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throughout the similarly defined past domain of

dependence D



(H)). We find that points in the future

Cauchy horizon H

(H), which is the future boundary

of D

(H) defined by

ðHÞ¼D

ðHÞI



ðD

ðHÞÞ;

has properties similar to the boundary of a past set, in

accordance with the above lemma, and also for the

past Cauchy horizon H



(H), defined correspondingly.

Singularity Theorems

and Related Questions

Now, applying our lemma to @I

(S), for a trapped

surface S  intD

(H), we find that every one of its

points lies on a null-geodesic segment  on @I

(S),

with past endpoint on S (for if  did not terminate at S

it would have to reach H, which is impossible).

Assuming future-null completeness and weak energy

(  0), we conclude that if extended far enough into

the future, the family of such null geodesics  must

encounter a caustic, and therefore they must leave

(S) and enter I

(S). We finally conclude that

(S) must be a compact topological 3-manifold.

Using basic theorems, we construct an everywhere

timelike vector field in intD

(H) which provides a

(1–1) continuous map from the compact @I

(S)toH,

yielding a contradiction if H is noncompact, thereby

establishing the following (Penrose 1965, 1968):

Theorem The requirement that there be a trapped

surface which, together with its closed future, lies in the

interior of the domain of dependence of a noncompact

spacelike hypersurface, is incompatible with future null

completeness and the weak energy condition.

We notice that this ‘‘singularity theorem’’ gives no

indication of the nature of the failure of future null

completeness in a spatially open spacetime subject to

weak positivity of energy and containing a trapped

surface. The natural assumption is that in an actual

physical situation of such gravitational collapse, the

failure of completeness would arise at places where

curvatures mount to such extreme values that

classical general relativity breaks down, and must be

replaced by the appropriate ‘‘quantum geometry’’ (see

Quantum Geometry and its Applications, etc.).

Hawking (1965) showed how this theorem (in time-

reversed form) could also be applied on a cosmolo-

gical scale to provide a strong argument that the

Big-Bang singularity of the standard cosmologies is

correspondingly stable. He subsequently introduced

techniques from ‘‘Morse theory’’ which could be

applied to timelike rather than just null geodesics

and, using arguments applied to Cauchy horizons,

was able to remove assumptions concerning domains

of dependence (e.g., Hawking (1967)). A later

theorem (Hawking and Penrose 1970)encompassed

most of the earlier ones and had, as one of its

implications, that virtually all spatially closed uni-

verse models, satisfying a reasonable energy condition

and without closed timelike curves, would have to be

singular, in this sense of ‘‘incompleteness,’’ but again

the topological-type arguments used give little indica-

tion of the nature or location of the singularities.

Another issue that is not addressed by these

arguments is whether the singularities arising from

gravitational collapse are inevitably ‘‘hidden,’’ as in

Figure 1, by the presence of a horizon – a conjecture

referred to as ‘‘cosmic censorship’’ (see Penrose

(1969, 1998)). Without this assumption, one cannot

deduce that gravitational collapse, in which a trapped

surface forms, will lead to a black hole, or to the

alternative which would be a ‘‘naked singularity.’’

There are many results in the literature having a

bearing on this issue, but it still remains open.

A related issue is that of strong cosmic censorship

which has to do with the question of whether

singularities might be observable to local observers.

Roughly speaking, a naked singularity would be one

which is ‘‘timelike,’’ whereas the singularities in black

holes might in general be expectedtobespacelike

(or future-null), and in the Big Bang, spacelike (or past-

null). There are ways of characterizing these distinctions

purely causally, in terms of past sets or future sets (sets Q

for which Q = I



(Q)orQ = I

(Q)); see Penrose (1998).

If (strong) cosmic censorship is valid, so there are no

timelike singularities, the remaining singularities would

be cleanly divided into past-type and future-type. In the

observed universe, there appears to be a vast difference

between the structure of the two, which is intimately

connected with the second law of thermodynamics,

there appearing to be an enormous constraint on

theWeylcurvature(see General Relativity: Overview)

in the initial singularities but not in the final ones.

Despite the likelihood of singularities arising in their

time evolution, it is possible to set up initial data for the

Einstein vacuum equations for a wide variety of

complicated spatial topologies (see Einstein Equations:

Initial Value Formulation). On the observational side,

however, there seems to be little evidence for anything

other than Euclidean spatial topology in our actual

universe (which includes black holes). Speculation on

the nature of spacetime at the tiniest scales, however,

where quantum gravity might be relevant, often

involves non-Euclidean topology, however. It may be

noted that an early theorem of Geroch established that

the constraints of classical Lorentzian geometry do not

permit the spatial topology to change without viola-

tions of causality (closed timelike curves).

622 Spacetime Topology, Causal Structure and Singularities

See also: Asymptotic Structure and Conformal Infinity;

Boundaries for Spacetimes; Computational Methods in

General Relativity: The Theory; Cosmology:

Mathematical Aspects; Critical Phenomena in

Gravitational Collapse; Einstein Equations: Exact

Solutions; Einstein Equations: Initial Value Formulation;

General Relativity: Overview; Geometric Analysis and

General Relativity; Lorentzian Geometry; Quantum

Cosmology; Quantum Geometry and its Applications;

Spinors and Spin Coefficients; Stationary Black Holes.

Further Reading

Arno’ld, Beem JK, and Ehrlich PE (1996) Global Lorentzian

Geometry, (2nd edn). New York: Marcel Dekker.

Eddington AS (1924) A comparison of Whitehead’s and Einstein’s

formulas. Nature 113: 192.

Finkelstein D (1958) Past–future asymmetry of the gravitational

field of a point particle. Physical Review 110: 965–967.

Hawking SW (1965) Physical Review Letters 15: 689.

Hawking SW (1967) The occurrence of singularities in cosmology

III. Causality and singularities. Proceedings of the Royal

Society (London) A 300: 187–.

Hawking SW and Ellis GFR (1973) The Large-Scale Structure of

Space-Time: Cambridge: Cambridge University Press.

Hawking SW and Penrose R (1970) The singularities of

gravitational collapse and cosmology. Proceedings of the

Royal Society (London) A 314: 529–548.

Oppenheimer JR and Snyder H (1939) On continued gravitational

contraction. Physical Review 56: 455–459.

Penrose R (1965) Gravitational collapse and space-time singula-

rities. Physics Review Letters 14: 57–59.

Penrose R (1972) Techniques of Differential Topology in

Relativity. CBMS Regional Conference Series in Applied

Mathematics, no.7. Philadelphia: SIAM.

Penrose R and Rindler W (1984) Spinors and Space-Time, Vol. 1:

Two-Spinor Calculus and Relativistic Fields. Cambridge:

Cambridge University Press.

Penrose R and Rindler W (1984) Spinors and Space-Time, Vol. 2:

Spinor and Twistor Methods in Space-Time Geometry. Cam-

bridge: Cambridge University Press.

Penrose R (1968) Structure of space-time. In: DeWitt CM and

Wheeler JA (eds.) Battelle Rencontres, 1967 Lectures in

Mathematics and Physics. New York: Benjamin.

Penrose R (1969) Gravitational collapse: the role of general

relativity. Rivista del Nuovo Cimento Serie I, Numero

Speciale 1: 252–276.

Penrose R (1998) The question of cosmic censorship. In: Wald

RM (ed.) Black Holes and Relativistic Stars, (reprinted in

(1999) Journal of Astrophysics and Astronomy 20, 233–248.)

Chicago, IL: University of Chicago Press.

Special Lagrangian Submanifolds see Calibrated Geometry and Special Lagrangian Submanifolds

Spectral Sequences

P Selick, University of Toronto, Toronto, ON, Canada

Introduction

Spectral sequences are a tool for collecting and

distilling the information contained in an infinite

number of long exact sequences. Their most

common use is the calculation of homology by

filtering the object under study and using a spectral

sequence to pass from knowledge of the homology

of the filtration quotients to that of the object itself.

This article will discuss the construction of spectral

sequences and the notion of convergence including

conditions sufficient to guarantee convergence.

Some sample applications of spectral sequences are

given.

A differential on an abelian group G is a self-map

d : G !G such that d

= 0. A morphism of differ-

ential groups is a map f : G !G

such that d

f = fd.

The condition d

= 0 guarantees that Im d  Ker d,

so to the differential group (G , d) we can associate

its homology, H(G, d):= Ker d=Im d. Often G has

extra structure and we require d to satisfy some

compatibility condition in order that H(G, d) should

also have this structure. For example, a differential

graded Lie algebra (L, d) requires a differential d

which satisfies the condition d[x, y] = [dx, y] þ

(1)

jxj

[x, dy]. While, for simplicity, throughout this

article we will always assume that G is an abelian

group, the concepts are readily extended to the case

where G is an object of some abelian category and

generalizations to nonabelian situations have also

been studied.

An important example of extra structure is the

case where G =

n = 1

is a graded abelian

group. The appropriate compatibility condition for

a differential graded group is that d should be

homogeneous of degree 1. That is, d(G

)  G

n1

In many contexts it is more natural to use super-

scripts and regard d as having degree þ1; the two

concepts are equivalent via the reindexing conven-

tion G

:= G

n

. Another important example is that

Spectral Sequences 623

where G forms a graded algebra, meaning that it has

a multip lication G

 G

n þk

. To form a

differential graded algebra, in addition to having

degree 1, d is required to satisfy the Leibniz rule

d(xy) = d(x)y þ(1)

jxj

xd(y) (where jxj denotes the

degree of x) familiar from the differentiation of

differential forms.

In many cases, G itself is not the main object of

interest, but is a relatively large and complicated

object, G = G(X), formed by applying some functor

G to the object X being studied. For example, X

might be some manifold and G could be the set of

all differential forms on X with the exterior

derivative as d. The presumption is that H(G(X))

carries the information we want about X in a much

simpler form than the whole of G(X).

A spectral sequenc e (Leray 1946) is defined

simply as a sequence ((E

, d

))

r = n

, n

þ1,...,

of diff er-

ential abelian groups such that E

rþ1

= H(E

, d

). By

reindexing, we could always arrange that n

= 1, but

sometimes it is more natural to begin with some

other integer. If all terms (E

, d

) of the spectral

sequence have the appropriate additional structure,

we might refer, for example, to a spectral sequence

of Lie algebras. If there exists N such that E

= E

for all r  N (equivalently d

= 0 for all r N), the

spectral sequence is said to ‘‘collapse’’ at E

The definition of spectral sequence is so broad

that we can say almost nothing of interest about

them without putting on some additional condi-

tions. We will begin by considering the most

common type of spectral sequence, historically the

one that formed the motivat ing example: the

spectral sequence of a filtered chain complex.

Filtered Objects

To study a complicated object X, it often helps to

filter X and study it one filtration at a time. A

filtration F

of a group X is a nested collection of

subgroups

:¼...F

X  F

nþ1

X X 1< n <1

A morphism f : F

of filtered groups is a

homomorphism f : X !Y such that f (F

(X))  F

(Y).

The groups F

X=F

n1

X are called the ‘‘filtration

quotients’’ and their direct sum Gr(F

):=

n1

X is called the associated graded group of the

filtered group F

. In cases where X has addit ional

structure, we might define special types of filtra-

tions satisfying some compatibility conditions so

that Gr(F

) inherits the additional structure. For

example, an algebra filtration of an algebra X is

defined as one for which ( F

X)(F

X)  F

n þk

Since our plan is to study X by computing

Gr(F

), the first question we need to consider is

what conditions we need to place on our filtration

so that Gr(F

) retains enough information to

recover X. Our experience from the ‘‘5-lemma’’

suggests that the appropriate way to phrase the

requirement is to ask for conditions on the filtra-

tions which are sufficient to conclude that f : X !Y

is an isomorphism whenever f : F

is a

morphism of filtered groups for which the induce d

Gr(f ) : Gr(X) !Gr(Y) is an isomorphism.

It is clear that GrF

can tell us nothing about

X  ([X

) so we require that X = [X

. Similarly

we need that \X

= 0. However, the latter condition

is insufficient as can be seen from the following

example.

Example 1 Let X :=

k = 1

Z and Y :=

k = 1

Z. Set

X :¼

X if n  0

k¼n

Z if n < 0



Y :¼

Y if n  0

k¼n

Z if n < 0



and let f : X !Y be the inclusion. Then Gr(f )isan

isomorphism but f is not.

To phrase the appropriate condition we need the

concept of algebraic limits. Given a sequence of

objects {X

}

n2Z

and morphisms f

: X

nþ1

some category, the ‘‘direct limit’’ or ‘‘colimit’’ of the

sequence, written lim

!

X, is an object X together

with morphisms g

: X

!X satisfying g

nþ1

 f

= g

having the universal property that given any object

together with maps g

: X

satisfying g

nþ1



= g

, there exists a unique morphism h : X !X

such that g

= h  g

for all n. By the usual

categorical argument the object X, if it exists, is

unique up to isomorphism. The dual concept,

‘‘inverse limit’’ or simply ‘‘limit’’ of the sequence,

written lim



X, is obtained by reversing the

directions of the morphisms. For intuition, we note

that these notions share, with the notion of limits of

sequences in calculus, the properties that changing

the terms X

only for n < N does not affect

lim

!

X, and if the sequence stabilizes at N (i.e.,

the morphisms f

are isomorphisms for all n  N),

then lim

!

X ﬃ X

. Similarly lim



X depends

only upon behavior of the sequence as n !1.

Limits over partially ordered sets other than Z can

also be taken but we shall not need them in this

article. Although limits need not exist in general, in

the cate gory of abelian groups, both the direct and

inverse limit exist for any sequence and are given

explicitly by the following constructions. lim

!

= where, letting i

be the

624 Spectral Sequences

canonical inclusion, the equivalence relation is gener-

ated by i

(x)  i

nþ1

f (x)forx 2X

. lim



{(x

) 2X

)= x

nþ1

8n}.

The condition needed is that our filtrations should

be bicomplete, defined as follows. F

is called

‘‘cocomplete’’ if the canonical map X !lim

!

is an isomorphism and F

is called ‘‘complete’’ if

X !lim



X=F

X is an isomorphism. F

is called

bicomplete if it is both complete and cocomplete.

Note that F

cocomplete is equivalent to [F

X = X

but F

complete is stronger than \F

X = 0.

Theorem 1 (Comparison theorem). Let F

bicomplete and let F

be cocomplete with

Y = 0. Suppose that f : F

is a morphism

such that Gr(f ) : Gr(X) !Gr( Y) is an isomorphism.

Then f : X !Y is an isomorphism.

Filtered Chain Complexes

A chain complex (C, d) of abelian groups consists of

abelian groups C

for n 2 Z together with homo-

morphisms d

: C

n1

such that d

 d

nþ1

= 0 for

all n. To the chain complex (C, d) we can associate

the differential (abelian) group (C



, d):=

n = 1

with dj

induced by d

. We often write simply C if

the differential is understood. The dual notion in

which d has degree þ1 is called a cochain complex

and the concepts are equivalent through our

convention C

:= C

n

Theorem 2 (Homology commutes with direct

limits). H(lim

!

) = lim

!

H(C

As we shall see later, failure of homo logy to

commute with inverse limits is a source of great

complication in working with spectral sequences.

Let F

be a filtered chain complex. In many

applications, our goal is to compute H



knowledge of H



C=F

n1

C) for all n. The overall

plan, which is not guaranteed to be successful in

general, would be:

1. use the given filtration on C to define a filtration

on H



(C),

2. use our knowledge of H



(Gr C) to compute

Gr H



(C),

3. reconstruct H



(C).

To begin, set F



C):= Im(s

)



, where s

filtration. The spectral sequence which we will

define for this situation can be regarded as a method

of keeping track of the information contained in

the infinite collection of long exact homology

sequences coming from the short exact sequences

0 !F

n1

C !F

C=F

n1

C !0. When working

with a long exact sequence, knowledge of two of

every three terms gives a handle on computing the

remaining terms but does not, in general, completely

determine those terms, which explains intuitively

why we have some reason to hope that a spectral

sequence might be useful and also why it is not

guaranteed to solve our problem.

Before proceeding with our motivating example,

we digress to discuss spectral sequences formed from

exact couples.

Exact Couples

In this section, we will define exact couple s, show

how to associate a spectral sequence to an exact

couple, and discuss some properties of spectral

sequences coming from exact couple s. As we shall

see, a filtered chain complex gives rise to an exact

couple and we will examine this spectral sequence in

greater detail.

Exact couples were invented by Massey and many

books use them as a convenient method of con-

structing spectral sequences. Other books bypass

discussion of exact couples and define the spectral

sequence coming from a filtered chain complex

directly.

Definition 1 An ‘‘exact couple’’ con sists of a

triangle

!

containing abelian groups D, E, and together with

homomorphisms i, j, k such that the diagram is

exact at each vertex.

In the following, to avoid conflicting notation

considering the many superscripts and subscripts

which will be needed, we use the convention that an

n-fold composition will be written f

n

rather than

the usual f

Given an exact couple, set d := jk : E ! E.By

exactness, kj = 0, so d

2

= jkjk = 0 and therefore

(E, d) form s a differe ntial group. To the exact

couple we can associate another exact couple, called

its derived couple, as follows. Set D

:= Im i  D and

:= H(E, d). Define i

:= ij

and let j

: D

given by j

(iy):= j(y), where x denotes the equiva-

lence class of x. The map k

: E

is defined by

(



z):= kz. One checks that the maps j

and k

are

well defined and that (D

, E

, i

, j

, k

) forms an exact

couple. Therefore, from our original exact couple,

we can inductively form a sequence of exact couples

, E

, i

, j

, k

)

r = 1

with D

:= D, E

:= E, D

:= (D

r1

)

Spectral Sequences 625

and E

:= (E

r1

)

. This gives a spectral sequence

, d

)

r = 1

with d

= j

To the filtered chain complex F

, we can

associate an exact couple as follows. Set D :=

p, q

where D

p, q

= H

pþq

C) and E :=

p, q

where E

p, q

= H

pþq

C=F

p1

C). The long exact

homology sequences coming from the sequences

0 ! F

p1

C !

C=F

p1

C ! 0 give rise, for

each p and q, to maps a



: D

p1, qþ1

! D

p, q

, b



p, q

! E

p, q

,and@ : E

p, q

! D

p1, q

.Definei : D ! D

to be the map whose restriction to D

p1, qþ1

is the

composition of a



with the canonical inclusion

p, q

! D. Similarly, define j : D ! E and k : E ! D

to be the maps whose restrictions to each

summand are the compositions of b



and @ with

the inclusions. The indexing scheme for the bigrada-

tions is motivated by the fact that in many

applications it causes all of the nonzero terms to

appear in the first quadrant, so it is the most

common choice, although one sometimes sees other

conventions.

There is actually a second exact couple we could

associate to F

, which yields the same spectral

sequence: use the same E as above but replace D by

p, q

with D

p, q

= H

pþqþ1

(C=F

C), and define i, j,

and k in a manner similar to that above.

When dealing with cohomology rather than

homology, the usual starting point would be a

system of inclusions of cochain complexes F

nþ1

 F

C  F

n1

C. This can be reduced to the

previous case by replacing the cochain complex C by

a chain complex C



using the convention C

:= C

p

and filtering the result by F



:= F

n

C. The usual

practice, equivalent to the above followed by a

rotation of 180



, is to leave the original indices and

instead reverse the arrows in the exact couple. In

this case, it is customary to write D

p, q

and E

p, q

for

the terms in the exact couple and spectral sequence.

In applications, it is often the case that E

known and that our goal includes computing D

The example of the filtered chain complex with the

assumption that we know H



C=F

p1

C) for all p

is fairly typical.

Since each D

is contained in D

r1

and each E

a subquotient of E

r1

, the terms of these exact

couples get smaller as we progress. To get properties

of the spectral sequence, we need to examine this

process and, in particular, analyze that which

remains in the spectral sequence as we let r go to

infinity.

For x 2 E,ifdx = 0 then



x belongs to E

and so

(



x) is defined. In the following, we shall usually

simplify the notation by writing simply x in place of



x and writing d

x = 0 to mean ‘‘d

x is defined and

equals 0.’’

If dx = 0, ..., d

r1

x = 0, then x represents an

element of E

and d

x is defined. Set Z

:= {x 2

E jd

x = 0 8m  r}. Then E

rþ1

ﬃ Z

= where x  y

if there exists z 2 E such that for some t  r we

have d

z = 0 for m < t (thus d

z is defined) and

z = x  y. With this as motivation, we set Z

= {x 2 E jd

x = 0 8m} (known as the ‘‘infinite

cycles’’) and define E

:= Z

=  where x  y if

there exists z 2 E such that for some t we have

z = 0 for m < t and d

z = x  y.

Notice that D

rþ1

= Imi

r

ﬃ D=Ker i

r

. There is no

analog of this statement for r = 1. Instead we have

separate concepts so we set D

:= D= [

Ker i

r

and

D := \

Imi

r

. The analog of the rth-derived

exact couple when r = 1 is the following exact

sequence.

Theorem 3 There are maps induced by i, j, and k

producing an exact sequence

0 ! D

!

D !

The fact that we were able to add the 0 term to

the left of this sequence but not the right can be

traced to the fact that lim

!

preserves exactness but

lim



does not.

In our motivating example, the terms of the initial

exact couple came with a bigrading D =

p, q

and

E =

p, q

and writing jf j for the bidegree of a

morphism f we had: jij= (1, 1); jjj= (0, 0); jkj=

(1, 0); d = (1, 0). It follows that ji

j= (1,1); jj

(r þ 1, r  1); jk

j= (1, 0); jd

j= (r , r  1) which

is considered the standard bigrading for a bigraded

exact couple. Similarly, the standard bigrading for a

bigraded spectral sequence is one such that

j= (r , r  1).

We observed earlier that terms of an exact couple

and its corresponding spectral sequence get smaller

as r !1as each is a subquotient of its predecessor.

Note that the bigrading is such that this applies to

each pair of coordinates individually (e.g., E

rþ1

p, q

a subquotient of E

p, q

) and so in particular if the

p, q-position ever becomes 0 that position remains 0

forevermore.

Convergence of Graded Spectral

Sequences

As noted earlier, the definition of spectral sequence

is so broad that we need to put some conditions on

our spectral sequences to make them useful as a

computational tool. From now on, we will restrict

attention to spectral sequences arising from exact

couples in which D =

and E =

are

graded with ij

D

pþ1

, jj

E

, and kj

D

p1

All the spectral sequences which have been studied

626 Spectral Sequences

to date satisfy this condition and in fact most also

have a second gradation as in the case of our

motivating example. To see how to proceed, we

examine that case more closely.

For a filtered chain complex F

with structure

maps s

: F

C ! C we defined F



(C)) = Im s



.If

x = i

(r1)

y belongs to

p; q

= Im i

ðr1Þ

: H

pþq

ðF

prþ1

CÞ!H

pþq

ðF

CÞ

then (s

)



x = (s

)



(r1)

y = (s

pþ1

)



r

y = (s

pþ1

)



ix.

Therefore, we have a commutative diagram

p;q

! F

ðH

pþq

ðCÞÞ

rþ1

pþ1;q1

! F

pþ1

ðH

pþq

ðCÞÞ

yielding a map

rþ1

pþ1; q1

p; q

! F

pþ1

pþq

ðCÞ



pþq

ðCÞ



= Gr

pþ1

ðH

pþq

ÞC

Letting r go to infinity, we get an induced map

 : D

) !Gr(H(C)).

Theorem 4 If F

(i) D

= F

(H(C));

(ii)  : D

) !Gr(H(C)) is an isomorphism;

(iii) There is an exact sequence 0 !Gr(H(C))

!

We say that the spectral sequence (E

) ‘‘abuts’’ to

if there is an isomorphism GrL ! E

. Here we

mean an isomorphism of graded abelian groups,

which makes sense since under our assumptions E

inherits a grading from E

for each r. If in addition

the filtration on L is cocomplete, we say that (E

)

‘‘weakly converges’’ to F

and if it is bicomplete we

say that (E

) ‘‘converges’’ (or strongly converges)to

. The notation (E

) )F

(or simply (E

) ) L

when the filtration on L is either understood or

unimportant) is often used in connection with

convergence but there is no universal agreement as

to which of the three concepts (abuts, weakly

converges, or converges) it refers to! In this article,

we will also use the expression (E

) ‘‘quasicon-

verges’’ to F

to mean that the spectral sequence

weakly converges to F

with \

L = 0. (Note: the

terminology quasiconverges is nonstandard although

the concept has appeared in the li terature, some-

times under the name converges.)

While it would be overstating things to claim that

convergence of the spectral sequenc e shows that E

determines H(C ), it is clear that convergence is what

we need in order to expect that E

contains enough

information to possibly reconstruct H(C). The sense

in which this is true is stated more precisely in the

following theorem.

Theorem 5 (Spectral sequence comparison

theorem). Let f = ( f

):(E

) !

be a morphism

of spectral sequences.

(i) If f : E

is an isomorphism for some N,

then f

is an isomorphism for all r  N (includ-

ing r = 1).

(ii) Suppose in addition that (E

) converges to F

and (

) quasiconverges to F

. Let  : F

be a morphism of filtered abelian groups which

is compatible with f. (i.e., there exist isomorph-

isms  :GrX ﬃ E

and

 :Gr

X ﬃ

such that

  =

 Gr(f )). Then f : X !

Xisan

isomorphism.

Within the constraints provided by Theorem 5,a

spectral sequence might have many limits. A typical

calculation of some group Y by means of spectral

sequences might proceed as an application of

Theorem 5 along the lines of the following plan.

1. Subgroups F

Y forming a filtration of Y are

defined, although usually not computable at this

point. The subgroups are chosen in a manner that

seems natural bearing in mind that to be useful it

will be necessary to show convergence properties.

2. Directly or by means of an exact couple, a

spectral sequence is defined in a manner that

seems to be related to the filtration.

3. Some early term of the spectral sequence (usually

or E

) is calculated explicitly and the

differentials d

are calculated successively result-

ing in a computation of E

4. With the aid of the knowledge of E

conjecture Y = G is formulated for some G.

5. A suitable filtration on G and a map of filtrations

or F

are defined.

6. The spectral sequence arising from F

is demon-

strated to converge to G.

7. The original spectral sequence is demonstrated to

converge to Y and Theorem 5 is applied.

The hardest steps are usually (3) and (7). For step

(3), in most cases the calculations require knowledge

which cannot be obtained from the spectral sequence

itself, although the spectral sequence machinery plays

its role in distilling the information and pointing the

way to exactly what needs to be calculated. Steps

(4)–(6) are frequently very easy, and often not stated

explicitly, with ‘‘by construction of G’’ being the

most common justification of (6). We now discuss

the types of considerations involved in step (7).

Convergence of a spectral sequence to a desired L

can be difficult to verify in general partly because

Spectral Sequences 627

the conditions are stated in terms of some filtration

(usually understood only in a theoretical sense ) on

an initially unknown L rather than in terms of

properties of the spectral sequence itself or an exact

couple from whi ch it arose. Theorem s 2 and 4(ii)

give us the following extremely important special

case in which we can conclude convergence to H(C)

of the spectral sequence for F

based on conditions

that are often easily checked.

Theorem 6 If F

is a filtered chain complex such

that F

is cocomplete and there exists M such that

H(F

C) = 0 for n < M, then the spect ral sequence

for F

converges to H(C).

Although the second hypothesis, which implies

that

D = 0, is very strong it handles the large

numbers of commonly used filtrations which are 0

in negative degrees.

Under the conditions of Theorem 6, inserting the

bigradings into Theorem 4 gives a short exact

sequence 0 ! D

p1, qþ1

! D

p, q

! E

p, q

! 0 with

p, q

ﬃ F

pþq

(X)); equivalently

ðCÞðÞ=F

k1

ðCÞðÞﬃE

k; nk

Thus, the only E

-terms relevant to the computa-

tion to H

the important case of a first quadrant spectral

sequence (E

p, q

= 0ifp < 0orq < 0), the number

of nonzero terms on any diagonal is finite so the

-terms on the diagonal p þ q = n give a finite

composition series for each H

(C).

Here is an elementary example of an application

of a spectral sequence.

Example 2 Let S



( ) denote the singular chain

complex, let H



():= H



( )) denote singular

homology, and let H

cell



( ) denote cellular homology.

Let X be a CW-complex with n-skeleton X

(n)

. The

inclusions S



(n)

) ! S



(X) yield a filtration on



(X). In the associated spectral sequence,

p;q

¼H

pþq

ðpÞ

ðp1Þ



ﬃ

free abelian group on the p-cells of X if q ¼0

0ifq 6¼0



The differential

p; 0

: H

ðpÞ

ðp1Þ



! H

p1

ðp1Þ

ðp2Þ



is the definition of the differential in cellular

homology. Therefore,

p;q

cell

ðXÞ if q ¼ 0

0ifq 6¼ 0



Looking at the bidegrees, the domain or range of d

p, q

is zero for each p and q so d

= 0, and similarly

= 0 for all r > 2. Therefore, the spectral sequence

collapses with E

= E

. The spectral sequence con-

verges to H



(X) so the terms on the diagonal

p þ q = n form a composition series for H

(X).

Since the (n, 0) term is the only nonzero term on

this diagonal, H

(X) ﬃ H

cell

(X). That is, ‘‘cellular

homology equals singular homology.’’

Returning to the general situation, set L

lim

!

and L

1

:= lim



. Filter L

by F

Im(D

! L

) and filter L

1

by F

1

:= Ker

1

! D

). It follows from the definitions that

= D

and so D

n1

) = Gr

.Atthe

other end, the canonical map L

1

! D

lifts to

yielding an injection L

1

. Therefore,

for each n there is an injection Gr

1

! K

where

= Ker(

n1

). In general, the map

1

need not be surjective (an element

could be in the image of i

r

for each finite r without

being part of a consistent infinite sequence), although

it is surjective in the special case when

sþ1

is surjective for each s. In the latter case we get

Gr L

1

ﬃ K. As we will see in the next section, the

exact sequence of Theorem 3 extends to the right

(Theorem 8) giving lim



= 0 as a sufficient condition

that

sþ1

be surjective for each s, where lim



is described in that section and (Z

) refers to the system

of inclusions Z

rþ1

 Z

r1

.Thus,

lim



= 0 is a sufficient condition for Gr L

1

ﬃ K.

Taking into account the short exact sequence

0 ! D

) ! E

! K ! 0 coming from

Theorem 3, the preceding discussion yields two

obvious candidates for a suitable F

: F

or F

1

In theory there are other possibilities, but in

practice one of these tw o cases usually occurs. We

examine them individually and see what additional

conditions are required for convergence.

Case I: Conditions for convergence to F

It is

easily checked from the definitions that lim

!

lim

!

so F

is always cocomplete. Therefore,

besides Gr L

ﬃ E

(equivalently, K = 0), it is

required to verify that F

is complete. As we will

see in the next section, the completeness condition can

be restated as \D

= 0andlim



= 0. According to

the preceding discussion, under the assumption that

1

= \ D

= 0, which we need anyway as part of the

requirement that F

be complete, lim



X = 0is

sufficient to show K = 0.

Case II: Conditions for convergence to F

1

Any

inverse limit is complete in its canonical filtration, so

1

is always complete and the issues are whether

GrL

1

ﬃ E

and whether F

1

is cocomplete.

1

is cocomplete if and only if every element of

628 Spectral Sequences

1

lies in Ker(L

1

! D

) for some n, for which a

sufficient condition is that L

= 0 or equivalently

ﬃ K. Therefore, if the reason for the isomorph-

ism Gr L

1

ﬃ E

is that the maps E



 K and

Gr L

1

 K are isomorphisms, then the rest of the

convergence conditions are automatic. In particular,

to deduce convergence to F

1

it suffices to know

that L

= 0 and lim



= 0.

Derived Functors

The left and right derived functors L

T, R

T of a

functor T provide a measure of the amount by which

the functor deviates from preserving exactness.

The category Inv of inverse systems indexed over Z

(i.e., the category whose objects are diagrams

of abelian groups !A

n1

! A

nþ1

!)

forms an abelian category in which a sequence of

morphisms A

! A ! A

is exact if and only if the

sequence A

! A

of abelian groups is exact

for each n. The functor of interest to us is lim



: Inv !

AB where AB denotes the category of abelian groups.

Let T :

be an additive functor between

abelian categories. Suppose that X in Obj

has an

injective resolution I

. The definition of additive

functor implies that T takes zero morphisms to zero

morphisms, so TI

forms a cochain complex in

The right derived functors of T are defined by

T)(X):= H

(TI

). The result is independent of

the choice of injective resolution (assuming one

exists) and satisfies:

1. If T is ‘‘left exact’’ (meaning that T preserves

monomorphisms), then R

T(X) = T(X);

2. If T preserves exactness, then (R

T)(X) = 0 for

n > 0.

Theorem 7 Let 0 ! X

! X ! X

! 0 be a short

exact sequence in

. Suppose T is left exact and that

all the objects have injective resolutions. Then there

is a (long) exact sequence

0 ! TðX

Þ!TðXÞ!TðX

Þ!ðR

TÞðX

Þ!

!ðR

n1

TÞðX

Þ!ðR

TÞðX

Þ!ðR

TÞðXÞ!

ðR

TÞðX

Þ!

Similarly, the left derived functors of T are defined

by using projective resolutions and have similar

properties with respect to the obvious duality.

The functor lim



is left exact and in the category

Inv every object has an injective resolution. There-

fore lim



is defined and lim



= lim



, where

lim



denotes the derived functor R

(lim



). It turns

out that lim



is 0 for q > 1, but we are particularly

interested in lim



Let (X

) be an inverse system with structure maps

n1

: X

n1

! X

. An explicit construction for lim



is as follows. Define  :

letting (x

) be the sequence whose nth component

is (x

 i

n1

). Then lim



ﬃ Coker . Observe

that Ker  ﬃ lim



according to the explicit for-

mula for lim



given earlier.

Recall that we defined

D = \

Im i

r

ﬃ lim



The exact sequence of Theorem 3 can be extended

to give:

Theorem 8 There is an exact sequence

0 ! D

!

D !

!

lim



!

lim



!

lim



! 0

It is clear from the explicit construction that if the

system (X

) stabilizes with X

= G for all sufficiently

small n, then lim



X = G and lim



X = 0. If the

spectral sequence collapses at any stage then the

system (Z

) stabilizes at that point, an d so for a

spectral sequence which collapses, the condition

lim



= 0, which arose in the discussio n of

convergence in the previous section, is automatic.

Let F

be a filtered abelian group. Applying

Theorem 7 to the short exact sequence 0 ! F

X !

X ! X=F

X ! 0 of inverse systems gives an exact

sequence

0 ! lim



X ! lim



X ! lim



X=F

! lim



X ! lim



Since lim



X = X and lim



X = 0, we get

Theorem 9 F

is complete if and only if

lim



X = 0 and lim



X = 0.

When working with lim



the following sufficient

condition for its vanishing, known as the Mittag–

Leffler condition, is often useful.

Theorem 10 Suppose A is an inverse system in

which for each n there exists k(n)  n such that

Im(A

! A

) equals Im(A

k(n)

! A

) for all i  k(n).

Then lim



A = 0.

Of course, this will not be (directly) useful in

establishing lim



X = 0 since the structure maps in

that system are all monomorphisms.

Some Examples of Standard Spectral

Sequences and Their Use

To this point we have considered the general theory

of spectral sequences. The properties of the spectral

sequences arising in many specific situ ations have

Spectral Sequences 629

been well studied. Usually the spectral sequence

would be defined either directly, through an exact

couple, or by giving some filtration on a chain

complex. This defines the E

-term. Typically, a

theorem would then be proved giving some formula

for the resulting E

-term. In many cases, conditions

under which the spectral sequence converges may

also be well known.

In this section, we shall take a brief look at the

Serre spectral sequence, Atiyah–Hirzebruch spectral

sequence, spectral sequence of a double complex,

Grothendieck spectral sequence, change of ring

spectral sequence, and Eilenberg–Moore spectral

sequence, and carry out a few sample calculations.

Serre Spectral Sequence

Let F ! X!



B be a fiber bundle (or more generally

a fibration) in which the base B is a CW-complex.

Define a filtration on the total space by

X := 

1

(n)

. This yields a filtration on H



(X)by

setting F



(X):= Im(H



X) ! H



(X)). The spec-

tral sequence coming from the exact couple in which

p, q

:= H

pþq

X)andE

p, q

:= H

pþq

X, F

p1

X)is

called the ‘‘Serre spectral sequence’’ of the fibration.

Theorems from topology guarantee that this filtra-

tion is cocomplete and that E

p, q

= 0 if either p < 0

or q < 0. Therefore, the Serre spectral sequence is

always a first quadrant spectral sequence converging

to H



(X).

Theorem 11 (Serre). In the Serre spectral sequence

of the fibration F ! E ! B there is an isomorphism

p, q

ﬃ H

(B;

(F)).

Here



(F ) denotes a ‘‘twisted’’ or ‘‘local’’

coefficient system in which the differential is

modified to take into account the action, coming

from the fibration, of the fundamental groupoid of

the base B on the fiber F. In the special case where B

is simply connected and Tor(H



(B), H



(F)) = 0, the

‘‘universal coefficient theorem’’ says that the

-term reduces to E

p, q

ﬃ H

(B)  H

(F).

The Serre spectral sequence for cohomology,

p, q

ﬃ H

(B;

(F )) ) H

pþq

(X), has the advantage

that it is a spectral sequence of algebras which

greatly simplifies calculation of the differentials d

which are restricted by the requirement that they

satisfy the Leibniz rule with respect to the cup

product on H



(B) and H



(F ), and which also allows

the computation of the cup product on H



(X). Since

it is a first quadrant spectral sequ ence, convergence

is not an issue.

Frequently in ap plications of the Serre spectral

sequence, instead of using the spectral sequence to

calculate H



(X) from knowledge of H



(F ) and H



(B)

it is instead H



(X) and one of the other two

homologies which is known, and one is working

backwards from the spectral sequence to find the

homology of the third space.

Example 3 The universal S

-bundle is the bundle

! S

! CP

where S

is contractible. We will

calculate H



(CP

) from the Serre spectral sequence

of this bundle, taking H



) and H



) as known.

We also take as known that CP

is path connected,

so H

(CP

) ﬃ Z.

p;q

ﬃ H

ðCP

ÞH

ðS

ﬃ

ðCP

Þ if q ¼ 0or1

0 otherwise



-terms on the diagonal p þ q = n form a compo-

sition series for H

) which is zero for n 6¼ 0.

Therefore E

p, q

= 0 unles s p = 0 and q = 0, with

0, 0

ﬃ Z. Because all nonzero terms lie in the first

quadrant, the bidegrees of the differentials show

that d

1,0

) = 0 for all r  2, so 0 = E

1,0

= H

(CP

). Since E

1,q

ﬃ E

1,0

 E

0,q

, it follows

that E

1,q

= 0 for all q. Taking into the account the

known zero terms, the bidegrees of the differentials

show that E

0,1

ﬃ Ker(d

: E

0,1

2,0

) and E

0,1

. Similarly, E

2,0

= E

2,0

ﬃ Coker(d

: E

0,1

2,0

Therefore, the vanishing of these E

-terms shows

that d

: E

0,1

ﬃE

2,0

and in particular H

(CP

) ﬃ

0,1

= H

) ﬃ Z. It follows that E

2,q

ﬃ Z  E

0,q

ﬃ

0, q

for all q. With the aid of the fact that we

showed E

1,1

= 0, we can repeat the argument used to

show E

1,q

= 0 for all q to conclude that E

3,q

= 0 for

all q. Repeating the procedure, we inductively find

that E

p,q

ﬃ E

p2,q

for all p > 0 and all q and in

particular

ðCP

Þﬃ

Z if n is even

0ifn is odd



The cup products in H



(CP

) can also be

determined by taking advantage of the fact that the

spectral sequence is a spectral sequence of algeb ras.

Let a 2 E

2,0

ﬃ Z be a generator and set x := d

a.By

the preceding calculation, d

is an isomorphism so x

is a generator of H

(CP

). Therefore, x  a is a

generator of E

2, 2

and the isomorphism d

gives

that d

(x  a) is a generator of H

(CP

). However,

(x  a) = d

(x  1)(1  a) = 0 1 þ(1)

(x  1)d

a = x

1 and thus, x

is a generator of H

(CP

Inductively, it follows that x

is a generator of

(CP

) for all n and so H



(CP

) ﬃ Z[x].

When working backwards from the Serre or

other first quadrant spectral sequences in which

p, q

ﬃ E

p,0

 E

0, q

the following analog of the

comparison theorem (Theorem 5) is often useful.

630 Spectral Sequences

Theorem 12 (Zeeman comparison theore m). Let

E and E

be first quadrant spectral sequences such

that E

p, q

= E

p,0

 E

0, q

and E

p, q

= E

p,0

 E

0, q

. Let

f : E ! E

be a homomorphism of spectral sequences

such that f

p, q

= f

p,0

 f

0, q

. Suppose that f

p, q

: E

p, q

is an isomorphism for all p and q. Then the

following are equivalent:

(i) f

p,0

is an isomorphism for p n 1;

(ii) f

0, q

: E

0, q

! E

0, q

is an isomorphism for q  n.

There is a version of the Serre spectral sequence

for generalized homology theories coming from

the exact couple obtained by applying the

generalized homology theory to the Serre filtra-

tion of X.

Theorem 13 (Serre spectral sequence for generalized

homology). Let F ! X ! B be a fibration and let

Ybean(unreduced) homology theory satisfying the

Milnor wedge ax iom. Then there is a (right half-

plane) spectral sequence with E

p, q

ﬃ H

(B;

(F ))

converging to Y

pþq

(X).

Cocompleteness of the filtration follows from the

properties of generalized homology theories satisfy-

ing the wedge axiom (Milnor 1962), and the rest of

the convergence conditions are trivial since the

filtration is 0 in negative degrees. Here, unlike

the Serre spectral sequence for ordinary homology,

the existence of terms in the fourth quadrant opens the

possibility for composition series of infinite length,

although in the case where B is a finite-dimensional

complex all the nonzero terms of the spectral

sequence will live in the strip between p = 0 and

p = dim B and so the filtrations will be finite.

The special case of the fibration !X ! X

yields what is known as the ‘‘Atiyah–Hirzebruch

spectral sequence’’.

Theorem 14 (Atiyah–Hirzebruch spectral sequence).

Let X be a CW-complex and let Y be an (unreduced)

homology theory satisfying the Milnor wedge

axiom. Then there is a (right half-plane) spectral

sequence with E

p, q

ﬃ H

(X; Y

()) converging to

pþq

(X).

In the coho mology Serre spectral sequence for

generalized cohomology (including the cohomol ogy

Atiyah–Hirzebruch spectral sequence), convergence

of the spectral sequence to Y



(X) is not guaranteed.

Convergence to lim





X), should that occur,

would be of the type discussed in case II in the

section ‘‘Con vergence of g raded spect ral sequenc es’’.

Since X

= ; for n < 0, the system defining L

stabilizes to 0. Therefore, L

= 0 and, by the

discussion in that section, lim



X = 0 becomes a

sufficient condition for convergence to lim





X).

However since the real object of study is usually



(X), the spectral sequence is most useful when one

is also able to show lim





X) = 0 in which case

the Milnor exact sequence (M ilnor 1962)

0!lim





ðF

XÞ!Y



ðXÞ

!lim





ðF

XÞ!0

gives Y



(X) ﬃ lim





X).

If Y



( ) has cup products then the spectral

sequence has the extra structure of a spectral

sequence of Y



()-algebras. In the case where B is

finite dimensional, all convergence problems disap-

pear since the spectral sequence lives in a strip and

the filtrations are finite.

Example 4 Let K



( ) be complex K-theory. S ince



() ﬃ Z[z, z

1

] with jzj= 2, in the Atiyah–

Hirzebruch spectral sequence for K



(CP

)wehave

p;q

Z if q is even and p is even with 0 p 2n

0 otherwise



Because CP

is a finite complex, the spectral

sequence converges to K



(CP

). Since all the non-

zero terms have even total degree and all the

differentials have total degree þ1, the spectral

sequence collapse s at E

and we conclude that

(CP

)=0ifq is odd and that it has a composition

series consisting of (n þ1) copies of Z when q is

even. Since Z is a free abelian group, this uniquely

identifies the group structure of K

even

(CP

)asZ

nþ1

To find the ring structure we can make use of the

fact that this is a spectral sequence of K



()-

algebras. The result is K



(CP

) ﬃK



()[x]=(x

nþ1

where jxj= 2.

In the Atiyah–Hirzebruch spectral sequence for



(CP

) again all the terms have even total degree

so the spectral sequence collapses at E

. We noted

earlier that collapse of the spectral sequence implies

that lim



X = 0 and so the spectral sequence

convergences to lim





(CP

), where we used

= CP

. Since our preceding calculation

shows that K



(CP

) ! K



(CP

n1

) is onto, Mittag–

Leffler (Theorem 10) implies that lim





(CP

) = 0.

Therefore, the spectral sequence converges to



(CP

) and we find that K



(CP

) ﬃ lim





(CP

), which is isomorphic to the power series

ring K



()[[x]], where jxj= 2.

In topology one might be interested in the Atiyah–

Hirzebruch spectral sequence in the case where X is

a spectrum rather than a space (a spectrum being a

generalization in which cells in negative degrees are

allowed including the possibil ity that the dimensions

Spectral Sequences 631