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9h
comment $K$ theory and singular cohomology
An example where the Chern character of a vector bundle doesn't belong to the direct sum is $X = \mathbb{CP}^{\infty}$ and $E$ the tautological line bundle. Then $\operatorname{ch}_k(E) = c_1(E)^k/k! \neq 0$, so $\operatorname{ch}(E) \not\in H^*(\mathbb{CP}^{\infty}; \mathbb{Q})$, but rather $\operatorname{ch}(E) \in \prod H^k(\mathbb{CP}^{\infty};\mathbb{Q})$.
9h
revised $K$ theory and singular cohomology
edited title
9h
comment $K$ theory and singular cohomology
If $X$ is an arbitrary cell complex, then $\operatorname{ch}(E)$ could be an infinite sum of non-zero terms, and hence not belong to $H^*(X; \mathbb{Q})$ which is the direct sum of the groups $H^k(X; \mathbb{Q})$. Instead the $\operatorname{ch}(E)$ would belong to the direct product of the groups. Note, if one assumes $X$ is a finite cell-complex, then there are only finitely many non-zero cohomology groups, there is no problem (the direct sum and direct product coincide).
Feb
3
reviewed No Action Needed tree properties on $\omega_1$ and $\omega_2$
Feb
2
reviewed No Action Needed Correspondences as generalized morphism between $C^*$-algebras
Feb
2
revised The Operator $\overline{\partial} + \overline{\partial}^*$ on an Hermitian Manifold
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Jan
26
revised some terminologies on limiting mixed hodge structures or rather Derived categories
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Jan
25
reviewed Edit Is there an example of a Killing vector field on a complete Riemannian manifold with finite volume?
Jan
25
revised Is there an example of a Killing vector field on a complete Riemannian manifold with finite volume?
Fixed mathjax
Jan
24
comment Stiefel-Whitney classes of closed topological manifolds with no smooth structure
Unless I'm mistaken, the proof you're referring to just uses contradiction (if the $E_8$ manifold had a smooth structure, then...). This proof doesn't require a notion of Stiefel-Whitney classes for closed topological manifolds which is what my question is about.
Jan
24
asked Stiefel-Whitney classes of closed topological manifolds with no smooth structure
Jan
24
revised Are the stiefel-Whitney classes of the tangent bundle determined by the mod 2 cohomology?
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Jan
23
revised Vanishing of Euler class
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Jan
18
reviewed No Action Needed Form of finite dimensional contractive projection in $L_p$
Jan
18
reviewed No Action Needed Non-reflexive Orlicz spaces
Jan
11
reviewed No Action Needed Reference request for Plancherel measure
Jan
7
comment embedding of quaternionic projective spaces
If I'm not mistaken, $12 \leq N \leq 16$, the lower bound coming from a standard calculation involving Stiefel-Whitney classes, and the upper bound coming from the Whitney embedding theorem.
Jan
5
revised Analogue to covering space for higher homotopy groups?
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Jan
5
revised Analogue to covering space for higher homotopy groups?
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Dec
30
reviewed No Action Needed On the universal property of the completion of an ordered field