Introduction to Algebraic Topology and Algebraic Geometry - U. Bruzzo
Thes e notes ass emble the contents of the intro ductory course s I have b ee n giving at
SISSA since 1995/96. Originally the c ourse was intende d as intro duc tion to (c omple x)
algebraic geome try for s tudents with an educ ation in theoretical phys ic s, to help them to
mas ter the basic algebraic ge ometric to ols ne ce ss ary for doing res earch in algebraically
inte grable sys tems and in the geome try of quantum fie ld theory and s tring theory. T his
motivation still transpire s from the chapters in the s ec ond part of thes e note s.
The firs t part on the contrary is a brie f but rathe r s ys tematic intro duc tion to two
topic s, singular homology (C hapter 2) and s he af the ory, inc luding their c ohom ology
(Chapte r 3). Chapter 1 as se mbles som e basic s fact in homologic al algebra and deve lops
the firs t rudim ents of de R ham cohomology, w ith the aim of providing an example to
the various abstract construc tions.
Chapte r 4 is an intro duction to s p e ctral se que nc es, a rathe r intricate but very p ower-ful c omputation to ol. The e xam ple s provided here are from sheaf theory but this c om-putational te chniques is als o very use ful in alge braic top ology.
I thank all my c olleagues and s tudents, in Trie ste and Genova and other lo cations ,
who have help ed me to clarify s ome is sues related to these notes , or have p ointed out
mis take s. I n this conne ction s p ec ial thanks are due to Fabio Pioli. Mos t of Chapte r 3 is
an adaptation of m aterial taken from [2 ]. I thank my friends and c ollab orators Claudio
Barto cc i and Daniel He rn´ande z Ruip´erez for granting p e rmiss ion to us e that material.
I thank Lothar G¨otts che for us eful s ugge stions and for p ointing out an error and the
stude nts of the 2002/2003 course for their interes t and c ons tant fe edback.
- PDF: 395 pages
- Language: English
Thes e notes ass emble the contents of the intro ductory course s I have b ee n giving at
SISSA since 1995/96. Originally the c ourse was intende d as intro duc tion to (c omple x)
algebraic geome try for s tudents with an educ ation in theoretical phys ic s, to help them to
mas ter the basic algebraic ge ometric to ols ne ce ss ary for doing res earch in algebraically
inte grable sys tems and in the geome try of quantum fie ld theory and s tring theory. T his
motivation still transpire s from the chapters in the s ec ond part of thes e note s.
The firs t part on the contrary is a brie f but rathe r s ys tematic intro duc tion to two
topic s, singular homology (C hapter 2) and s he af the ory, inc luding their c ohom ology
(Chapte r 3). Chapter 1 as se mbles som e basic s fact in homologic al algebra and deve lops
the firs t rudim ents of de R ham cohomology, w ith the aim of providing an example to
the various abstract construc tions.
Chapte r 4 is an intro duction to s p e ctral se que nc es, a rathe r intricate but very p ower-ful c omputation to ol. The e xam ple s provided here are from sheaf theory but this c om-putational te chniques is als o very use ful in alge braic top ology.
I thank all my c olleagues and s tudents, in Trie ste and Genova and other lo cations ,
who have help ed me to clarify s ome is sues related to these notes , or have p ointed out
mis take s. I n this conne ction s p ec ial thanks are due to Fabio Pioli. Mos t of Chapte r 3 is
an adaptation of m aterial taken from [2 ]. I thank my friends and c ollab orators Claudio
Barto cc i and Daniel He rn´ande z Ruip´erez for granting p e rmiss ion to us e that material.
I thank Lothar G¨otts che for us eful s ugge stions and for p ointing out an error and the
stude nts of the 2002/2003 course for their interes t and c ons tant fe edback.
