,rq(Y)=O,O<i<p).In

this case, by consecutive use of the lïrst exten-

sion theorem and the lïrst homotopy theorem,

we cari show that each @(f ‘) (i <p) consists of

a single element and OP+‘( f ‘) also consists of a

single element Pc1 (f ‘) E HP+‘(K, L; xP( Y)). The

element CP+’ (f ‘), called the primary obstruc-

tion off ‘, vanishes if and only if f’ cari be

extended to RP+1 (second extension theorem).

When Hi+’ (K, L; ai(Y)) = 0 for i > p (for exam-

ple, when ai(Y) = 0 for p ci < dim(K -L)),

f’ is extendable to K if and only if the first ob-

struction off’ vanishes (third extension

theorem).

Correspondingly, if H’(K, L; zi( Y))=

Hi-’ (K, L; rci( Y)) = 0 (0 < i < p), then for any

two mappings fo, fi : K-+ Y, f. (L = fi 1L,

Op(fo, fi) consists of a single element dP(fo, fi)

E HP(K, L; rcp( Y)), which we cal1 the primary

difference off0 and fi. This element vanishes

if and only if f0 1l@’ = fi 1&’ (rel L) (second

homotopy theorem). Moreover, when H’(K, L;

rci( Y)) = 0 (i > p), the primary difference is zero

if and only if f. E fi (rel L) (third homotopy

theorem).

Assume that the hypotheses of the second

extension theorem and second homotopy

theorem are satistïed. If we assign to each

element f P of @‘(f ‘) the primary difference of

f P and the tïxed element f{, then Gi”( f ‘) is in

one-to-one correspondence with HP(RP, L;

rcp( Y)) by the lïrst classification theorem (sec-

ond classification theorem). Similarly, assume

that the hypotheses of the third extension

theorem and third homotopy theorem are

satisfîed. Iff,:K+ Y, f’=& 1L, then homotopy

classes relative to L of extensions f off’ are

put in one-to-one correspondence with the

and for the inverse homotopy h;;’ : f, 1Z?‘-l T

f. 1R”-’ of h”,;‘, clearly

d”(f,,h;o’,fo)=

-d”(fo,h&‘,f,).

Therefore O”(fO,fO) forms a subgroup of

H”(K, L, nnn(Y)) that is determined by the

homotopy class off0 1em1 relative to L. In

305 D

Obstructions

1152

elements of HP(K, L; xp( Y)) by pairing dp(,f;fO)

with ,f (tbird classification theorem).

ferential equations by reducing the operations

of differentiation and integration into alge-

brait ones in a symbolic manner. The idea

was initiated by P. S. Laplace in his Théorie

analytique des probabilités (18 12), but the

method has acquired popularity since 0.

Heaviside used it systematically in the late

19th Century to solve electric-circuit problems.

The method is therefore also callec. Heaviside

calculus, but Heaviside gave only a forma1

method of calculus without bothering with

rigorous arguments. The mathematical foun-

dations were given in later years, tïrst in terms

of +Laplace transforms, then by ap:$ying the

theory of tdistributions. One of the motiva-

tions behind L. Schwartz’s creation of this

latter theory in the 1940s was to give a sound

foundation for the forma1 method, but the

theory obtained has had a much larger range

of applications. Schwartz’s theory was based

on the newly developed theory of +topological

linear spaces. On the other hand, J. Mikusin-

ski gave another foundation, based only on

elementary algebrdic notions and on Titch-

marsh’s theorem, whose proof has recently

been much simplifïed.

In this article, we fïrst explain the simple

theory established by Mikusinski [2] and later

discuss its relation to the classical Laplace

transform method.

D. Secondary Obstructions

For simplicity, assume that ni(Y) = 0 (i< p and

p < i < q). If the primary obstruction CP+I (f’)~

H P+I (K, L; 7~,,(Y)) off’: L-t Y vanishes, we cari

detïne 04+’ (,f’) c H4+’ (K, L; 7rIq(Y)), which we

cal1 the secondary obstruction off’. When Y =

SP, q = p + 1, p > 2, the secondary obstruction

Op”(f’) coincides with a coset of HpfZ(K, L;

Z,) modulo the subgroup Sq2(HP(K, L; Z)),

where Sq’ denotes the +Steenrod square

operation [S]. In this case, if L = KP, then

Op”(,f’) reduces to a cohomology class,

Sq’(i*)-‘f’*(o) with i: L+K, where o is a gen-

erator of HP(SP, Z) (in this case (i*)m’f’*(a) #

0 is equivalent to ?“(f’)=O) [S]. Moreover,

if Sq2f’*(a) = 0, then there exists a suitable

extension fp+2:lZp+2 + Y = SP of ,f’. The set of

obstruction cocycles of all such fp” defïnes

the tertiary obstruction Op+3( f ‘), which coin-

cides wit h a coset of Hp+3(K, L; Z,) modulo the

subgroup SqZ(HP”(K, L; Z,)). By using the

tsecondary cohomology operation 0 of J.

Adem, it cari be expressed as @((i*)-‘f’*(o))

(- 64 Cohomology Operations).

All the propositions in this article remain

true if we take +CW complexes instead of

polyhedra K.

References

B. The Operational Calculus of Mikusibski

[l] S. Eilenberg, Cohomology and continuous

mappings, Ann. Math., (2) 41 (1940), 231-251.

[2] P. Olum, Obstructions to extensions and

homotopies, Ann. Math., (2) 52 (1950) l-50.

[3] P. Olum, On mappings into spaces in

which certain homotopy groups vanish, Ann.

Math., (2) 57 (1953) 561-574.

[4] N. E. Steenrod, The topology of fibre

bundles, Princeton Univ. Press, 1951.

[5] N. E. Steenrod, Products of cocycles and

extensions of mappings, Ann. Math., (2) 48

(1947), 290-320.

[6] E. H. Spanier, Algebraic topology,

McGraw-Hill.

1966.

The set % of all continuous complex-valued

functions a = {a(t)} defïned on t 2 0 is a tlinear

space with the usual addition and scalar multi-

plication. %?is a +Commutative algebra with

multiplication a. b detïned by the tconvolution

{S;u(t-s)b(s)ds}. Th e ring W has no +zero

divisors (Titchmarsh’s theorem). (There have

been several interesting proofs of Titchmarsh’s

theorem since the tïrst demonstratilon given

by Titchmarsh himself [3]. For example, a

simple proof has been published by C. Ryll-

Nardzewski (1952).) Hence we cari construct

the tquotient field -2 of the ring %. An element

of 2 is called a Mikusitiski operator, or simply

an operator. If we deiïne a(t) = 0 for t < 0 for

the elements {a(t)} in ??, then V? is a subalgebra

of %VI,which is the set of all locally integrable

(locally L,) functions in (-a, a) uhose +sup-

port is bounded below. Here we identify two

functions that coincide almost everywhere.

The algebra J& has no zero divisor, and its

quotient tïeld is also 2.

The unity element for multiplication in %,

denoted by 6 = b/b (b #O), plays the role of the

+Dirac &function. It is sometimes called the

impulse function. The operator 1= { 1) ~‘6 is the

306 (X11.20)

Operational Calculus

A. General Remarks

The term “operational calculus” in the usual

sense means a method for solving tlinear dif-

1153

306 C

Operational Calculus

function that takes the values 0 and 1 accord-

ing as t < 0 or t > 0. This operator is Heavi-

side’s function and is sometimes called the

unit function. Usually it is denoted by l(t)

or simply 1. The value l(0) may be arbitrary,

but usually it is detïned as 1/2, the mean of the

limit values from both sides. The operator I is

an integral operator, because, as an operator

carrying a into I. a, it yields

sented by

m M(ni)

i=1Zj7Je

where we assume that Âo, 1,). . . , i, exhaust the

roots of the equation o(L) = 0, 1, is a multiple

root of degree 1, and a11other roots are simple

(WI = n - 1). The above formula is called the

expansion theorem.

More generally, the operator {t”-‘/I(n)} (ReÂ.

> 0) gives the Âth-order integral. The operator

s = 611, which is the inverse operator of 1, is a

differential operator. If a E +Zis of tclass C’ ,

then we have

a(s)ds = the integral of a over [0, t].

C. Limits of Operators

A sequence a, of operators is said to converge

to the limit a = b/q if there exists an operator

q( # 0) such that q. a, EV and the sequence

of functions q. a, converges tuniformly to b

on compact sets. The limit a is determined

uniquely without depending on the operator q.

Based on this notion of limits of operators, we

cari construct the theory of series of operators

and differential and integral calculus of func-

tions of an independent variable i whose

values are operators. They are completely

parallel to the usual theories of elementary

calculus (- 106 Differential Calculus; 216

Integral Calculus; 379 Series). A linear partial

differential equation in the function V(X, t) of

two variables, in particular its initial value

problem, reduces to a linear ordinary differen-

tial equation of an operational-valued

s~a=a’+a(+0)6=a’+{a(+O)}/{l}.

Similarly, if UEW is of class C”, we have

S”.a=a’“)+a’“-“(0)6+a(“-2)(0)s

+ . +a(O)s”-1.

(2)

The operator U+S. a cari be applied to func-

tions a that are not differentiable in the ordi-

nary sense, and considering the application of

s to be the operation of differentiation, we cari

treat the differential operator algebraically in

the tïeld 02. In particular, we have s. 1 = 6, and

this relation is frequently represented by the

formula

dl(t)/dt=S(t). (3)

A rational function of s whose numerator is

of lower degree than its denominator is an

telementary function of t. For example, we

have the relations

func-

tion of an independent variable x.

For a given operator w, the solution (if

it exists) of the differential equation v’(1) =

w. <p(Â) with the initial condition C~(O)= 6 is

unique, is called the exponential function of an

operator w, and is denoted by ~(1) = e”“. If the

power series

l/(s-~)“={t”-‘ea’/(n-l)!},

s/(s2 +/32) = {cos~t}.

(4)

j. I”w”/n!

(6)

The solution of an ordinary linear differ-

ential equation with constant coefficients

C:=,, a,<p(‘)(t) =f(t) under the tinitial condition

C~“)(O) = yi (0 < i <n - 1) is thus reduced to an

equation in s by using formulas (1) and (2), and

is computed by decomposing the following

operator into partial fractions:

converges, the limit is identical to the exponen-

tial function e”‘“. However, there are several

cases in which eAWexists even when the series

(6) of operators does not converge.

For example, for w = - &, we have

e -d=

{(Â/2&P)exp(

-L’/4t)},

, (7)

(5)

and for w = -s, we have

whereLi=~,+l~o+~,+2~l + . .. +w~-,-~,

0 < r < n - 1. The general solution is repre-

sented by (5) if we consider the constants yo,

. . . ,yn-i or fio, . ,jI-i as arbitrary parameters.

If the rational function in the right-hand side

of (5) is M@)/L)(s) and the degree of the numer-

ator is less than that of the denominator, then

the right-hand side of (5) is explicitly repre-

e-“S=h”=s.H,(t), (8)

where the function H,(t) takes the values 0 and

1 according as t < 1 or t > 1. H,(t) belongs to

the ring @ and is called the jump function at Â.

For f(t)e%, we have

~A~u(t)~=u(t-4~9

and hence we cal1 (8) the translation operator

l.a=

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