Serre-Swan Theorem and some K groups

This is an overview on Serre-Swan theorem and some ideas on the construction of K-groups for a Banach category. Serre-Swan theorem establishes equivalences between the categories of topological vector bundles over a compact Hausdorff space X, the category of finitely generated projective C(X)-modules and the categories of algebraic vector bundles of finite rank over
the affine scheme \mathrm{Spec}C(X). This theorem connects different objects of interest in K-theory.
It also introduces some ideas on the construction of K-groups for a Banach category and
in particular for compact topological spaces and Banach algebras.

This is my course thesis for Algebraic Topology II at UIUC.

Calculate (co)limits as (co)equalisers (two examples)

There is a general formulation for constructing limits as equalisers: see Theorem 1 in Section V.2, Maclane. For the dual version, see Theorem A.2.1 in Appendix A written by me.

The constructions look like these (see the links above for details):

limitscolimits

But in practice, these diagrams may not be helpful to see what the equalisers should be. Now I give proofs for the (co)equalisers in two examples: the connected component of a simplicial set and the sheaf condition.

The connected components [Background]

For definitions and other backgrounds, see Subsection 00G5. For the record, see [P12, DLOR07] for the cosimplicial identities and Tag 000G for simplicial identities. (These identities are used in my proofs.)

Two examples of (co)limits as (co)equalisers

Pdf here: Two examples of (co)limits as (co)equalisers

exampleexample1example2example3example4example5

[Short Notes] Non-compactness of the closed unit ball in an infinite-dimensional Banach space

This is about an exercise in [Bass]:

Exercise 19.5. Prove that if H is infinte-dimensional, that is, it has no finite basis, then the closed unit ball in H is not compact.

Proof. Choose an orthonormal basis \{x_i\}, then ||x_i-x_j||^2=||x_i||^2+||x_j||^2=2. This means the sequence is not Cauchy hence has no convergent subsequence.

For a Banach space, by Riesz’s lemma to find a non-Cauchy sequence.

 

[Bass] Bass, R. F. (2013). Real analysis for graduate students. Createspace Ind Pub.

Lusin’s Theorem and Continuous Extension

Here we give proofs for two versions of Lusin’s Theorem, one from Exercise 44, Ch2 in Folland’s Real Analysis and the other from the textbook used for my first year undergraduate mathematical analysis course in Beijing.  The latter version is a stronger result which in addition discusses the condition for a real-valued function defined on a subset of \mathbb{R}^n to be extended to the whole of \mathbb{R}^n. A more general result in topology is the Tietze Extension Theorem. 

See the full post here: Lusin’s Theorem and Continuous Extension

Here we let \mu denote the Lebesgue measure on \mathbb{R}.

Lusin’s Theorem (Version 1)[Exercise 2.44, Folland]. Suppose E\subset \mathbb{R}^n is Lebesgue  measurable, f: E\to \mathbb{R} is Lebesgue measurable and \epsilon> 0, there is a compact set F\subset E such that \mu(F^c)<\epsilon and f|_F is continuous.

Lusin’s Theorem(Version 2)[Huan]. Suppose E\subset \mathbb{R}^n is Lebesgue measurable and f: E\to \bar{\mathbb{R}} is a Lebesgue measurable extended real valued function with \mu(|f|=+ \infty)=0, then  \forall \epsilon >0, \exists g\in C(E) such that \mu(f\neq g)<\epsilon, where C(E) denotes the space of continuous function on E

Continuous Extension Theorem[Huan]. Suppose E\subset \mathbb{R}^n, then f can be extended to a continuous function on \mathbb{R}^n if and only if f can be extended to a continuous function on the closure \bar{E} of E.

Tietze Extension Theorem. Let X be normal and F \subset X be closed and let f: F \to R be continuous. Then there is a map g: X \to R such that
g(x) = f(x) for all x\in F. (Note that in topology, by a map we mean a continuous function. )

 

Krull’s Principal Ideal Theorem in Dimension Theory and Regularity

This post is about some applications of Krull’s Principal Ideal Theorem and regular local rings in dimension theory and regularity of schemes [Part IV, Vakil], with the aim of connecting the 2018-2019 Warwick course MA4H8 Ring Theory with algebraic geometry. The lecture notes/algebraic references are here:  2018-2019 Ring Theory.  Note that the algebraic results included here follow the notes. Alternatively, one can also find them in [Vakil] either as exercises or proved results for which I have included the references.

Besides including results in both their geometric and algebraic statements, I have given proofs to a selection of exercises in Part IV, [Vakil] to illustrate more applications and other connections to the contents in the Ring Theory course. The indexes for exercises follow those in [Vakil].

See here for the full post: Application of Krull’s Principal Ideal Theorem

Please also let me know if you find any errors or have suggestions on any of my posts.

The “Dunce Cap” Space Is Contractible

Here is the exercise 6 on P. 50 in the book Topology and Geometry by Glen Bredon. I put it here because I found the drawing of this cap very lovely. Indeed I like that most of the pictures in this book are lovely sketches.

Question. The “dunce cap” space is the quotient of a triangle (and interior) obtained by identifying all three edges in an inconsistent manner. That is, if the vertices of the triangle are p, q, r then we identify the line segment (p, q) with (q, r) and with (p, r) in the
orientation indicated by the order given of the vertices. (See Figure 1-6.) Show that
the dunce cap is contractible.the dunce cap

Following the development in the book, I will use the following theorem that the homotopy type of a mapping cylinder or cone
depends only on the homotopy class of the map [Theorem 14.18, Topology and Geometry by Glen Bredon]. The idea is to identify the dunce cap as a mapping cone.

Theorem 14.18. If f_0\simeq f_1:X\to Y \text{ then } M_{f_0 } \simeq M_{f_1}\text { rel } X+Y \text{ and }C_{f_0}\simeq C_{f_1}\text{ rel } Y+\mathrm{vertex}.

Proof of the Qustion. Suppose f: S^1\to S^1 is a map from S^1 to itself. The cone C_f=M_f/S^1 \times\{1\} for f is obtained by pinching the top of the mapping cylinder to a point. As M_f  is the cylinder S_1\times [0,1] with the bottom pasted to S_1 by the map f, C_f is D_2 with \partial D_2 pasted to S_1 by the map f. So the dunce cap is just C_f \text{ with } f: S^1\to S^1 defined as

f(e^{2\pi i t})= \begin{cases} e^{2\pi i (3t)}, 0\leq t\leq 2/3\\ e^{2\pi i(2- 3t)}, 2/3\leq t\leq 1. \end{cases}

which is homotopic to the identity by a linear homotopy (note that we make the choice of f for an easy definition of the homotopy)

H(e^{2\pi i t},s)= \begin{cases} e^{2\pi i (3t(1-s)+st}, 0\leq t\leq 2/3\\ e^{2\pi i[(2- 3t)(1-s)+st]}, 2/3\leq t\leq 1. \end{cases}.

So the dunce cap is homotopic to C_{id}\simeq D^2 which is contractible.