David Roberts scratch 2016

This page is intended to be a dumping ground for thoughts. No guarantee is made for correctness. On the other hand, I would be happy to discuss stuff from here if you have ideas. See David Roberts for my contact details. See also: scratch 2014, scratch 2015, scratch 2021

---

2016

December

2

For $\lambda$-rings with positive structure, see these notes by Grinberg, Appendix X. ‘Positive structure on $\lambda$-rings’. This is needed for the 17 August comment below.

November

28

Any torsion extension of $B//G$, for $B$ a ball in an orthogonal representation of $G$ (a compact Lie group) is isomorphic to one of the form $B//\tilde{G}$, for $\tilde{G} \to G$ a torsion extension.

27

Here’s a collection of things I’ve written about class forcing, set theory and the like:

• Here’s an interesting fact… (Google+ 5 Dec 2012)

• I’m teaching myself bits and pieces of forcing at the moment… (Google+ 17 Dec 2012)

• I’ll admit it: I don’t understand the definition of the Easton product. (MO 17 Dec 2012)

• As it turns out, the answer was staring me in the face… (18 Dec 2012)

• I learned something last night that I find somewhat marvellous, and I’m surprised I haven’t seen it mentioned before. (Google+ 12 May 2013)

• From the product lemma to to a result about powersets (MO 8 Jan 2013) G+ post

• My notes from my talk at Category Theory 2013 (Google+ 16 April 2014)

• Class forcing in intuitionistic (structural) set theory (Google+ 25 April 2014)

• More on class forcing (Google+ 27 April 2014)

• Tame forcings (Google+ 1 May 2014)

• Pretameness to the rescue! (Google+ 30 May 2014)

• The corollaries to König’s theorem (Google+ 10 November 2015)

• How close to being well-orderable does this make my powerset? (MO 27 November 2015), G+ posting, Related Blog post by Asaf Karagila

September

29

Some papers about the Heisenberg group:

• EIGENVALUES OF THE HODGE LAPLACIAN ON THE HEISENBERG GROUP http://homepage.sns.it/fricci/papers/elescorial.pdf
• EXISTENCE OF ISOPERIMETRIC REGIONS IN CONTACT SUB-RIEMANNIAN MANIFOLDS http://arxiv.org/abs/1011.0633
• SUB-LAPLACIAN EIGENVALUE BOUNDS ON SUB-RIEMANNIAN MANIFOLDS http://arxiv.org/abs/1407.0358

August

17

Husemöller chapter (8/9) ‘Stability properties of vector bundles’ section 1 (‘Trivial summands of vector bundles’). In particular, for a complex vector bundle $E\to M$ over an $n$-dimensional manifold, for $rk(E) = m \gt \lfloor(n-1)/2\rfloor$, then $E \simeq E'\oplus \underline{\mathbb{C}^{m-\lfloor(n-1)/2\rfloor}}$, for some vector bundle $E'$. Here $\underline{V}$ means the trivial bundle with fibre $V$.

If $E_1,E_2$ are two vector bundles on $X$ of rank $k \geq \lfloor n/2\rfloor$, and which are stably isomorphic (i.e. $E_1 \oplus \underline{\mathbb{C}^l} \simeq E_2 \oplus \underline{\mathbb{C}^l}$ for some $l$, then $E_1 \simeq E_2$.

For $G$ a compact (Lie) group and $E\to X$ a $G$-vector bundle, then

$$\bigoplus_{\rho\in \widehat{G}} Hom(\underline{V_\rho},E)\otimes \underline{V_\rho}$$

for $\widehat{G}$ the set of isomorphism classes of irreducible representations $\rho\colon G\to V_\rho$ of $G$.

Weibel’s K-book, Theorem 4.6: augmented $\lambda$-rings such that every element has finite $\gamma$-dimension have their augmentation ideal contained in the nilradical (i.e. every element is nilpotent). For $K$-theory of a compact space (or of a paracompact connected space) then the augmentation ideal is the nilradical. Note that this sort of thing doesn’t hold for the representation ring of a finite group in general (e.g. $G=C_2$). See also ‘positive structures’ on $\lambda$-rings.

There’s a version of this proof in Atiyah’s book K-Theory, specifically for topological K-theory, but it is really about the filtration and the operations.