A concise course in algebraic topology - May J.P.
The first year graduate program in mathe matics at the Unive rs ity of Chic ago
consis ts of three thre e-quarte r c ourse s, in analysis, alge bra, and top ology. The first
two quarters of the top ology se que nc e fo cus on manifold theory and diffe re ntial
geom etry, inc luding diffe re ntial forms and, usually, a glimpse of de Rham c ohom ol-ogy. The third quarte r fo c us es on algebraic top ology. I have b een te aching the
third quarte r off and on s ince around 1970. Be fore that, the top ologists , inc luding
me , thought that it would b e imp ossible to squee ze a s erious intro duc tion to al-gebraic top ology into a one quarte r course , but we were overruled by the analysts
and alge braists, who fe lt that it was unac ce ptable for graduate s tudents to obtain
the ir PhDs w ithout having some contact with alge braic top ology.
This raise s a c onundrum . A large numb er of stude nts at Chic ago go into top ol-ogy, alge braic and geom etric. The intro duc tory c ourse should lay the foundations
for the ir later work, but it s hould als o b e viable as an intro duction to the s ub je ct
suitable for those going into other branches of mathem atic s. T he se note s re flec t
my e fforts to organize the foundations of algebraic top ology in a way that cate rs
to b oth p edagogical goals . There are e vident defe cts from b oth p oints of view . A
treatme nt more close ly attune d to the ne eds of alge braic ge ome te rs and analysts
would inc lude ˇCe ch cohomology on the one hand and de Rham c ohomology and
p e rhaps Morse homology on the othe r. A treatme nt more close ly attuned to the
needs of algebraic top ologists would include sp ec tral s equences and an array of
calc ulations w ith them . I n the end, the overriding p e dagogic al goal has b ee n the
intro duction of bas ic ideas and me tho ds of thought.
Our unders tanding of the foundations of alge braic top ology has undergone sub-tle but se rious changes sinc e I b e gan te aching this cours e. Thes e changes re flec t
in part an e normous internal de ve lopm ent of algebraic top ology ove r this p erio d,
one w hich is large ly unknown to most other m athem atic ians , eve n those working in
such close ly re late d fie lds as geom etric top ology and algebraic geom etry. More over,
this deve lopme nt is p o orly reflec te d in the textb o oks that have app eared over this period.
Let me give a s mall but technically imp ortant example . T he s tudy of gen-eralize d homology and cohomology the orie s p ervades mo dern alge braic top ology.
Thes e theorie s s atis fy the e xc ision axiom. One cons tructs mos t such the ories ho-motopically, by c ons tructing re pres enting ob je cts called s p ec tra, and one must then
prove that e xc ision holds. T he re is a way to do this in general that is no more dif-ficult than the standard verification for s ingular homology and c ohom ology. I find
this pro of far m ore conce ptual and illuminating than the s tandard one e ven when
sp ec ialize d to s ingular homology and cohomology. (It is bas ed on the approxim a-tion of e xc isive triads by we akly e quivalent CW triads .) T his should by now b e a
1
2 IN TRODUC TION
standard approach. Howe ver, to the b e st of my knowle dge , the re e xis ts no rigorous
exp osition of this approach in the lite rature, at any leve l.
More c entrally, there now exist axiom atic treatme nts of large swaths of hom o-topy theory bas ed on Quillen’s the ory of close d m o del c ategorie s. While I do not
think that a first course s hould intro duce such abs trac tions, I do think that the e x-p os ition should give em phasis to those fe atures that the axiomatic approach shows
to b e fundam ental. For example , this is one of the re asons , although by no me ans
the only one, that I have dealt with c ofibrations, fibrations, and weak e quivalences
much more thoroughly than is usual in an intro ductory course .
Some parts of the the ory are dealt with quite class ic ally. The the ory of fun-damental groups and c overing s pac es is one of the few parts of algebraic top ology
that has probably re ached definitive form, and it is we ll tre ate d in many s ource s.
Neve rthele ss, this m ate rial is far to o imp ortant to all branche s of mathe matics to
b e omitted from a first course. For variety, I have m ade more use of the funda-me ntal group oid than in s tandard tre atments, 1 and my use of it has some nove l
features . For conce ptual inte re st, I have e mphasiz ed diffe rent categorical ways of
mo deling the top ologic al s ituation alge braically, and I have take n the opp ortunity
to intro duc e s ome ide as that are c entral to equivariant alge braic top ology.
Poincar´e duality is als o to o fundame ntal to om it. T he re are more elegant ways
to tre at this topic than the c lassic al one given he re, but I have pre fe rre d to give the
the ory in a quick and standard fas hion that re aches the des ire d c onc lus ions in an
ec onom ic al way. Thus he re I have not pre se nte d the truly mo dern approach that
applies to generalized homology and c ohom ology theories. 2
The reader is warne d that this b o ok is not des igned as a textb o ok, although
it c ould b e us ed as one in exce ptionally s trong graduate programs . Even then, it
would b e imp os sible to cove r all of the m aterial in de tail in a quarter, or e ve n in a
year. There are se ctions that s hould b e om itte d on a firs t reading and others that
are intende d to w he t the stude nt’s app e tite for further de velopm ents. In prac tic e,
when teaching, my lec tures are re gularly inte rrupte d by (purp oseful) digres sions,
mos t often dire ctly prom pte d by the ques tions of s tudents. T hes e intro duce m ore
advanc ed topic s that are not part of the formal intro ductory cours e: c ohomology
op e rations, characte ristic c las se s, K-theory, c ob ordism , etc., are ofte n firs t intro-duc ed e arlier in the le cture s than a line ar deve lopme nt of the sub ject would dic tate.
Thes e digres sions have b een expande d and writte n up he re as sketche s without
com plete pro ofs , in a logically c ohe re nt order, in the last four chapte rs . The se
are topics that I fee l mus t b e intro duce d in s om e fashion in any se rious graduate
leve l intro duction to alge braic top ology. A defect of ne arly all existing texts is
that they do not go far enough into the sub je ct to give a fee l for really subs tantial
applications : the reade r se es sphe re s and pro jec tive space s, mayb e le ns spaces , and
applications acc es sible with knowle dge of the hom ology and cohomology of such
space s. That is not e nough to give a re al fee ling for the s ub jec t. I am aware that
this treatme nt s uff ers the same defec t, at least b efore its sketchy last chapters.
Most chapters end with a se t of proble ms. Mos t of the se ask for com puta-tions and applications base d on the m aterial in the text, s ome extend the theory
and intro duc e furthe r c onc epts, som e as k the reader to furnis h or com plete pro ofs
1 But see R. Brow n’s b o ok c ited in § 2 o f the s ugge stion s for furth er rea ding .
2 Th at ap proach d erive s Poinc ar´e du ality a s a con sequenc e of Span ier-Wh itehe ad an d Atiyah
du ality, via the Tho m is omo rphism for or iented vect or bu ndle s.
omitted in the te xt, and som e are es say que stions w hich implicitly as k the reader
to s ee k ans we rs in othe r sourc es. Proble ms marked ∗ are more difficult or more
p e riphe ral to the main ideas . Most of the se problem s are included in the wee kly
problem s ets that are an integral part of the cours e at Chic ago. In fact, doing the
problems is the heart of the cours e. (The re are no exam s and no grade s; stude nts
are s trongly e nc ouraged to work toge ther, and more work is ass igned than a s tudent
can re asonably b e exp ec ted to complete working alone.) The read er is urged to try
most o f the problems: th is is the w ay to learn t he materia l. T he lecture s fo cus on
the ideas ; their as similation requires m ore c alc ulational example s and applications
than are include d in the te xt.
I have e nded with a brie f and idios yncratic guide to the lite rature for the reader
inte reste d in going further in algebraic top ology.
Thes e notes have e volved over m any years, and I claim no originality for m ost
of the material. In partic ular, m any of the problems , es p ec ially in the m ore class ical
chapters, are the sam e as , or are variants of, problem s that app e ar in othe r texts.
Perhaps this is unavoidable: interes ting problem s that are doable at an early s tage
of the de velopm ent are fe w and far b etwee n. I am e sp e cially aware of my debts to
earlier texts by Mas se y, Gre enb e rg and Harp er, Dold, and Gray.
I am very grate ful to John Gree nle es for his careful reading and suggestions,
es p e cially of the las t three chapters. I am als o grate ful to Igor Kriz for his sugges-tions and for trying out the b o ok at the University of Michigan. B y far my greates t
debt, a c umulative one , is to s eve ral ge ne rations of students, far to o nume rous to
name. They have c aught countle ss infelicities and outright blunde rs , and they have
contribute d quite a fe w of the details. You know who you are . Thank you.
- Paperback: 254 pages
- Publisher: University Of Chicago Press; 1 edition (September 1, 1999)
- Language: English
- ISBN-10: 0226511839
- ISBN-13: 978-0226511832
- Product Dimensions: 9.1 x 6 x 0.6 inches
The first year graduate program in mathe matics at the Unive rs ity of Chic ago
consis ts of three thre e-quarte r c ourse s, in analysis, alge bra, and top ology. The first
two quarters of the top ology se que nc e fo cus on manifold theory and diffe re ntial
geom etry, inc luding diffe re ntial forms and, usually, a glimpse of de Rham c ohom ol-ogy. The third quarte r fo c us es on algebraic top ology. I have b een te aching the
third quarte r off and on s ince around 1970. Be fore that, the top ologists , inc luding
me , thought that it would b e imp ossible to squee ze a s erious intro duc tion to al-gebraic top ology into a one quarte r course , but we were overruled by the analysts
and alge braists, who fe lt that it was unac ce ptable for graduate s tudents to obtain
the ir PhDs w ithout having some contact with alge braic top ology.
This raise s a c onundrum . A large numb er of stude nts at Chic ago go into top ol-ogy, alge braic and geom etric. The intro duc tory c ourse should lay the foundations
for the ir later work, but it s hould als o b e viable as an intro duction to the s ub je ct
suitable for those going into other branches of mathem atic s. T he se note s re flec t
my e fforts to organize the foundations of algebraic top ology in a way that cate rs
to b oth p edagogical goals . There are e vident defe cts from b oth p oints of view . A
treatme nt more close ly attune d to the ne eds of alge braic ge ome te rs and analysts
would inc lude ˇCe ch cohomology on the one hand and de Rham c ohomology and
p e rhaps Morse homology on the othe r. A treatme nt more close ly attuned to the
needs of algebraic top ologists would include sp ec tral s equences and an array of
calc ulations w ith them . I n the end, the overriding p e dagogic al goal has b ee n the
intro duction of bas ic ideas and me tho ds of thought.
Our unders tanding of the foundations of alge braic top ology has undergone sub-tle but se rious changes sinc e I b e gan te aching this cours e. Thes e changes re flec t
in part an e normous internal de ve lopm ent of algebraic top ology ove r this p erio d,
one w hich is large ly unknown to most other m athem atic ians , eve n those working in
such close ly re late d fie lds as geom etric top ology and algebraic geom etry. More over,
this deve lopme nt is p o orly reflec te d in the textb o oks that have app eared over this period.
Let me give a s mall but technically imp ortant example . T he s tudy of gen-eralize d homology and cohomology the orie s p ervades mo dern alge braic top ology.
Thes e theorie s s atis fy the e xc ision axiom. One cons tructs mos t such the ories ho-motopically, by c ons tructing re pres enting ob je cts called s p ec tra, and one must then
prove that e xc ision holds. T he re is a way to do this in general that is no more dif-ficult than the standard verification for s ingular homology and c ohom ology. I find
this pro of far m ore conce ptual and illuminating than the s tandard one e ven when
sp ec ialize d to s ingular homology and cohomology. (It is bas ed on the approxim a-tion of e xc isive triads by we akly e quivalent CW triads .) T his should by now b e a
1
2 IN TRODUC TION
standard approach. Howe ver, to the b e st of my knowle dge , the re e xis ts no rigorous
exp osition of this approach in the lite rature, at any leve l.
More c entrally, there now exist axiom atic treatme nts of large swaths of hom o-topy theory bas ed on Quillen’s the ory of close d m o del c ategorie s. While I do not
think that a first course s hould intro duce such abs trac tions, I do think that the e x-p os ition should give em phasis to those fe atures that the axiomatic approach shows
to b e fundam ental. For example , this is one of the re asons , although by no me ans
the only one, that I have dealt with c ofibrations, fibrations, and weak e quivalences
much more thoroughly than is usual in an intro ductory course .
Some parts of the the ory are dealt with quite class ic ally. The the ory of fun-damental groups and c overing s pac es is one of the few parts of algebraic top ology
that has probably re ached definitive form, and it is we ll tre ate d in many s ource s.
Neve rthele ss, this m ate rial is far to o imp ortant to all branche s of mathe matics to
b e omitted from a first course. For variety, I have m ade more use of the funda-me ntal group oid than in s tandard tre atments, 1 and my use of it has some nove l
features . For conce ptual inte re st, I have e mphasiz ed diffe rent categorical ways of
mo deling the top ologic al s ituation alge braically, and I have take n the opp ortunity
to intro duc e s ome ide as that are c entral to equivariant alge braic top ology.
Poincar´e duality is als o to o fundame ntal to om it. T he re are more elegant ways
to tre at this topic than the c lassic al one given he re, but I have pre fe rre d to give the
the ory in a quick and standard fas hion that re aches the des ire d c onc lus ions in an
ec onom ic al way. Thus he re I have not pre se nte d the truly mo dern approach that
applies to generalized homology and c ohom ology theories. 2
The reader is warne d that this b o ok is not des igned as a textb o ok, although
it c ould b e us ed as one in exce ptionally s trong graduate programs . Even then, it
would b e imp os sible to cove r all of the m aterial in de tail in a quarter, or e ve n in a
year. There are se ctions that s hould b e om itte d on a firs t reading and others that
are intende d to w he t the stude nt’s app e tite for further de velopm ents. In prac tic e,
when teaching, my lec tures are re gularly inte rrupte d by (purp oseful) digres sions,
mos t often dire ctly prom pte d by the ques tions of s tudents. T hes e intro duce m ore
advanc ed topic s that are not part of the formal intro ductory cours e: c ohomology
op e rations, characte ristic c las se s, K-theory, c ob ordism , etc., are ofte n firs t intro-duc ed e arlier in the le cture s than a line ar deve lopme nt of the sub ject would dic tate.
Thes e digres sions have b een expande d and writte n up he re as sketche s without
com plete pro ofs , in a logically c ohe re nt order, in the last four chapte rs . The se
are topics that I fee l mus t b e intro duce d in s om e fashion in any se rious graduate
leve l intro duction to alge braic top ology. A defect of ne arly all existing texts is
that they do not go far enough into the sub je ct to give a fee l for really subs tantial
applications : the reade r se es sphe re s and pro jec tive space s, mayb e le ns spaces , and
applications acc es sible with knowle dge of the hom ology and cohomology of such
space s. That is not e nough to give a re al fee ling for the s ub jec t. I am aware that
this treatme nt s uff ers the same defec t, at least b efore its sketchy last chapters.
Most chapters end with a se t of proble ms. Mos t of the se ask for com puta-tions and applications base d on the m aterial in the text, s ome extend the theory
and intro duc e furthe r c onc epts, som e as k the reader to furnis h or com plete pro ofs
1 But see R. Brow n’s b o ok c ited in § 2 o f the s ugge stion s for furth er rea ding .
2 Th at ap proach d erive s Poinc ar´e du ality a s a con sequenc e of Span ier-Wh itehe ad an d Atiyah
du ality, via the Tho m is omo rphism for or iented vect or bu ndle s.
omitted in the te xt, and som e are es say que stions w hich implicitly as k the reader
to s ee k ans we rs in othe r sourc es. Proble ms marked ∗ are more difficult or more
p e riphe ral to the main ideas . Most of the se problem s are included in the wee kly
problem s ets that are an integral part of the cours e at Chic ago. In fact, doing the
problems is the heart of the cours e. (The re are no exam s and no grade s; stude nts
are s trongly e nc ouraged to work toge ther, and more work is ass igned than a s tudent
can re asonably b e exp ec ted to complete working alone.) The read er is urged to try
most o f the problems: th is is the w ay to learn t he materia l. T he lecture s fo cus on
the ideas ; their as similation requires m ore c alc ulational example s and applications
than are include d in the te xt.
I have e nded with a brie f and idios yncratic guide to the lite rature for the reader
inte reste d in going further in algebraic top ology.
Thes e notes have e volved over m any years, and I claim no originality for m ost
of the material. In partic ular, m any of the problems , es p ec ially in the m ore class ical
chapters, are the sam e as , or are variants of, problem s that app e ar in othe r texts.
Perhaps this is unavoidable: interes ting problem s that are doable at an early s tage
of the de velopm ent are fe w and far b etwee n. I am e sp e cially aware of my debts to
earlier texts by Mas se y, Gre enb e rg and Harp er, Dold, and Gray.
I am very grate ful to John Gree nle es for his careful reading and suggestions,
es p e cially of the las t three chapters. I am als o grate ful to Igor Kriz for his sugges-tions and for trying out the b o ok at the University of Michigan. B y far my greates t
debt, a c umulative one , is to s eve ral ge ne rations of students, far to o nume rous to
name. They have c aught countle ss infelicities and outright blunde rs , and they have
contribute d quite a fe w of the details. You know who you are . Thank you.
