Articles – Konrad Voelkel's Blog – Mathemagically

If you happened to buy books from Amazon.com (or, in my case, Amazon.de) and maybe used the recommendation engine and the wishlist (and and and ...) then there will be lots of data about your books on the Amazon website. Have you ever thought about organizing your library with a different tool? May it be Google Books or LibraryThing or Shelfari, you will have to export this precious big amount of data from Amazon to the other service. Luckily, some intelligent people invented ISBN, so you basically need to extract a list of ISBNs to identify the books (neglecting your reviews and tags for now). Not that luckily, Amazon doesn't offer such export functionality to the layman. Searching the internet yields a Greasemonkey script that enables you to export wishlist content - but no ISBNs, so import into other services is not so easy.

The solution is to save each website of "your purchases" (or other such lists of books) as HTML file and let a smart script do the extraction work. This way, you're not violating Amazon's terms of service (which most likely don't allow any robots scraping the website) and on the positive side, it works.

Here is my python script, which you can also download here (in a better version):
import sys, re
asinRegExString = "<tr valign=middle id=\"iyrListItem([A-Z0-9]{10})\">"
asinRegEx = re.compile(asinRegExString)
filename = sys.argv[1]
f = open(filename,'r')
asinlist = []
for line in f.readlines():
    match = asinRegEx.match(line)
    if match != None:
        asinlist+=[match.group(1)]
f.close()
print "\n".join(asinlist)

To run this script, you need a Python interpreter. On most common GNU/Linux systems, those are installed or easily installable, for example by "apt-get install python" on Debian-based systems.

I have tested it with Amazon.de and the "purchased books" website but I guess it would work equally well with Amazon.co.uk and Amazon.com. As always, leave a comment if it worked for you or not. If it doesn't work or if you have different needs (like, extracting ISBN and name and price) this will be easily possible by altering the regular expressions in the script (easy for a programmer, not that easy for anyone else).

I used this to import all books I bought via Amazon into my Google Books library which I use to maintain a list of all books I own. The nice thing about Google Books, on the other hand, is their XML export feature, which I commented on earlier.

Now I'll explain a little bit what essential manifolds are and what they're good for.

Definition
A (connected closed orientable topological) n-manifold $M$ is called essential, if there exists a continuous map $f : M \to K(\pi_1(M,\ast),1)$ such that the induced morphism on the top homology $f_\ast : H_n(M,\mathbb{Z}) \to H_n(K(\pi_1(M,\ast),1),\mathbb{Z})$ maps the fundamental class $[M] \in H_n(M,\mathbb{Z})$ to some non-zero element $f_\ast([M]) \neq 0 \in H_n(K(\pi_1(M,\ast),1),\mathbb{Z})$.

To have a very explicit example, take a n-torus $M$, that is a manifold of dimension n which is homotopy equivalent to a product of n copies of $S^1$. Each such $S^1$ yields a different non-contractible loop on $M$, so there are n non-homotopic loops $\gamma_1,...,\gamma_n$ and the fundamental group is just $\pi_1(M,\ast) = \mathbb{Z}[\gamma_1,...,\gamma_n]$, the free abelian group generated by the $\gamma_i$. The homology is the exterior algebra over the fundamental group. The cohomology is the exterior algebra over the dual of the fundamental group, i.e. $H^\bullet(M,\mathbb{Z}) = \mathbb{Z}[\gamma_1^\ast,...,\gamma_n^\ast]$. The fundamental class is just $\gamma_1 \wedge ... \wedge \gamma_n \in H_n(M,\mathbb{Z})$. The universal cover of a n-torus is n-dimensional euclidean space, which is contractible, so $M$ has a contractible universal cover, thus it is acyclic, in other words, a $K(\pi_1(M,\ast),1)$. Taking the identity map $f := id_M$, this induces on top homology the identity map (since homology is functorial) and thus maps the fundamental class to itself, a non-zero element. So we have seen that any torus is essential. Note that we haven't looked at metric properties at all, because essentialness is a purely homotopy theoretic notion.

If you look closer, you see that we haven't actually used that the space $M$ was a torus - we just used that it is an aspherical space, so every aspherical manifold is essential.

The Borel conjecture predicts that closed aspherical manifolds are topologically rigid. The most common examples of non-topologically rigid spaces are lens spaces - there are many non-homeomorphic lens spaces of the same homotopy type. Lens spaces are closed, and they are good examples of non-aspherical essential manifolds, so they don't disprove the Borel conjecture.

Definition
Let $p$ and $q_1,...,q_n$ be integers (for some $n \geq 2$), with $q_i$ coprime to $p$ for each $i$. Define $\ell_k := 2\pi i q_k/p$. Take the unit sphere in $\mathbb{C}^n$, which is a $S^{2n-1}$ and let $\mathbb{Z}/p$ act on it by

$[1].(z_1,...,z_n) := (e^{\ell_1}z,...,e^{\ell_n}z).$

The quotient of $S^{2n-1}$ by this action is denoted $L(p;q_1,...,q_n)$, the {lens space associated to $(p;q_1,...,q_n)$.

This is a $(2n-1)$-dimensional closed manifold with fundamental group $\mathbb{Z}/p$. The universal cover is given by the quotient map $S^{2n-1} \to L(p;q_1,...,q_n)$, so the universal cover is clearly non-contractible and in fact very spherical. This shows that lens spaces are never aspherical.

In the literature on homology and homotopy, you'll often find 3-dimensional lens spaces $L(p,q) := L(p;1,q)$. For these, there exists a nice classification of homeomorphism types via Reidemeister torsion (or: simple homotopy type), ultimately boiling down the question to arithmetic relation between different $q$, modulo $p$.

To see that lens spaces are essential, we have to produce a map $f : L(p;q_1,...,q_n) \to K(\mathbb{Z}/p,1)$ which on top homology maps the fundamental class to a non-zero element. The homology of $K(\mathbb{Z}/p,1)$ is well-known, it is

$H_k(\mathbb{Z}/p,\mathbb{Z}) = \begin{cases} \mathbb{Z} & k=0,\\ \mathbb{Z}/p & k \text{ odd},\\ 0 & k \text{ even}. \end{cases}$

The dimension of a lens space is $2n-1$, so it is odd - phew!

Now we need an explicit model for $K(\mathbb{Z}/p,1)$. One such model is given by the infinite lens space $L^\infty(p) := S^\infty/_{\mathbb{Z}/p}$, where $S^\infty := \lim S^n$ is seen as the union of spheres where the n-sphere sits inside the (n+1)-sphere as equator. The group $\mathbb{Z}/p$ acts by multiplication with p-th roots of unity in each coordinate, which is possible by putting the $S^\infty$ in a $\mathbb{C}^\infty := \lim \mathbb{C}^n$ by taking the limit over the embeddings $S^{2n-1} \to \mathbb{C}^n$.
We can modify this construction slightly, by starting with the lens space $L(p;q_1,...,q_n)$ and taking the limit over all $L(p;q_1,...,q_n,q'_1,...,q'_k)$ for $k \to \infty$ and $q'_i = q_n$ for all i. This yields the same $L^\infty(p)$ up to homotopy and even better, it admits an inclusion map from $L(p;q_1,...,q_n)$. On homology, the inclusion map maps the fundamental form to a generator of $\mathbb{Z}/p$, which is non-zero. Therefore, lens spaces are essential.

With a very similar idea, one can prove that real projective spaces $\mathbb{R}P^n$ are essential, by looking at the inclusion into $\mathbb{R}P^\infty = \lim \mathbb{R}P^k$, which is aspherical with the same fundamental group $\mathbb{Z}/2$.

In general, it suffices to find a continuous map of non-zero degree from a manifold $M$ onto an essential manifold to deduce that $M$ is essential.

To give a counter-example, look at the spherical space $S^n$ (for $n \geq 2$) with trivial fundamental group. It is certainly not aspherical (its higher homotopy groups are quite interesting) but there is an inclusion map $S^n \to S^\infty$ (as above). This inclusion map has to be the zero map on top degree homology, since $H_n(S^\infty,\mathbb{Z}) = 0$ for all $n \geq 1$ (because $S^\infty$ is contractible). This shows that spheres are never essential.

Finally, you might ask
What are essential manifolds good for?
In his 1983 paper "Filling Riemannian Manifolds", Gromov defined essential manifolds the first time, to state (and prove) his "main isosystolic inequality".
To formulate it, we have to say what a systole is first:

Definition
Let $M$ be a Riemannian manifold. Then the systole of $M$ is $sys_1(M) := \inf_{\gamma} length(\gamma)$, where the infimum goes over all non-contractible loops $\gamma$ in $M$ (in fact it is a minimum).

Theorem (Gromov)
Let $M$ be a closed essential Riemannian manifold of dimension $n$. Then

$sys_1(M) \leq C_n \sqrt[n]{Vol(M)}$

with some constant $C_n$ not depending on $M$ which satisfies

$0 < C_n < 6(n+1) n \sqrt[n]{(n+1)!}.$

So the job of essential manifolds is to be the domain where Gromov's theorem holds. As far as I know, it is not so clear whether there exist larger classes of manifolds that satisfy such a systolic inequality.

The theorem is a generalisation of a theorem on tori:
Theorem (Loewner)
Let $\gamma$ be a shortest closed geodesic in a flat torus $T^n$. Then

$sys_1T^n = length(\gamma) \leq C_n \sqrt[n]{Vol(T^n)}.$

Let $M$ be a 2-torus (with arbitrary metric), then

$sys_1M \leq C_2 \sqrt{Area(M)}$

and $C_2 = \sqrt{\frac{2}{\sqrt{3}}}$.
The 2-torus realising equality in this inequality is the quotient of $\mathbb{R}^2$ by the hexagonal lattice spanned by the 3rd roots of unity.

Pu proved a similar systolic inequality on $\mathbb{R}P^2$, so it is very reasonable to look for a class of closed manifolds that contain tori and real projective space and furthermore allow systolic inequalities.

Well, that's enough for today!

In this post I want to sketch the idea of aspherical manifolds - manifolds which don't admit higher homotopically non-trivial spheres - and the related concepts of Eilenberg-MacLane-spaces and classifying spaces for groups.

Definition
A topological space $M$ is called aspherical if all higher homotopy groups vanish, i.e. $\pi_n(M,m_0) = 0 \quad \forall n > 1$ where $m_0 \in M$ is an arbitrary basepoint and $M$ is assumed to be connected.

Since manifolds admit universal covers, you could equivalently define a manifold to be aspherical if and only if its universal cover is contractible.

Just one example illustrating how rich this class of spaces is:
Metric spaces that are of non-positive curvature (i.e. locally CAT(0)-spaces), for example the Bruhat-Tits building of a simple algebraic group over a field with a discrete valuation, are aspherical.

A good survey on aspherical manifolds was given by Wolfgang Lück.

Definition
A connected topological space $X$ is called Eilenberg-MacLane-space for a group $G$ and a natural number n if its nth homotopy group is exactly $G$ and all other homotopy groups vanish, i.e.
$\pi_k(X,x_0) = \begin{cases} G & k=n \\ 0 & else.\end{cases}$
Then one calls $X$ also $K(G,n)$.

The standard examples of $K(G,1)$ spaces are $S^1$, which is a $K(\mathbb{Z},1)$ and $\mathbb{R}P^\infty$, which is a $K(\mathbb{Z}/2,1)$.
Of course, every $K(G,1)$ is aspherical and every aspherical space is a $K(G,1)$ for $G$ being its fundamental group.

One can also define a functorial construction of a $K(G,1)$ which gives a CW-complex model for every group $G$ and transforms group homomorphisms into continuous maps of spaces.

For this, we need the functorial nerve construction.
Definition
The nerve $N(G)$ of a (discrete) group $G$ is the simplicial $G$-set with n-simplices being the (n+1)-fold cartesian product of sets $G \times G \times \cdots \times G$, face maps just omitting one factor in the cartesian product, degeneracies adding the identity element of $G$ in one factor.
By construction, seen as a discrete simplicial group, $G$ embeds into $N(G)$ as the 0-skeleton. Observe that $N(G)$ is contractible, since every n-simplex $(g_0,...,g_n) \in N(G)_n$ is the face of $(e,g_0,...,g_n) \in N(G)_{n+1}$ which also has the face $(e,g_1,...,g_n) \in N(G)_n$, thus allowing to move every point to the identity $(e,...,e) \in N(G)_m$ which is just a degeneracy of $e \in N(G)_0 = G$.
The group $G$ acts diagonally on $N(G)$, i.e. it acts on an n-simplex by the formula $(g,(g_0,...,g_n)) \mapsto (gg_0,...,gg_n) \in N(G)_n$. This action is compatible with face and degeneracy maps, thus making $N(G)$ into a simplicial $G$-set. The action is free, i.e. no two elements of $G$ operate in the same way.

Using the nerve construction, we now define the classifying space:
Definition
The classifying space $BG$ of a group $G$ is the quotient $BG := |N(G)|/G$ of the geometric realisation $|N(G)$ of the nerve construction by the group action described above. It turns out that $G$ operates on $|N(G)|$ like a deck transformation group, thus giving $BG$ the structure of a CW-complex with universal cover $|N(G)|$ and fundamental group $G$.
A group homomorphism $\phi : G \to H$ gives rise to a morphism of simplicial sets $N(\phi) : N(G) \to N(H)$ by pointwise application. Geometric realisation is also functorial, and due to $\phi$ being a homomorphism, the continuous map $|N(\phi)| : |N(G)| \to |N(H)|$ descends to a continuous map of classifying spaces $B\phi : BG \to BH$.

If you are not into simplicial sets and geometric realisation, you can look for a more hands-on approach in Hatcher's book "Algebraic Topology", on page 87, chapter 1.B, more specifically Example 1B.7 on page 89.

Now back to our first definitions: An aspherical manifold is just a manifold which happens to be a $K(G,1)$ for $G$ being its fundamental group. The classifying space is just an explicit (functorial!) construction which gives a $K(G,1)$ for every group $G$ (although most authors would call our $BG$ just one explicit model for $BG$...).

One would like to work only with CW-complexes, if possible, since they allow induction over the skeleton and cell-by-cell arguments. Is every manifold homeomorphic to a CW-complex - long time ago there was the "Hauptvermutung" (main conjecture) which asked this, but it's wrong. While compact manifolds admit a homotopy equivalent CW-model (by Kirby and Siebenmann), this is not true for topological manifolds in general. Let us look what one could do with a CW-model:

Proposition
Let $X$ be a connected CW complex and $Y$ be a $K(G,1)$ (for example, your favourite aspherical manifold). Let $\phi : \pi_1(X,x_0) \to \pi_1(Y,y_0) = G$ be a homomorphism of groups. Then there is a continuous map $\Phi : X \to Y$ mapping $x_0$ to $y_0$ which induces $\phi$ on fundamental groups; furthermore, the map $\Phi$ is unique up to homotopy relative $x_0$.

The proof of this proposition goes roughly like that: First, let $\Phi$ map $x_0$ to $y_0$. Now, for each 1-cell $\gamma$, take a representative of $\phi([\overline{\gamma}]) \in \pi_1(Y,y_0)$ to define $\Phi$ on $\gamma$. Then one has to extend the map given on the 1-skeleton to $X$, using the fact that $Y$ has no higher homotopy.

Corollary
Every two CW-complexes $X,Y$ which are both $K(G,1)$-spaces are homotopy equivalent ("of the same homotopy type").

To prove this, just take isomorphisms $f : \pi_1(X,x_0) \to G$ and $g : \pi_1(Y,y_0) \to G$ and define $\phi := f \circ g^{-1}$ which gives $\Phi : Y \to X$ with inverse up to homotopy given by $\Psi : X \to Y$ induced by $\psi := g \circ f^{-1}$.

This justifies that every invariant of $BG$ that depends only on the homotopy type, is actually an invariant of $G$ - a very useful idea. One can define group homology with integer coefficients of $G$ by the formula $H_n(G,\mathbb{Z}) := H_n(BG,\mathbb{Z})$.

One drawback of the classifying space via the nerve construction is that it is usually very large - there are simplices in arbitrary high dimensions. For example, the circle $S^1$, given as example of a $K(\mathbb{Z},1)$, is much more efficient than $B\mathbb{Z}$.

Of course, talking about aspherical manifolds, we don't want to forget the manifold structure. Given a group $G$, one could expect that many non-homeomorphic aspherical manifolds with fundamental group $G$ exist - even many non-homotopy equivalent ones. At least we can say that such non-homotopy equivalent aspherical manifolds are not of CW homotopy type. There is an old conjecture on this theme:

Conjecture (Borel)
Let M and N be closed aspherical manifolds, and let $f : M \to N$ be a homotopy equivalence. Then $f$ is homotopic to a homeomorphism.

Together with the result of Kirby and Siebenmann (that every closed manifold is of CW homotopy type), this would imply that closed aspherical manifolds are classified by their fundamental group up to homeomorphism.

The property that every homotopy equivalence is homotopic to a homeomorphism is called topological rigidity.

Now this is a slightly corrected (although still somewhat messy) version of my diploma thesis - in german:
Matsumotos Satz und A¹-Homotopietheorie.

You can read something about the content in this blog post, containing an extended abstract in english.

For the german-speaking mathematicians, here is some abstract:

Matrizengruppen über den reellen oder komplexen Zahlen sind topologische Gruppen und ihre Fundamentalgruppen sind interessante Invarianten. Während die Fundamentalgruppe über stetige Schleifen am Basispunkt und Homotopien definiert ist, können wir uns fragen, welche Gruppe man erhält, wenn man nur polynomiale Schleifen am Basispunkt und polynomiale Homotopien zulässt. Während die Fundamentalgruppe eine Klassifikation der Überlagerungen erlaubt, so erhalten wir durch die “polynomiale Fundamentalgruppe” eine Klassifikation der zentralen Gruppenerweiterungen - ein homologietheoretisches Analogon von Überlagerungen.

Für eine Überlagerung X → X mit Faser F über dem Basispunkt und punktiertem Schleifenraum ΩX gibt es eine Liftungsabbildung L : ΩX → F, die im Fall einer topologischen Gruppe X = G ein Gruppenhomomorphismus ist. Ist X die universelle Überlagerung, so liefert L unter π_0 einen Isomorphismus π_1(X,∗) → F.
Jede Überlagerung topologischer Gruppen ist auch eine zentrale Erweiterung - jedoch nicht umgekehrt. Die zentralen Erweiterungen einer perfekten Gruppe G werden klassifiziert durch den Schur-Multiplikator H_2(G,Z); das ist der Kern der universellen zentralen Erweiterung. Wir betrachten den Fall X = G(k), wobei G(k) die k-rationalen Punkte einer einfach zusammenhängenden Chevalley-Gruppe G mit Wurzelsystem Φ für einen unendlichen Körper k ist. Dann ist der Schur-Multiplikator H_2(G(k),Z) =: K_2(Φ,k), die zweite instabile K-Theorie bezüglich Φ und die universelle zentrale Erweiterung ist die Steinberg-Gruppe St(Φ, k).

Wir definieren eine simpliziale Gruppe SingG(k), deren n-Simplizes genau die Matrizen aus G(k[t1,...,tn]) sind und zeigen, dass ihre simpliziale Fundamentalgruppe genau die instabile K-Theorie ist, indem wir (unter einer gewissen Regularitätsvoraussetzung an K_2(Φ,k)) die simpliziale Überlagerung
K_2(Φ,k) → SingSt(Φ,k) → SingG(k)
und ihren Liftungshomomorphismus L studieren. Wir geben eine explizite Umkehrabbildung zu L an und können somit alle Schleifen in SingG(k) bis auf Homotopie explizit beschreiben.
Die genannte Regularitätsvoraussetzung ist Homotopieinvarianz von K_2(Φ,·) in einer Variablen über einem Körper k, d.h. K_2(Φ,k[t])=K_2(Φ,k).
Für Φ = A_n mit n ≥ 3 ist die Aussage bereits durch Sätze von Quillen und van der Kallen bekannt. Für rk Φ ≥ 3 ist dies eine Vermutung von Wendt.
Wir arbeiten mit der Steinberg-Präsentation der universellen zentralen Erweiterung St(Φ,k) von G(k) und mit der Matsumoto-Präsentation ihres Kerns, der instabilen zweiten K-Theorie von k:
K_2(Φ,k)=KSp(k), falls Φ symplektisch, K_2(Φ,k)=KM(k) (Milnor-K-Theorie) sonst.

Matsumotos Beweis dieser Präsentation wird in dieser Arbeit ausführlich nachgerechnet. Die Ergebnisse über die Fundamentalgruppe von SingG(k) liefern eine Verallgemeinerung eines Satzes von Jardine auf instabile K-Theorie und unendliche Körper, die nicht notwendig algebraisch abgeschlossen sind:
π_1(SingG(k))=K_2(Φ,k).

Unter Verwendung von Resultaten von Morel und Wendt
π_1^A¹(G)(k)=π_1(SingG(k))
erhalten wir schließlich eine Aussage über π_1^A¹, die motivische Fundamentalgruppe im Sinne von Morel und Voevodsky.
Damit ist die ursprüngliche Fragestellung der Arbeit beantwortet:
“Wie sehen die Schleifen in der A¹-Homotopietheorie von G(k) aus?”.

...und weil beim herauskopieren des Abstracts aus der PDF-Datei bestimmt das ein oder andere schief gegangen ist, sollte man lieber gleich das PDF lesen, wenn man denn überhaupt etwas lesen möchte.