Reference on complex cobordismTextbook or lecture notes in topological K-TheoryHeat kernel estimates and...
Reference on complex cobordism
Textbook or lecture notes in topological K-TheoryHeat kernel estimates and Gaussian estimates for semigroups, good reference?How is the differential in complex cobordism defined?Algebraic De Rham cup product versus Betti cup productPullback of the tautological vector bundle and the nubmer of trivializationsSuitable reference on smooth manifolds for qualifying exam study?$C_2$-equivariant Betti realization of MGLQuestions about the exceptional zeros of Dirichlet $L$-functionsHow would you organize a cycle of seminars aimed at learning together some basics of Derived Algebraic Geometry?Is the $hat{A}$-genus invariant under crepant birational maps between smooth algebraic varieties?
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I am trying to study a little of algebraic cobordism and I lack background from the classic setting. Hence, I am looking for a textbook or expository writing covering the basics of complex cobordism.
Do you know any suitable reference for the basics of complex cobordism?
If possible, I would like the reference to cover a particular result. Let $E$ be a spectrum representing a cohomology theory and $MU$ the spectrum representing complex cobordism. As an analogy with the algebraic case, it should hold that
$$
E^*(MU)simeq E^*(pt)[[c_1,c_2, ldots]]
$$
where $c_i$ are the universal Chern classes. In other words, $E^*(MU)simeq E^*(Gr)$ where $Gr$ denotes the infinite Grassmanian. If possible, I would like the reference to cover such result and its surroundings.
reference-request at.algebraic-topology stable-homotopy cobordism orientation
asked 1 hour ago
TintinTintin
1,200823
$endgroup$
add a comment |
$begingroup$
I am trying to study a little of algebraic cobordism and I lack background from the classic setting. Hence, I am looking for a textbook or expository writing covering the basics of complex cobordism.
Do you know any suitable reference for the basics of complex cobordism?
If possible, I would like the reference to cover a particular result. Let $E$ be a spectrum representing a cohomology theory and $MU$ the spectrum representing complex cobordism. As an analogy with the algebraic case, it should hold that
$$
E^*(MU)simeq E^*(pt)[[c_1,c_2, ldots]]
$$
where $c_i$ are the universal Chern classes. In other words, $E^*(MU)simeq E^*(Gr)$ where $Gr$ denotes the infinite Grassmanian. If possible, I would like the reference to cover such result and its surroundings.
reference-request at.algebraic-topology stable-homotopy cobordism orientation
asked 1 hour ago
TintinTintin
1,200823
$endgroup$
add a comment |
$begingroup$
I am trying to study a little of algebraic cobordism and I lack background from the classic setting. Hence, I am looking for a textbook or expository writing covering the basics of complex cobordism.
Do you know any suitable reference for the basics of complex cobordism?
If possible, I would like the reference to cover a particular result. Let $E$ be a spectrum representing a cohomology theory and $MU$ the spectrum representing complex cobordism. As an analogy with the algebraic case, it should hold that
$$
E^*(MU)simeq E^*(pt)[[c_1,c_2, ldots]]
$$
where $c_i$ are the universal Chern classes. In other words, $E^*(MU)simeq E^*(Gr)$ where $Gr$ denotes the infinite Grassmanian. If possible, I would like the reference to cover such result and its surroundings.
reference-request at.algebraic-topology stable-homotopy cobordism orientation
asked 1 hour ago
TintinTintin
1,200823
$endgroup$
I am trying to study a little of algebraic cobordism and I lack background from the classic setting. Hence, I am looking for a textbook or expository writing covering the basics of complex cobordism.
Do you know any suitable reference for the basics of complex cobordism?
If possible, I would like the reference to cover a particular result. Let $E$ be a spectrum representing a cohomology theory and $MU$ the spectrum representing complex cobordism. As an analogy with the algebraic case, it should hold that
$$
E^*(MU)simeq E^*(pt)[[c_1,c_2, ldots]]
$$
where $c_i$ are the universal Chern classes. In other words, $E^*(MU)simeq E^*(Gr)$ where $Gr$ denotes the infinite Grassmanian. If possible, I would like the reference to cover such result and its surroundings.
reference-request at.algebraic-topology stable-homotopy cobordism orientation
reference-request at.algebraic-topology stable-homotopy cobordism orientation
asked 1 hour ago
TintinTintin
1,200823
asked 1 hour ago
TintinTintin
1,200823
asked 1 hour ago
TintinTintin
1,200823
asked 1 hour ago
TintinTintin
1,200823
asked 1 hour ago
TintinTintin
1,200823
1,200823
add a comment |
add a comment |
1 Answer
1
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oldest
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$begingroup$
This is worked out in part 2 of
Adams, J. F., Stable homotopy and generalised homology, Chicago Lectures in Mathematics. Chicago - London: The University of Chicago Press. X, 373 p. £ 3.00 (1974). ZBL0309.55016.
(note that to understand part 2 you need to have read part 3 first. Yeah, I know)
In particular the result you want is true only for complex orientable cohomology theories (those theories for which we can define $c_1$). This is to be expected, since what you want is essentially a version of the Thom isomorphism, that should hold only if the vector bundle is orientable for your cohomology theory.
Also, Adams works on homology rather than cohomology (it is a lot more convenient when having non-finite spaces like $BU_n$), but of course once you've proven you have a Thom class, the Thom isomorphism theorem holds in cohomology as well, thus proving the statement you're after.
answered 1 hour ago
Denis NardinDenis Nardin
8,45723360
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add a comment |
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1 Answer
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1 Answer
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oldest
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oldest
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$begingroup$
This is worked out in part 2 of
Adams, J. F., Stable homotopy and generalised homology, Chicago Lectures in Mathematics. Chicago - London: The University of Chicago Press. X, 373 p. £ 3.00 (1974). ZBL0309.55016.
(note that to understand part 2 you need to have read part 3 first. Yeah, I know)
In particular the result you want is true only for complex orientable cohomology theories (those theories for which we can define $c_1$). This is to be expected, since what you want is essentially a version of the Thom isomorphism, that should hold only if the vector bundle is orientable for your cohomology theory.
Also, Adams works on homology rather than cohomology (it is a lot more convenient when having non-finite spaces like $BU_n$), but of course once you've proven you have a Thom class, the Thom isomorphism theorem holds in cohomology as well, thus proving the statement you're after.
answered 1 hour ago
Denis NardinDenis Nardin
8,45723360
$endgroup$
add a comment |
$begingroup$
This is worked out in part 2 of
Adams, J. F., Stable homotopy and generalised homology, Chicago Lectures in Mathematics. Chicago - London: The University of Chicago Press. X, 373 p. £ 3.00 (1974). ZBL0309.55016.
(note that to understand part 2 you need to have read part 3 first. Yeah, I know)
In particular the result you want is true only for complex orientable cohomology theories (those theories for which we can define $c_1$). This is to be expected, since what you want is essentially a version of the Thom isomorphism, that should hold only if the vector bundle is orientable for your cohomology theory.
Also, Adams works on homology rather than cohomology (it is a lot more convenient when having non-finite spaces like $BU_n$), but of course once you've proven you have a Thom class, the Thom isomorphism theorem holds in cohomology as well, thus proving the statement you're after.
answered 1 hour ago
Denis NardinDenis Nardin
8,45723360
$endgroup$
add a comment |
$begingroup$
This is worked out in part 2 of
Adams, J. F., Stable homotopy and generalised homology, Chicago Lectures in Mathematics. Chicago - London: The University of Chicago Press. X, 373 p. £ 3.00 (1974). ZBL0309.55016.
(note that to understand part 2 you need to have read part 3 first. Yeah, I know)
In particular the result you want is true only for complex orientable cohomology theories (those theories for which we can define $c_1$). This is to be expected, since what you want is essentially a version of the Thom isomorphism, that should hold only if the vector bundle is orientable for your cohomology theory.
Also, Adams works on homology rather than cohomology (it is a lot more convenient when having non-finite spaces like $BU_n$), but of course once you've proven you have a Thom class, the Thom isomorphism theorem holds in cohomology as well, thus proving the statement you're after.
answered 1 hour ago
Denis NardinDenis Nardin
8,45723360
$endgroup$
This is worked out in part 2 of
Adams, J. F., Stable homotopy and generalised homology, Chicago Lectures in Mathematics. Chicago - London: The University of Chicago Press. X, 373 p. £ 3.00 (1974). ZBL0309.55016.
(note that to understand part 2 you need to have read part 3 first. Yeah, I know)
In particular the result you want is true only for complex orientable cohomology theories (those theories for which we can define $c_1$). This is to be expected, since what you want is essentially a version of the Thom isomorphism, that should hold only if the vector bundle is orientable for your cohomology theory.
Also, Adams works on homology rather than cohomology (it is a lot more convenient when having non-finite spaces like $BU_n$), but of course once you've proven you have a Thom class, the Thom isomorphism theorem holds in cohomology as well, thus proving the statement you're after.
answered 1 hour ago
Denis NardinDenis Nardin
8,45723360
edited 39 mins ago
answered 1 hour ago
Denis NardinDenis Nardin
8,45723360
answered 1 hour ago
Denis NardinDenis Nardin
8,45723360
answered 1 hour ago
Denis NardinDenis Nardin
8,45723360
8,45723360
add a comment |
add a comment |
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