If is a connected topological manifold, and
is a point in
, the (topological) fundamental group
of
at
is traditionally defined as the space of equivalence classes of loops starting and ending at
, with two loops considered equivalent if they are homotopic to each other. (One can of course define the fundamental group for more general classes of topological spaces, such as locally path connected spaces, but we will stick with topological manifolds in order to avoid pathologies.) As the name suggests, it is one of the most basic topological invariants of a manifold, which among other things can be used to classify the covering spaces of that manifold. Indeed, given any such covering
, the fundamental group
acts (on the right) by monodromy on the fibre
, and conversely given any discrete set with a right action of
, one can find a covering space with that monodromy action (this can be done by “tensoring” the universal cover with the given action, as illustrated below the fold). In more category-theoretic terms: monodromy produces an equivalence of categories between the category of covers of
, and the category of discrete
-sets.
One of the basic tools used to compute fundamental groups is van Kampen’s theorem:
Theorem 1 (van Kampen’s theorem) Let
be connected open sets covering a connected topological manifold
with
also connected, and let
be an element of
. Then
is isomorphic to the amalgamated free product
.
Since the topological fundamental group is customarily defined using loops, it is not surprising that many proofs of van Kampen’s theorem (e.g. the one in Hatcher’s text) proceed by an analysis of the loops in , carefully deforming them into combinations of loops in
or in
and using the combinatorial description of the amalgamated free product (which was discussed in this previous blog post). But I recently learned (thanks to the responses to this recent MathOverflow question of mine) that by using the above-mentioned equivalence of categories, one can convert statements about fundamental groups to statements about coverings. In particular, van Kampen’s theorem turns out to be equivalent to a basic statement about how to glue a cover of
and a cover of
together to give a cover of
, and the amalgamated free product emerges through its categorical definition as a coproduct, rather than through its combinatorial description. One advantage of this alternate proof is that it can be extended to other contexts (such as the étale fundamental groups of varieties or schemes) in which the concept of a path or loop is no longer useful, but for which the notion of a covering is still important. I am thus recording (mostly for my own benefit) the covering-based proof of van Kampen’s theorem in the topological setting below the fold.
— 1. Proof of van Kampen theorem —
The proof of van Kampen’s theorem boils down (after using the above-mentioned equivalence of categories between covers of a manifold , and sets with an action of the fundamental group) to the following fact about covers:
Proposition 2 (Gluing of covers) Let
be connected open sets covering a connected topological manifold
with
also connected, and let
be an element of
. If
and
are covering maps which become isomorphic upon restricting the base to
, then there is a covering map
which becomes isomorphic to
on restricting the base to
, and isomorphic to
on restricting the base to
(and with all four isomorphisms forming a commuting square).
This proposition is easily verified by gluing together and
as topological spaces along the indicated isomorphism between
and
, and checking that the resulting space is still a covering space.
Now we can prove van Kampen’s theorem. Suppose that one has group homomorphisms ,
to a target group
which form a commuting square with the canonical homomorphisms from
to
and
. It will suffice to show that there is a unique homomorphism
such that
factors as the composition of
with the canonical homomorphism from
to
for
.
We first demonstrate existence. Let be a universal cover for
, thus
is simply connected, and if we pick a base point
, then every other point
in that fibre there is a unique element
for which
, where
is the (right) action of
by monodromy on
. This gives a left action of
on
by deck transformations
for each
, which maps
to
for any
:
The fibres of the universal cover are copies of
. We can now form a new cover
whose fibres are copies of
, by first forming the Cartesian product
(which still covers
) and then quotienting out by the equivalence
for all
,
,
. This is a (possibly disconnected) covering space for
, whose fibre above
can be identified with
by identifying
with the equivalence class
. The monodromy (right) action of
on this fibre is then identified with the right action of
on
induced by
. Furthermore, the group
acts on
on the left by deck transformations, with each
mapping
to
.
Similarly, we can form a covering of
whose monodromy action of
on the fibre
can be identified with the right action of
on
induced by
. When both of these covering spaces are restricted to
, the monodromy action of
on the fibre above
are then isomorphic to each other, and thus the two restrictions are isomorphic to each other too. By Proposition 2, we can then glue these two covers together to obtain a cover
which is isomorphic to
or
upon restricting the base to
or
respectively, with all four isomorphisms forming a commuting square; in particular, the fibre
is still identified with
. The monodromy right action of
on
then restricts to the previously described action of
on
and of
on
. Also, because
acted on the left by deck transformations on both
and
, in a manner which can be seen to be compatible with the isomorphism on restriction to
,
continues to act by deck transformations on the gluid cover
. As monodromy actions commute with deck transformations, we conclude that the right action of an element
of
on
is given by right multiplication by an element
of
. It is then routine to verify that
is a homomorphism with the required properties.
Now we prove uniqueness. It suffices to show that is generated as a group by the images of
and
. Suppose this were not the case, so that the images of
and
generate a proper subgroup
of
. Let
be a universal cover of
, so that the fibre
may be identified with
(after fixing a reference point in that fibre as before). Let
be the restriction of
to
. The monodromy (right) action of
on the fibre above
is induced by the canonical homomorphism from
to
. In particular,
is preserved by this action, and so one can find a proper subcover
of
whose fibre corresponds to
. Similarly, we can find a proper subcover
of the restriction
of
to
with the same fibre. By Proposition 2, we may glue these two covers together to obtain a proper subcover
of
. But such a proper subcover cannot exist because
is connected, and the claim follows.
Remark 1 The arguments above relied heavily on the universal cover and its attendant deck transformations and monodromy actions, mostly in order to give a concrete equivalence between coverings of
and discrete sets with actions of the fundamental group
. It is also possible to proceed without constructing this cover, working instead with a directed family of Galois covers as a substitute for the universal cover to obtain this equivalence of categories; this is the approach in Grothendieck’s Galois theory, which among other things can be used to construct the étale fundamental group, as this is a context in which universal covers need not exist, but one still has plenty of Galois covers. See for instance Szamuely’s text for more details.
10 comments
Comments feed for this article
28 October, 2012 at 11:41 am
chorasimilarity
Have you seen the book Topology and Groupoids by Ronald Brown?
28 October, 2012 at 11:44 am
JuanPi
“such as locally path connected spaces, … to avoid path-ologies.”
That’s a funny one!
28 October, 2012 at 1:35 pm
Rodolfo Niborski
In the first sentence, did you mean to say “with two loops considered equivalent to each other if they are homotopic” ? [Corrected, thanks – T.]
28 October, 2012 at 1:58 pm
Approximate groupoids may be useful « chorasimilarity
[…] by way of groupoids, of the Van Kampen theorem on the fundamental group of a union of spaces, Terence Tao has a post about this subject. I wonder if he is after some applications of his results on approximate groups in the realm of […]
28 October, 2012 at 9:37 pm
Ryan Reich
I enjoyed this presentation! Your argument has the following conceptual summary (which I do not intend as a criticism, but just a road map):
is the same as a covering space of M that is a principal G-bundle (with G considered as a discrete group).
to
.
1. Given the universal cover of M, a map
2. (Proposition 2) The pushout of two covering spaces over an open covering of M is a covering space of M, and if they are both principal G-bundles, then so is the pushout.
3. The restriction of a $G$-bundle to an open subspace U of M corresponds to the restriction of
The proof is then: by Point 1 we can do the construction entirely with G-bundles; by Point 2 we can create the map to G, which by Point 3 has the desired restriction properties and is unique because pushouts are unique.
This is, I guess, the Grothendieck-Galois-theoretic proof of the theorem. I think you could summarize even further by using the word “stack”, but I won’t :)
1 November, 2012 at 5:58 pm
Anonymous
I think the proof in the book of Douady (Theories Galoisiennes) is slightly smoother because it is 98% categorical (in fact there is a topos version due to Ronnie Brown and company). Their proof is roughly like this. If one has a pushout diagram of groups G,H,K where K has maps into G,H respectively, then by the universal property for the pushout, call it A, an A-set is the same thing as a G-set X, an H-set Y and a K-set isomorphism between these. The gluing argument above then implies the category if pi_1(M)-sets is equivalent to the category of A-sets for the pushout A of the fundamental groups via the natural maps. It is an easy exercise in category theory to deduce that a group homomorphism which induces an equivalence of action categories is an isomorphism.
14 November, 2012 at 9:42 am
Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets « What’s new
[…] an étale version of the van Kampen theorem for the fundamental group, which I discussed in this previous blog post. With this fact (and another, rather deep fact, about the étale fundamental group in zero […]
20 November, 2012 at 11:57 am
pablocecil
In his book Algebraic Topology. A First Course, Bill Fulton attributes this proof to Grothedieck.
29 July, 2015 at 7:42 am
brydustin
In van Kampen’s theorem, the intersection is assumed to be connected…. shouldn’t it be assumed to be simply connected?
[van Kampen’s theorem can handle the situation in which the intersection is not simply connected (the free product becomes an amalgamated free product over the fundamental group of the intersection. – T.]
25 November, 2018 at 6:09 am
Week 9 - Introduction to Topology
[…] alternative derivation of the Seifert-van Kampen Theorem is sketched in Terence Tao’s blog. The derivation is based on covering […]