Understanding the cohomology ring of the Grassmannian
Some background first: I'm trying to understand the solution of some enumerative geometry problems, such as proving that a smooth cubic contains $27$ lines. I know that this becomes easier once one understands the cohomology of the Grassmannian.
I know that the Grassmannian can be given a CW-complex structure, but I don't understand how to compute the actual cohomology ring. I think that is the subject of Schubert calculus, and names like Pieri's or Giambelli's formulas often pop up. But I have also read elsewhere, such as in Hatcher's book Vector Bundles and K-Theory, that one can use Chern classes to describe the cohomology ring.
My question is, how are the two approaches related, and, most importantly, what is a comprehensive textbook on the subject?
reference-request homology-cohomology grassmannian schubert-calculus
edited 2 days ago
Matt Samuel
37.5k63665
asked Mar 25 '17 at 8:23
un umile appassionatoun umile appassionato
367316
add a comment |
Some background first: I'm trying to understand the solution of some enumerative geometry problems, such as proving that a smooth cubic contains $27$ lines. I know that this becomes easier once one understands the cohomology of the Grassmannian.
I know that the Grassmannian can be given a CW-complex structure, but I don't understand how to compute the actual cohomology ring. I think that is the subject of Schubert calculus, and names like Pieri's or Giambelli's formulas often pop up. But I have also read elsewhere, such as in Hatcher's book Vector Bundles and K-Theory, that one can use Chern classes to describe the cohomology ring.
My question is, how are the two approaches related, and, most importantly, what is a comprehensive textbook on the subject?
reference-request homology-cohomology grassmannian schubert-calculus
edited 2 days ago
Matt Samuel
37.5k63665
asked Mar 25 '17 at 8:23
un umile appassionatoun umile appassionato
367316
1
Milnor does not have a book called Vector Bundles and K-Theory. There is a book in progress by Hatcher of that name.
– Michael Albanese
Mar 25 '17 at 15:04
Thank you. I have been reading Milnor's Characteristic classes and got the two mixed up.
– un umile appassionato
Mar 26 '17 at 12:13
1
Eisenbud and Harris 3264 and All That: A Second Course in Algebraic Geometry.
– Jan-Magnus Økland
Mar 27 '17 at 9:18
add a comment |
Some background first: I'm trying to understand the solution of some enumerative geometry problems, such as proving that a smooth cubic contains $27$ lines. I know that this becomes easier once one understands the cohomology of the Grassmannian.
I know that the Grassmannian can be given a CW-complex structure, but I don't understand how to compute the actual cohomology ring. I think that is the subject of Schubert calculus, and names like Pieri's or Giambelli's formulas often pop up. But I have also read elsewhere, such as in Hatcher's book Vector Bundles and K-Theory, that one can use Chern classes to describe the cohomology ring.
My question is, how are the two approaches related, and, most importantly, what is a comprehensive textbook on the subject?
reference-request homology-cohomology grassmannian schubert-calculus
edited 2 days ago
Matt Samuel
37.5k63665
asked Mar 25 '17 at 8:23
un umile appassionatoun umile appassionato
367316
Some background first: I'm trying to understand the solution of some enumerative geometry problems, such as proving that a smooth cubic contains $27$ lines. I know that this becomes easier once one understands the cohomology of the Grassmannian.
I know that the Grassmannian can be given a CW-complex structure, but I don't understand how to compute the actual cohomology ring. I think that is the subject of Schubert calculus, and names like Pieri's or Giambelli's formulas often pop up. But I have also read elsewhere, such as in Hatcher's book Vector Bundles and K-Theory, that one can use Chern classes to describe the cohomology ring.
My question is, how are the two approaches related, and, most importantly, what is a comprehensive textbook on the subject?
reference-request homology-cohomology grassmannian schubert-calculus
reference-request homology-cohomology grassmannian schubert-calculus
edited 2 days ago
Matt Samuel
37.5k63665
asked Mar 25 '17 at 8:23
un umile appassionatoun umile appassionato
367316
edited 2 days ago
Matt Samuel
37.5k63665
asked Mar 25 '17 at 8:23
un umile appassionatoun umile appassionato
367316
edited 2 days ago
Matt Samuel
37.5k63665
edited 2 days ago
Matt Samuel
37.5k63665
edited 2 days ago
Matt Samuel
37.5k63665
37.5k63665
asked Mar 25 '17 at 8:23
un umile appassionatoun umile appassionato
367316
asked Mar 25 '17 at 8:23
un umile appassionatoun umile appassionato
367316
asked Mar 25 '17 at 8:23
un umile appassionatoun umile appassionato
367316
367316
1
Milnor does not have a book called Vector Bundles and K-Theory. There is a book in progress by Hatcher of that name.
– Michael Albanese
Mar 25 '17 at 15:04
Thank you. I have been reading Milnor's Characteristic classes and got the two mixed up.
– un umile appassionato
Mar 26 '17 at 12:13
1
Eisenbud and Harris 3264 and All That: A Second Course in Algebraic Geometry.
– Jan-Magnus Økland
Mar 27 '17 at 9:18
add a comment |
1
Milnor does not have a book called Vector Bundles and K-Theory. There is a book in progress by Hatcher of that name.
– Michael Albanese
Mar 25 '17 at 15:04
Thank you. I have been reading Milnor's Characteristic classes and got the two mixed up.
– un umile appassionato
Mar 26 '17 at 12:13
1
Eisenbud and Harris 3264 and All That: A Second Course in Algebraic Geometry.
– Jan-Magnus Økland
Mar 27 '17 at 9:18
1
1
Milnor does not have a book called Vector Bundles and K-Theory. There is a book in progress by Hatcher of that name.
– Michael Albanese
Mar 25 '17 at 15:04
Milnor does not have a book called Vector Bundles and K-Theory. There is a book in progress by Hatcher of that name.
– Michael Albanese
Mar 25 '17 at 15:04
Thank you. I have been reading Milnor's Characteristic classes and got the two mixed up.
– un umile appassionato
Mar 26 '17 at 12:13
Thank you. I have been reading Milnor's Characteristic classes and got the two mixed up.
– un umile appassionato
Mar 26 '17 at 12:13
1
1
Eisenbud and Harris 3264 and All That: A Second Course in Algebraic Geometry.
– Jan-Magnus Økland
Mar 27 '17 at 9:18
Eisenbud and Harris 3264 and All That: A Second Course in Algebraic Geometry.
– Jan-Magnus Økland
Mar 27 '17 at 9:18
add a comment |
1 Answer
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The two texts I have most often been referred to for studying the cohomology of the Grassmannian are Young Tableaux by Fulton and Symmetric Functions, Schubert Polynomials, and Degeneracy Loci by Manivel. From my perspective, Manivel takes a fairly combinatorial perspective while Part III of Fulton gives an excellent treatment of this question from a more geometric perspective.
answered Oct 29 '17 at 23:09
ZachZach
748313
add a comment |
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The two texts I have most often been referred to for studying the cohomology of the Grassmannian are Young Tableaux by Fulton and Symmetric Functions, Schubert Polynomials, and Degeneracy Loci by Manivel. From my perspective, Manivel takes a fairly combinatorial perspective while Part III of Fulton gives an excellent treatment of this question from a more geometric perspective.
answered Oct 29 '17 at 23:09
ZachZach
748313
add a comment |
The two texts I have most often been referred to for studying the cohomology of the Grassmannian are Young Tableaux by Fulton and Symmetric Functions, Schubert Polynomials, and Degeneracy Loci by Manivel. From my perspective, Manivel takes a fairly combinatorial perspective while Part III of Fulton gives an excellent treatment of this question from a more geometric perspective.
answered Oct 29 '17 at 23:09
ZachZach
748313
add a comment |
The two texts I have most often been referred to for studying the cohomology of the Grassmannian are Young Tableaux by Fulton and Symmetric Functions, Schubert Polynomials, and Degeneracy Loci by Manivel. From my perspective, Manivel takes a fairly combinatorial perspective while Part III of Fulton gives an excellent treatment of this question from a more geometric perspective.
answered Oct 29 '17 at 23:09
ZachZach
748313
The two texts I have most often been referred to for studying the cohomology of the Grassmannian are Young Tableaux by Fulton and Symmetric Functions, Schubert Polynomials, and Degeneracy Loci by Manivel. From my perspective, Manivel takes a fairly combinatorial perspective while Part III of Fulton gives an excellent treatment of this question from a more geometric perspective.
answered Oct 29 '17 at 23:09
ZachZach
748313
answered Oct 29 '17 at 23:09
ZachZach
748313
answered Oct 29 '17 at 23:09
ZachZach
748313
answered Oct 29 '17 at 23:09
ZachZach
748313
748313
add a comment |
add a comment |
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1
Milnor does not have a book called Vector Bundles and K-Theory. There is a book in progress by Hatcher of that name.
– Michael Albanese
Mar 25 '17 at 15:04
Thank you. I have been reading Milnor's Characteristic classes and got the two mixed up.
– un umile appassionato
Mar 26 '17 at 12:13
1
Eisenbud and Harris 3264 and All That: A Second Course in Algebraic Geometry.
– Jan-Magnus Økland
Mar 27 '17 at 9:18