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user69091
user69091
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There are definitely lots of ways of thinking about Chern classes.

Let me just make the following a bit imprecise statement:

Up to tensoring with $\mathbb Q$, a K-theory class (i.e. for compact spaces: a vector bundle up to stable equivalence) is the same as a collection of cohomology classes (i.e. Chern classes).

Edit: Integral results are much harder. Sometimes they are deduced from index theorems, where on one side, you have a characteristic number (the coefficients of the monomials in the Chern classes can be complicated, e.g. Bernoulli numbers show up regularly), and on the other side the index of an operator which must be an integer. See the other answers. This is one way to prove the result for even-dimensional spheres. This relates all three answers present at the moment.

There are definitely lots of ways of thinking about Chern classes.

Let me just make the following a bit imprecise statement:

Up to tensoring with $\mathbb Q$, a K-theory class (i.e. for compact spaces: a vector bundle up to stable equivalence) is the same as a collection of cohomology classes (i.e. Chern classes).

Edit: Integral results are much harder. Sometimes they are deduced from index theorems, where on one side, you have a characteristic number, and on the other side the index of an operator which must be an integer. See the other answers. This is one way to prove the result for even-dimensional spheres.

There are definitely lots of ways of thinking about Chern classes.

Let me just make the following a bit imprecise statement:

Up to tensoring with $\mathbb Q$, a K-theory class (i.e. for compact spaces: a vector bundle up to stable equivalence) is the same as a collection of cohomology classes (i.e. Chern classes).

Edit: Integral results are much harder. Sometimes they are deduced from index theorems, where on one side, you have a characteristic number (the coefficients of the monomials in the Chern classes can be complicated, e.g. Bernoulli numbers show up regularly), and on the other side the index of an operator which must be an integer. This is one way to prove the result for even-dimensional spheres. This relates all three answers present at the moment.

added 64 characters in body
Source Link
user69091
user69091
  • 171
  • 2

There are definitely lots of ways of thinking about Chern classes.

Let me just make the following a bit imprecise statement:

Up to tensoring with $\mathbb Q$, a K-theory class (i.e. for compact spaces: a vector bundle up to stable equivalence) is the same as a collection of cohomology classes (i.e. Chern classes). To attack the integral question does involve number theory

Edit: Integral results are much harder. For example Sometimes they are deduced from index theorems, inwhere on one side, you have a characteristic number, and on the problemother side the index of an operator which must be an integer. See the other answers. This is one way to prove the result for even-dimensional spheres, Bernoulli numbers come up.

There are definitely lots of ways of thinking about Chern classes.

Let me just make the following a bit imprecise statement:

Up to tensoring with $\mathbb Q$, a K-theory class (i.e. for compact spaces: a vector bundle up to stable equivalence) is the same as a collection of cohomology classes (i.e. Chern classes). To attack the integral question does involve number theory. For example, in the problem for even-dimensional spheres, Bernoulli numbers come up.

There are definitely lots of ways of thinking about Chern classes.

Let me just make the following a bit imprecise statement:

Up to tensoring with $\mathbb Q$, a K-theory class (i.e. for compact spaces: a vector bundle up to stable equivalence) is the same as a collection of cohomology classes (i.e. Chern classes).

Edit: Integral results are much harder. Sometimes they are deduced from index theorems, where on one side, you have a characteristic number, and on the other side the index of an operator which must be an integer. See the other answers. This is one way to prove the result for even-dimensional spheres.

Source Link
user69091
user69091
  • 171
  • 2

There are definitely lots of ways of thinking about Chern classes.

Let me just make the following a bit imprecise statement:

Up to tensoring with $\mathbb Q$, a K-theory class (i.e. for compact spaces: a vector bundle up to stable equivalence) is the same as a collection of cohomology classes (i.e. Chern classes). To attack the integral question does involve number theory. For example, in the problem for even-dimensional spheres, Bernoulli numbers come up.