Intereting Posts

Does $\int_{1}^{\infty} g(x)\ dx$ imply $\lim\limits_{x\to \infty}g(x)=0$?
Cayley's Theorem – Questions on Proof Blueprint
About the definition of Cech Cohomology
Proving that the smooth, compactly supported functions are dense in $L^2$.
Field extension of composite degree has a non-trivial sub-extension
Show that $y = \frac{2x}{x^2 +1}$ lies between $-1$ and $1$ inclusive.
Show that if $G$ is simple a graph with $n$ vertices and the number of edges $m>\binom{n-1}{2}$, then $G$ is connected.
$x_1+x_2+\cdots+x_n\leq M$: Cardinality of Solution Set is $C(M+n, n)$
Reducibility of $P(X^2)$
Undergrad Student Trying to Figure Out What to Study
Integral 0f a function with range in convex set is in the convex set
Does the sum of reciprocals of primes converge?
Height one prime ideal of arithmetical rank greater than 1
A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language
If $n$ is a positive integer, Prove that $\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac{2329}{3600}.$

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?)

- Product of manifolds & orientability
- Fundamental solution to Laplace equation on arbitrary Riemann surfaces
- The graph of a smooth real function is a submanifold
- Topology on Klein bottle?
- Easier proof about suspension of a manifold
- How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?

- Does Hartshorne *really* not define things like the composition or restriction of morphisms of schemes?
- Geometric understanding of differential forms.
- Geodesic equations and christoffel symbols
- What is more elementary than: Introduction to Stochastic Processes by Lawler
- The only 1-manifolds are $\mathbb R$ and $S^1$
- Books, Video lectures, other resources to Teach Yourself Analysis
- The structure of $(\mathbb Z/525\mathbb Z)^\times$
- Is a simple curve which is nulhomotopic the boundary of a surface?
- Homology of a co-h-space manifold
- What is the best book to learn probability?

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.

- Why are projective spaces and varieties preferable?
- Difference between variables, parameters and constants
- References to integrals of the form $\int_{0}^{1} \left( \frac{1}{\log x}+\frac{1}{1-x} \right)^{m} \, dx$
- A subgroup of a cyclic group is cyclic – Understanding Proof
- Why would $(A^{\text T}A+\lambda I)^{-1}A^{\text T}$ be close to $A^{\dagger}$ when $A$ is with rank deficiency?
- Formal definition of conditional probability
- Can $S^4$ be the cotangent bundle of a manifold?
- How to interpret Hessian of a function
- Integer partition with fixed number of summands but without order
- Convergents of square root of 2
- Getting my number theoretic series straight
- How to compute rational or integer points on elliptic curves
- A lemma about extension of function
- maximum curvature of 2D Cubic Bezier
- Why must polynomial division be done prior to taking the limit?