An Introduction to Algebraic Topology

February 21, 2007

An Introduction to Algebraic Topology

Posted by John Baez

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This quarter, besides my seminars on Quantization and Cohomology and Classical vs. Quantum Computation, I’m also teaching the graduate qualifier course on algebraic topology. While a bit elementary for some Café regulars, it might be fun for other folks:

John Baez, Mike Stay and Christopher Walker, Algebraic Topology.

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The lectures are by John Baez, except for classes 2-4, which were taught by Derek Wise. The lecture notes are by Mike Stay, and until the course is finished in mid-March, his notes may be more up-to-date. The answers to homework problems are by Christopher Walker. The course used this book:

James Munkres, Topology, 2nd edition, Prentice Hall, 1999.

So, theorem numbers match those in this book whenever possible, and it’s best to read these notes along with the book. But, we deviate from Munkres at various points, and also skip many sections. We place more emphasis on concepts from category theory.

But, the star of the show is $\pi_1$ — the fundamental group!

Posted at February 21, 2007 1:50 AM UTC

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Re: An Introduction to Algebraic Topology

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I had a meeting this week, together with two emeriti at my university, with concerned teachers at the local mathematics-profiled secondary school. They have a handfull of school kids who simply are beyond the teacher’s capacity to handle satisfactory, and so wanted us to render some backup.

I got two very ambitious 9th-graders; and am toying with the idea of nudging them into algebraic topology over the Euler characteristic (which they probably know in the classical case).

One idea that occurred to me there was whether or not the fundamental groupoid may even be easier to start with than the fundamental group, if only you introduce it the right way. Thoughts?

Posted by: Mikael Johansson on February 21, 2007 1:40 PM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

For 101 good reasons to use groupoids see Ronnie Brown’s From groups to groupoids: a brief survey.

Posted by: David Corfield on February 21, 2007 3:51 PM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

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I strongly considered teaching this course using the fundamental groupoid instead of the fundamental group.

Especially now that I’ve hit the Seifert-van Kampen theorem, the advantages of the fundamental groupoid are obvious. If you look at Munkres’ proof of this theorem, you’ll see he’s actually using the fundamental groupoid, but then showing this groupoid is equivalent to the fundamental group for path-connected spaces. (This is why the usual statement of the Seifert-van Kampen theorem for a space $X = U \cup V$ requires that $U \cap V$ be path-connected.)

But, I decided to teach the course using the fundamental group, for four very practical reasons:

The students have all taken the first quarter of a standard 3-quarter algebra qualifier course. So, they know a lot of basic results about groups, but none about groupoids. They deserve to see some nice applications of group theory.
This course is a graduate qualifier course, so it has some duty to teach students ‘familiar stuff that all mathematicians should know’. It’s only 10 weeks long, so there’s barely enough time to cover the fundamental group carefully. If I’m lucky I’ll have time to give them a taste of homology theory. Fundamental groupoids, while wonderful, are less crucial.
I’m teaching them just enough category theory to see why it’s important to treat invariants of topological spaces as functors. I’m mainly illustrating this with the example of $\pi_1 : Top_* \to Grp$ but I’m also giving them a taste of $\pi_n : Top_* \to Grp$ so they can see many of the arguments depend less on the details of the functor than the fact that it is a functor. The moral: functors are good for you. If I started with the example $\Pi_1: Top \to Gpd$ where the objects of the category $Gpd$ are themselves categories, it might blow their little minds and make them miss this moral. So, I’ve decided to save this as a fun digression for a rainy day.
This is my first time teaching this course, and I figured I should learn how to teach algebraic topology the usual way before I try to revolutionize it.

If I were teaching some ambitious 9th-graders who had no previous knowledge of group theory and no particular need to learn ‘standard stuff’, I might teach them the fundamental groupoid first.

Or, if I were teaching a second course on algebraic topology, I might say “now that you know about the fundamental group, let’s do things right — let’s talk about the fundamental groupoid!”

Posted by: John Baez on February 21, 2007 9:42 PM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

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If I were teaching some ambitious 9th-graders who had no previous knowledge of group theory and no particular need to learn ‘standard stuff’, I might teach them the fundamental groupoid first.

To a certain extent precisely my point. They have (I’m told - I’ll meet them in a week) quite a large hunger for mathematics, really up to speed minds, and almost no formal training, and also almost no requirements to fulfill other than to get stimulated.

Thus, groupoids will be about as familiar as groups. And I’ll be bootstrapping them from scratch anyway. And for this particular case, working through “This is a group, and here’s how we get the fundamental group” might be only little less mindblowing than “This is a groupoid, and here is how we get the fundamental groupoid.”

Posted by: Mikael Johansson on February 22, 2007 2:31 PM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

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Mikael wrote:

And for this particular case, working through “This is a group, and here’s how we get the fundamental group” might be only little less mindblowing than “This is a groupoid, and here is how we get the fundamental groupoid.”

In fact, for people who haven’t already been brainwashed by the usual mathematics curriculum, groupoids should be less mindblowing than groups. Here’s what I’d say: a groupoid is a bunch of things (= objects) and a bunch of ‘ways of going between things’ (= morphisms) which we can compose, and which are all invertible.

A great example of a groupoid comes from a 16 puzzle. The different states of the puzzle are objects; clicking on the puzzle implements a morphism, and of course we get more complicated morphisms by composing these.

Another great example comes from a Rubik’s cube. Again the states of the cube are objects, and the ways of going between states are morphisms.

A somewhat trickier example is a square, which gives a groupoid with just one object (the square) and 4 morphisms (rotations of the square). This example is tricky because since there’s just one object, the idea of a morphism as ‘a way of going between things’ is obscured. This tricky degenerate sort of groupoid is called a ‘group’. You can tell them these tricky examples are covered in more advanced courses.

Another great example of a groupoid comes from a topological space: the fundamental groupoid!

All of this should be quite intuitive when explained patiently. You’re in a great position to do things right, that most of us are stuck in doing wrong.

Posted by: John Baez on February 22, 2007 10:47 PM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

I can see the 16-puzzle being more naturally associated with a groupoid. You can’t compose an arbitrary pair of moves. I’m not quite so certain with Rubik’s cube.

Actually, I just started a series of posts about the cube. Since any two maneuvers can be composed, it seems more natural to consider a group acting on a set of states.

Of course, any group acting on a set like this can be reinvented as a groupoid, but what does considering this groupoid gain you over the group point of view?

Posted by: John Armstrong on February 23, 2007 2:04 AM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

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I can see the 16-puzzle being more naturally associated with a groupoid. You can’t compose an arbitrary pair of moves. I’m not quite so certain with Rubik’s cube.

Since the objects in the Rubik’s cube groupoid are positions, it’s clear that you can’t compose, say, a quarter turn from (the starting position) to (two solid sides and four striped sides) and a quarter turn from (a thoroughly scrambled position) to (another scrambled position).

Posted by: Mike Stay on March 12, 2007 8:38 PM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

Yes, you can turn the G-set into a groupoid. I understood the construction he meant, and that’s not my question. It’s facile to claim I’m confounding the morphisms of the groupoid with the moves in the original group.

My point is that no matter what the current state of the cube is you can do the same things to it. This is not the case for the 16 puzzle. Rubik’s groupoid is tremendously redundant, since it comes from a group action on a set, while the 16 puzzle isn’t nearly so redundant since your available moves depend on your current state.

Maybe Baez is right that I’m simply too familiar with the G-set picture, but the groupoid just seems to obscure the underlying simplicity of Rubik’s group. It’s as if we decided to throw out symmetric groups in favor of symmetric groupoids. In fact, it’s exactly the same thing.

Posted by: John Armstrong on March 12, 2007 10:00 PM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

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I’m a bit hopeless when it comes to the Rubik’s cube (remove the darn stickers and paste them on in the right way!) and the 16-puzzle, but for what its worth here’s a little thing where groupoids seem to crop up naturally: characters of representations of groups.

So, I’m claiming that even if you only ever deal with ordinary groups and their representations, at some point one is “led” to introducing groupoids. Huh, you say?

Well, what is the character $\chi_\rho$ of a representation $\rho$ of a group $G$ ?

Answer 1 : It’s a class function on the group. Thus, a function $\chi_\rho : G \rightarrow \mathbb{C}$ satisfying $\chi_\rho(h g h^{-1}) = \chi_\rho(g)$ for all $g, h \in G$ .

Answer 2 : It’s a section of the trivial line bundle over the loop groupoid $\Lambda G$ .

(Recall that $\Lambda G$ is the groupoid which depicts $G$ acting on itself by conjugation. Objects are elements $g \in G$ , and morphisms are written $g \stackrel{h}{\rightarrow} h g h^{-1}$ , which compose in the obvious way.)

Aargh, you say… my head is so filled with groupoids I’m stubbornly insisting on putting them everywhere. Okay - everybody looks at things in their own favourite ways. I suppose this is just a very geometric way of thinking about characters.

But I would argue that thinking of a character as a section of a line bundle over the loop groupoid gives it a natural home.

For, suppose we start talking about twisted representations of the group $G$ . Thus, we look for $\rho(g)$ satisfying

(1)

\rho(g) \rho(h) = \phi(g,h) \rho(g h)

for some 2-cocycle $\phi \in Z^2(G, U(1))$ .

What is the character of a twisted representation?

Well, in the old terminology, it’s a certain function on the group, which behaves in a slightly awkward way under conjugation:

(2)

\chi_\rho (h g h^{-1}) = \frac{\phi(h,g)}{\phi(h g h^{-1}, g)} \chi_\rho (g).

In the groupoid terminology, we recognize these factors as saying precisely that $\chi$ is a flat section of the transgressed line bundle over $\Lambda G$ :

(3)

\chi_\rho \in \Gamma_{\Lambda G} (\tau(\phi)_{\mathbb{C}}).

The point is that this geometric language treats the untwisted and twisted pictures in an even-handed way, and actually “explains” the prefactors which pop out as the transgression of the 2-cocycle $\phi$ . As usual, I’m taking all this stuff from Simon’s paper. I feel silly to keep harping on about it :-) I guess I’m the “line bundles on a groupoid” fanatic around here :-)

Posted by: Bruce Bartlett on March 13, 2007 1:39 AM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

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Don’t worry, you’re not the only one … if you include yourself into the set of people obsessed with bundles of all sorts over groupoids.

Posted by: David Roberts on March 13, 2007 3:42 AM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

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I thought we were by now all obsessed even with line bundles on categories that need not be groupoids, like configuration spaces of a relativistic particle…

Posted by: urs on March 13, 2007 1:03 PM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

John Armstrong wrote:

Of course, any group acting on a set like this can be reinvented as a groupoid, but what does considering this groupoid gain you over the group point of view?

We’re talking about some 9th-graders who haven’t been brainwashed by the usual way of teaching math. So, they’d ask: “what does considering that group gain you over the groupoid point of view?”

Posted by: John Baez on February 23, 2007 3:19 AM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

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John Armstrong writes:

what does considering this groupoid gain you over the group point of view?

John Baez writes:

“what does considering that group gain you over the groupoid point of view?”

On the other hand, we are talking here not just about a group, but about a group with an action.

Maybe A good occasion to pause and think about the abstract way (i.e. not the standard definition in terms of elements) an action groupoid arises from the action of a group.

I talked about that with Igor Bakovic in Toronto, since he was writing down an action bigroupoid for the action of a bigroupoid on a category.

Let’s see, what was the answer… Ah, right:

in general, for an action of any category on any set, we get the action category as follows:

We think of the category as a monad in spans, and think of the set it acts on as a semi-span (i.e. a span with one leg to the terminal object).

Then draw the 2-morphism in spans that gives the action. In that diagram, regard the unique subdiagram which is a span with both legs ending at the given set. This is the span that defines the action category.

(Hm, okay, that’s a definition that does not mention elements, but it is probably still not nice enough. Certainly some expert reading this can give a nice abstract definition of the “action category” associated to the action of a category on some set.)

Posted by: urs on February 23, 2007 8:37 AM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

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Hi guys,

John B. wrote:

A great example of a groupoid comes from a 16 puzzle. The different states of the puzzle are objects; clicking on the puzzle implements a morphism, and of course we get more complicated morphisms by composing these.

I’m a bit confused! Isn’t it more “fundamental” to think of the 16 puzzle as an oriented graph? Then the groupoid is the free groupoid on that graph. The trouble is, as John mentions, that composing two moves is not a move. That would confuse me if I was a 9th grader!

John A. wrote:

What does considering this groupoid gain you over the group point of view?

Here’s my two cents worth for groupoids : thinking of the 16 puzzle as a groupoid $G$ makes it very natural to introduce a line-bundle-with-connection on $G$ ! In fact, this geometry is already there in the Java version of the game. You’ll notice there is a “move counter” : every move you make adds 1 to this box. This is nothing but a (rather simple) connection on the trivial line bundle over $G$ ! So the challenge is to proceed from your initial state $a$ to the final state $b$ (where all the letters are in alphabetical order) while minimizing your parallel transport.

A connection is nothing but the free assignment of a number to each edge in the oriented graph (a weighting on the graph).

More interesting connections will produce more interesting versions of the game. For instance, you might have that going from state $a$ to state $b$ has a value of +1 if the letter you moved has gone from being in alphabetical order to being in alphabetical order (or the anti version), and -1 otherwise. That will give you grief on a cold winter’s day!

Posted by: Bruce Bartlett on February 23, 2007 5:59 PM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

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Ye gods! Then I’m going to have to grok line bundles and connections too! Yikes!

Posted by: Mikael Johansson on February 23, 2007 10:21 PM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

Bruce wrote:

I’m a bit confused! Isn’t it more “fundamental” to think of the 16 puzzle as an oriented graph? Then the groupoid is the free groupoid on that graph.
The trouble is, as John mentions, that composing two moves is not a move.

Good point — but that’s no reason to avoid this example. And, there’s no need to mention “the free groupoid on a graph”: just draw some graphs and show how they give rise to groupoids. This makes a great warmup for explaining the fundamental groupoid, since this way of getting a groupoid from a graph is a very nice discrete analogue of the fundamental groupoid of a space!

Indeed, there’s an obvious way to turn any graph into a space, and then the fundamental groupoid of the space is equivalent to the groupoid of the graph! But that’s a bit sophisticated, since it involves the concept of ‘equivalence’ of groupoids. A simpler, more vivid point is that ‘playing the 16-puzzle game’ is really ‘tracing out a path in the 16-puzzle graph’ — and equivalence classes of these paths give ‘elements of the fundamental groupoid of the 16 puzzle’.

That would confuse me if I was a 9th grader!

If you were a 9th grader, and I were teaching you this stuff, we’d work our way through these subtleties, and have lots of fun doing it!

Posted by: John Baez on February 23, 2007 6:58 PM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

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Also, these ninth graders are not in any way your normal ninth graders. Apparently, they digested first few terms of university calculus just to have something to do.

I definitely think that with a nice geometric and non-panicked presentation, they have all the tools they’d need to fiddle with groupoid.

Posted by: Mikael Johansson on February 23, 2007 10:20 PM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

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Bruce wrote:

Here’s my two cents worth for groupoids : thinking of the 16 puzzle as a groupoid $G$ makes it very natural to introduce a line-bundle-with-connection on $G$ ! In fact, this geometry is already there in the Java version of the game.

Very nice!

But not quite true: undoing a move you just made still adds 1 to the count of moves. So, what we have is a connection, not on the free groupoid on the 16-puzzle graph, but on the free category on this graph.

In other words, we’ve got a functor from the free category on the 16-puzzle graph to $\mathbb{N}$ , regarded as a 1-object category. Every free category comes equipped with a god-given functor like this: the ‘move counter’.

Posted by: John Baez on February 23, 2007 7:09 PM | Permalink | Reply to this

16 puzzle revisited

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Hi guys,

The 16 puzzle recently came up in a conversation amongst my friends. Believe it or not, it has my mind going in circles! Seeing as we’re discussing groupoids and algebraic topology, it seems an interesting thing to mention.

Recall that its the puzzle that looks like this:

Although you actually move the white numbered squares, you can “dually” think that you’re really moving the hole . Around each “node” (i.e. interior intersection point in the grid) there is a “cube root” rotation thing going on:

That is, if the hole moves counterclockwise in a loop, the numbers rotate cyclically. You need to go round the loop three times to get back to where you started! Here the asterisk marks the base point of the loop.

Thus the 16 puzzle introduces the idea of fundamental group. You can even think of it as a fundamental groupoid , if you let the base point of the hole be the middle point of any of the 16 squares.

Finally, you can even draw a link between the 16-puzzle and Riemann surfaces! Around each node, we have a cube root behaviour. Place the puzzle on the table, so that the 9 interior nodes (marked “x”) are complex numbers $\alpha_i$ :

Then define the “16-function” $f(z)$ by

(1)

f(z) = \left((z-\alpha_1)(z-\alpha_2)\cdots(z-\alpha_9)\right)^{\frac{1}{3}}

[Ed : MathML couldn’t do a cube root for some reason] . Then the motion of the system as you play the game traces out a point on this Riemann surface. Right? I don’t know, this stuff has my mind going round in circles $pic$ ! What’s a nice way to think of it?

Posted by: Bruce Bartlett on March 26, 2007 1:24 AM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

Thanks a lot for so interesting lections.

I’m looking for some “natural ways” of obtaining most important mathematical notions from the brain, but not from practice (like physics). In other words I try to understand why some important mathematical notions may be so important. I try to look at practical sciences (like physics) as at sciences about our cognitive ability in mathematics. I am not mathematician, i am not a physicist, just a dilettante, so maybe my questioning is kind of stupidity for your point of view.

Anyway I have a question. I do not know any natural way of obtaining the interval of real numbers [0,1] “from the brain”, but I feel that it’s possible to formulate the notion of fundamental group without using [0,1] (but using more primitive constructions, like total orders for example). If somebody knows such a way then please let me know. Thank you.

Posted by: osman on May 22, 2007 10:15 AM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

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The only thing about [0,1] you -really- need is the topological structure - i.e. that you have a 1-dimensional building block terminated by two 0-dimensional building blocks.

This returns when you go all in in the algebraic direction - and formulate homotopy as something that deals with $C_*\otimes I$ , where $C_*$ is some chain complex representing your space and $I$ is the chain complex $0\rightarrow c\rightarrow a,b\rightarrow 0$ , given by $c\mapsto a-b$ .

This didn’t really say much - but at least points you in a direction for replacing [0,1] with $I$ .

Posted by: Mikael Johansson on May 25, 2007 11:15 AM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

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I have the following idea. Let’s describe the notion of the Abstract Path-Space. If $X$ is some set, then we say $\Theta(X)$ is the set of all total orderings of all subsets of $X$ . Now Abstract Path-Space (APS) is some $M \subseteq \Theta(X)$ closed by (1) taking concatenations of total orders (2) taking ‘pieces’ of total orders (such that concatenation of ‘pieces’ is initial total order). Total orders from $M$ are called prepaths. Now, what is path? I think it’s clear that all confined prepaths of our space (i.e. such that we can take minimal and maximal elements) build some graph. Now we say that paths of our APS are exactly morphisms of the free category over this graph.

Posted by: osman on May 28, 2007 7:59 AM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

Now we can describe several representation functors between Top ans APS. And we can describe the notions of path-homotopy and fundamental group without using [0,1].

Posted by: osman on May 28, 2007 8:07 AM | Permalink | Reply to this

Re: An Introduction to Algebraic Topology

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I feel I must refine the notion of concatentaion of prepaths. Two prepaths $a$ and $b$ are adjoined iff $max(a) = min(b)$ . Concatenation of $a$ and $b$ is defined iff thay are adjoined, and, by definition, $a + b = sup_{\Theta(M)}\{a,b\}$ is concatenation of $a$ and $b$ .

Posted by: osman on May 28, 2007 8:31 AM | Permalink | Reply to this

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