Intereting Posts

Does dividing by zero ever make sense?
List of Interesting Math Blogs
Relation between Borel–Cantelli lemmas and Kolmogorov's zero-one law
Prove that any two left cosets $aH, bH$ either coincide or are disjoint, and prove Lagrange's theorem
Are there real numbers that are neither rational nor irrational?
Fractals using just modulo operation
Useful techniques of experimental mathematics (reference request)
Continuous functions are differentiable on a measurable set?
Strictly convex Inequality in $l^p$
Get the last two digits of $16^{100}$ and $17^{100}$
If $n = 51! +1$, then find number of primes among $n+1,n+2,\ldots, n+50$
Minimum value of $2^{\sin^2x}+2^{\cos^2x}$
How can the following language be determined in polynomial time
Rearrangements that never change the value of a sum
Why is the restricted direct product topology on the idele group stronger than the topology induced by the adele group?

Suppose I have a smooth manifold $M$, and want to consider the $K$-theory $K^0(M)$. I could define this in the usual way (by taking the Grothendieck group of the monoid of equivalence classes of vector bundles) or in a ‘smooth’ way (considering *only smooth* vector bundles, and taking the Grothendieck group as usual).

I haven’t seen this discussed anywhere. Is there any difference between the two approaches? And are there any references in which this question is discussed?

(Even better: if $M$ is a $G$-space for a compact Lie group $G$, is the *equivariant* $K$-theory affected by taking only smooth bundles?)

- Smooth surfaces that isn't the zero-set of $f(x,y,z)$
- Why are we interested in closed geodesics?
- Topologies and manifolds
- How do I know when a form represents an integral cohomology class?
- Geodesics on the torus
- When is a fibration a fiber bundle?

- Good introductory probability book for graduate level?
- Explicit Riemann mappings
- Is there a different name for strongly Darboux functions?
- Algebraic independence via the Jacobian
- Open source lecture notes and textbooks
- Good book recommendations on trigonometry
- ArcTan(2) a rational multiple of $\pi$?
- Two-sided Laplace transform
- Smooth surfaces that isn't the zero-set of $f(x,y,z)$
- Books that develop interest & critical thinking among high school students

The $K^0(M)$ based on continuous vector bundles is the same as that based on smooth vector bundles because of the fundamental (and perhaps not sufficiently advertised) result:

**Theorem** Every continuous vector bundle on a $C^\infty$ manifold has a compatible $C^\infty$ vector bundle structure. Such a structure is unique up to $C^\infty$ isomorphism.

You can find a proof in Hirsch’s *Differential Topology* , Chapter 4, Theorem 3.5, page 101.

- Maximum of Polynomials in the Unit Circle
- Help with Induction problem?
- Effect the zero vector has on the dimension of affine hulls and linear hulls
- Does commutativity imply Associativity?
- Chain rule proof doubt
- Proof of infinitude of primes using the irrationality of π
- Two problems on real number series
- Series in Real Analysis
- show that if $n\geq1$, $(1+{1\over n})^n<(1+{1\over n+1})^{n+1}$
- Chinese Remainder Theorem and linear congruences
- Who invented $\vee$ and $\wedge$, $\forall$ and $\exists$?
- Trying to understand why circle area is not $2 \pi r^2$
- Expected revenue obtained by the Vickery auction with reserve price $1/2$
- Knight returning to corner on chessboard — average number of steps
- Non-standard models of arithmetic for Dummies (2)