Topologie en Meetkunde, 2012
Schedule
The hoorcollege of
WISB341 is on mondays
from 11:00-to 12:45 uur, in room BBL 169.
The werkcollege is wednesday from 9:00 to 10:45 in room BBL 165.
Teachers
The main teacher is André Henriques,
and the assistant is Reinier Storm (R.W.Storm - uu.nl)
Dictaat
We will be mainly following the dictaat of Marius Crainic,
available at his webpage.
Homework
There will be homework every other week, to be handed in during the werkcollege
9 days after the day when it is posted on the website.
The homework counts for 10% ofthe final grade.
Exams
There are two exams for this class: one written exam half-way,
and one oral exam at the end. Both count for 45% of the final grade.
Material covered:
Monday May 21st
Fundamental group of 2-dimensional CW-complexes: end of proof.
Construction of univesal covers for locally simply connected spaces.
Hand-in exercise 4 (hand in on Wednesday June 6th):
Wednesday May 16th
• Prove that the
Hawaiian earring space does
not have a universal cover.
• Prove using the path lifting property that any two universal covers
of a given space are homeomorphic.
• Goal: find a presentation of the group of quaternions
Q:={1,-1,i,-i,j,-j,k,-k}.
Method: use the fact that Q acts of S3 (by viewing S3 as the unit
sphere in the quaternions).
The quotient M:=S3/Q is a 3-manifold whose universal cover is S3,
and whose fundamental group is Q.
Write down an explicit CW-structure for M, and read off the presentation
from the attaching maps of the 2-skeleton of M.
Further hint in order to write down an explicit CW-structure of S3/Q:
write down a CW-structure of S3 that is invariant under Q.
• Write down a presentation of the fundamental group of the
Poincare dodecahedral space.
• Definition: a map p:E → B is called etale if for every point x in
E there is an open neighborhood U
of x such that the restriction map p:U → p(U) is a homeomorphism.
Show that covers are etale, but that not all etale maps are covers.
Show that etale maps satisfy the "uniqueness" part of the path lifting propetry,
but not the "existence" part.
Monday May 14th
The path lifting property for covers.
Example: ℝ2→T2.
The natural bijection between p-1(*) and π1(X) (end of proof).
Normal subgroups, quotient groups, presentations of groups.
Computation of π1(S1), and of π1(S1v...vS1).
π1 of 2-dimensional CW-complexes.
Wednesday May 9rd
• What is the universal cover of the n-dimensional torus (S1)n.
What is π1((S1)n), and what is the preimage of the base point * in (S1)n?
• What is the universal cover of S1 v Sn? (n>1)
Hint: what is the fundamental group of S1 v Sn?
• What is the universal cover of P2?
Hint: S2 is simply connected.
Use this to compute π1(P2).
• Let X be the space that is obtained by glueing a 2-cell on S1 via
the map z ↦ z3?.
Compute the universal cover of that space.
Answer: it's a 2-sphere equipped with an extra 2-disk glued along its equator.
• What is the universal cover of the genus two orientable surface Σ?
Hint: it's the n=8, k=8 case of this.
The graph drawn on this disc is the preimage of the standard cell
decomposition of Σ.
• What is the universal cover of the Klein Bottle?
Describe the map from the universal cover down to KB.
Monday May 7th
Proof that πnX(n+1) → πnX is an isomorphism.
Free groups and reduced words.
The natural map from the free group <a1,...,an> to π1(S1vS1v...vS1).
The definition of universal cover p:\tilde(X)→ X of a topological space X.
The natural bijection between p-1(*) and π1(X).
Wednesday May 2nd
• Given a space X, define its suspension ΣX to be the quotient of
X×[0,1]
by the equivalence relation that crushes X×{0} to a point, and also crushes
X×{1} to a point.
Show that there is a natural homomorphism from πn(X) to πn+1(ΣX).
[check that it is a well defined, and that it is a homomorphism]
• Given a topological group G, check that π1(G) is always abelian.
• show that there is a natural action of π1(X) on πn(X).
[Check that it is well defined, and that it is an action]
The space S1 v Sn is an example where that action is non-trivial.
Wednesday April 25th
• Show by an example that it is possible to construct a compact space
K and an increasing chain
of closed subspaces K_1 in K_2 in K_3... such that K = union of K_i.
Show however, that the colimit topology on union of K_i does not agree
with the original topology on K.
• define colimits in an orbitrary category.
Show that the functor πn commuted with colimits:
The πn of a colimit of spaces is the colimit (in the category of
groups) of the πn's.
• Construct a space whose π1 is ℚ in the following way:
write down a presentation of ℚ, and construct a 2-dimensional
CW-complex with attaching maps given by that presentation.
If the presentation is <a,b,c,d,e,f,g,h....|a=b2, b=c3, c=d4,
d=e5, ....>.
Hand-in exercise 3 (hand in on Wednesday May 16th):
Monday April 23rd
A map from a compact space to a CW-complex factors through some finite sub-complex.
A maps from something n-dimensional to a CW-complex can be homotoped to a map that lands in the n-skeleton.
Corollary: the inclusion of the n-skeleton induces a surjection at the level of πn.
Practice exam
Monday Apr 2nd (last hoorcollege before the exam)
Higher homotopy groups πn(X) are abelian for n≥2.
Homotopy groups are functors from the category of pointed topological spaces to the category of (abelian) groups.
If two maps f and g are homotopic, then πn(f)=πn(g).
Homotopy equivalent spaces have isomorphic homotopy groups.
If X is a CW complex, then the inclusion of the n-skeleton X(n) into X induces a surjection on πn,
and the inclusion of the (n+1)-skeleton into X induces a isomorphism of πn [without proof].
I finished with a quick-and-dirty explanation of how to read off a presentation of π1 of some 2-dimensional CW-complex
from the attaching maps of the 2-cells.
Wednesday Mar 28th
• Show that any graph is homotopy equivalent to a wedge of S1's.
• Let X and Y be two spaces that are homotopy equivalent.
Let f : Sn-1 → X be a map and let g: Sn-1 → Y be the
corresponding map. Show that X ∪f en ~ Y ∪g en.
• Consider the space obtained by attaching two 2-cells onto S1.
One of the 2-cells is attached by the map z ↦ z2, and the other
2-cell is attached by the map z ↦ z3. Show that the resulting
cell complex is homotopy equivalent to S2.
• Show that pi1(X×Y) = pi1(X) x pi1(Y)
• Prove that if two spaces are homotopy equivalent, then their fundamental groups are isomorphic.
• Let X be obtained from A by attaching an n-cell. Prove that for every point p in X - A the inclusion
i:A → X - {p} is a homotopy equivalence.
monday Mar 26th
Deformation rectracts. Changing the attanching map by a homotopy doesn't change the resulting space up to homotopy equivalence.
The fundamental group π1(X).
Wednesday Mar 21st
• If f:X→X and g:X→X are homotopic maps, then their n-th iterates
are also homotopic.
• Moebius band ~ S1 (here, the symbol ~ means homotopy equivalent)
• T2 minus a point ~ S1 v S1
• CP2 minus a point ~ S2
• Rn minus two points ~ Sn-1 v Sn-1
• Rn minus k points ~ k-fold wedge of copies of Sn-1
• Prove that if X ~ Y and U ~ Z, then X×U ~ Y×Z.
• Prove that Sinfty is contractible.
• show that it is not true that A ~ A', B ~ B', C ~ C'
implies that the pushout of A←B→C is h.e. to the pushout of
A'←B'→C'.
Monday Mar 19th
The colimit topology. The infinite dimensional sphere: the colimit topology is not the same as teh one coming from the metric.
The infinite wedge of intervals: that space is not first countable.
Homotopy between maps. Straight line homotopy. Star-shaped domains. Homotopy is an equivalence relation.
The notion of two spaces being homotopy equivalent; the fact that it's an equivalence relation.
Example: ℝ2\{0} is homotopy equivalent to S1.
Example: "B" is not homotopy equivalent to other letters of the alphabet.
Wednesday Mar 7th
• Prove that a CW complex is compact iff it has finitely many cells.
• Show that the join of infinitely many copies of S1 is not
homeomorphic to the Hawaiian earing space
by exhibiting a subset that is open in the infinite wedge but not open in the
Hawaiian earing space
• Prove that the space e0 ∪ e2 is homeomorphic to S1.
Hint: use the fact that a bijective continuous map between compact
spaces is always a homeomorphism.
• Write down the attaching map for the 4-cell in CP2 = e0 ∪ e2 ∪ e4.
• Write down the attaching map S1 → S1 v ... v S1 for the 2-cell
in a surface of genus g.
• How many cells does one need to build the real projective space RPn?
• write down a cell decomposition for the Klein bottle. What is the
attaching map of the 2-cell?
• Write down a CW-complex structure for S∞ = colim Sn.
Hint: use two n-cells for each n.
Monday Mar 5th
I compared the notion of triangulation that I used in class with the one given in Crainc's dictaat (let's call the latter a regular triangulation).
Exercise (that I didn't do): the smallest regular triangulation of T2 uses 14 triangles.
Examples of CW-complexes: Sn, SnvSm, Sn×Sm.
The notion of attaching map. The explicit form of the attaching map Sn+m-1→SnvSm for the (n+m)-cell of Sn×Sm.
The general definition of pushout. Existence and uniqueness of pushouts.
Attaching an n-cell: X∪fen.
General definition of a CW-complex.
Hand-in exercise 2 (hand in on Wednesday March 21st):
Wedneseday Feb 29th
• what surface do you get from the following two labelling schemes:
"abdee-1fd-1c-1cb-1a-1f-1", and "abcdefabcdef"
• compute the Euler characteristic of S2 in 5 different ways, using the 5 platonic solids.
Use the fact that χ(S2)=2 to show that "there exists a platonic solid with n regular k-gons per vertex" =>
(n,k) is either (3,3), (3,4), (4,3), (3,5), or (5,3). • compute the Euler characteristic of a connected orientable surface with genus g (note: the "genus" is the "number of holes", i.e. the number of tori that are connect-summed together). • prove that if M is a (triangulated) orientable n-dimensional manifold with boundary, then its boundary ∂M is an orientable (n-1)-dimensional manifold.
Monday Feb 27th
Standard forms for surfaces: S2 is given by "aa-1"; P2#...#P2 is given by "aabbcc...";
T2#...#T2 is given by "aba-1b-1cdc-1d-1efe-1f-1..."
Two invariant that can be used to distinguish surfaces: orientability, and the Euler characteristic χ(M).
Some reminders about the definition of orientations. Piecewise linear cellular decomposition of surfaces
(i.e. glueing convex polygons along edges).
Theorem: Let M1 and M2 be two compact (triangulated) surfaces that are (piecewiseliy linearly) homeomorphic,
then M1 is orientable iff M2 is orientable, and χ(M1) = χ(M2).
Wednesday Feb 22nd
• show that any orientable surface is the boundary of a 3-manifold with boundary.
• show that the Klein bottle is the boundary of a 3-manifold with boundary.
• Find a triangulation of the torus that uses only two triangles, and draw it on the torus (in its usual embedding in R3).
• Find a triangulation of the torus that uses a (2,3)-curve as one of its edges [a (2,3)-curve is a curve that runs 2 times along the
meridian, and 3 times along the parallel of the torus]
Monday Feb 20th:
Terminology: "handle", "cross-handle", "cross-cap", "hole".
Definition of manifolds with boundary ("closed" manifold = compact without boundary).
Proof that every compact triangulable 2-manifold with boundary is homeomorphic
to a finite disjoint union of S2, S2 with handles and holes, S2 with cross-caps and holes.
The proof followed largely Conway's ZIP proof.
Hand-in exercise 1 (hand in on Wednesday Feb 29th):
Wednesday Feb 15th:
• Write down a triangulation of the projective plane P2.
• Prove that a triangulation of a surface necessarily involves an even
number of triangles.
• prove (in detail!) that every compact connected one-dimensional
manifold is homeomorphic to S1. [Careful: if you didn't use
Hausdorfness in your proof, the proof is wrong, as there exist
many non-Hausdorff manifold in dimension 1]
• construct an example of a one dimensional non-Hausdorff manifold
that is not orientable.
(ok, that goes against my definition of the word "manifold" -- apologies)
• Which of the following surfaces is orientable: P2, KB, T2, S2,
T2#T2, T2#KB, T2#P2?
(Here KB = Klein bottle).
• How many orientations does the disconnected manifold S2 ⊔ ...
⊔ S2 admit?
Monday Feb 13th:
Definitions of connectednes and compactness.
Some examples of non-compact 2-manifolds: ℝ2 \ ℕ ≅ ℝ2 \ ℤ;
ℝ2 \ Cantor set. I mentionned the fact that one cannot expect to classify non-compact 2-manifolds.
Various names (projective plane; real projective plane; cross cap) and definitions for P2:
lines in ℝ3; 2-sphere modulo antipodal relation; 2-disc modulo glueing antipodal points of the boundary;
Moebius band with boundary crushed to a point. P2 # P2 ≅ Klein Bottle.
Definition of what it means for a (topological) manifold to admit a triangulation.
I mentionned the fact (without proof) that all manifolds of dimensions 1, 2, and 3 admit triangulations,
but that this is no longer the case in dimensions ≥ 4.
I defined an orientation on a simplex as an equivalence class of total ordernings of the set of vertices,
where two ordernings are equivalent if they differ by an element of the alternating group.
I defined (quickly) the induced orientation on a facet of a simplex, and what it means for a triangulated
manifold to be oriented: each simplex should be oriented, and for every (n-1)-dimensional simplex of the
triangulation, the orientations induced by the two n-simplices of which it is a facet should be different from
each other.
Wednesday Feb 8th:
• Exercises 1.1, 1.2, 1.3, 1.4 of the
dictaat
• Show that the connected sum of two P2's is a Klein bottle (see
figure on page 12 of the dictaat for some inspiration about how to prove that; see also Figure 7 of
ZIP)
• Show that the quotient of a disc by the equivalence relation that
collapses the boundary to a point is
homeomorphic to a sphere (for that one will need to recall the
definition of the quotient topology, and use that
the boundary of the disc is compact -- except that I forgot the say
what "compact" means).
• Show that if one starts with an arbitrary finite collection of
convex polygons whose total number of edges is even,
and one glues the edges in pairs, then the resulting topological space is always a manifold.
Note also that the same procedure in 3 dimensions can fail to produce
a manifold (think of the cone on a torus: it can be
obtained by glueing tetrahedra, but it has a non-manifold point at the top).
Monday Feb 6th: I recalled the definition of topological spaces, product spaces, quotient spaces, disjoint unions, the subspace topology. Example: the reals ℝ,
the n-dimensional Eucledian space ℝn. The definition of manifolds: topological space that are locally homeomorphic to ℝn and that are
furthermore Hausdorff and second countable. Discussed some examples that fail Hausdorfness (the line with two origins), and that fail second countability (the long line).
Examples of curves: ℝ, and the circle S1. Examples of surfaces: the sphere S2, the torus T2, the Kein bottle,
the projective plane P2. I discussed how to get S2 as a quotient of a dics where one collapses the whole boundary to a point, as two discs
with identified boundary, and as one disc with half of the boundary identified to the other half. I discussed how to obtain the 2-torus by glueing opposite sides of a square,
or by glueing opposite sides of a hexagon, and explained why those two constructions produce homeomorphic manifolds. Finally, I stated the classification of compact connected surfaces
(in class, I forgot to say "compact"): the are all connected sums of tori or connected sums of copies of P2.
Results of first midterm exam:
3470814: 3.0
3350398: 7.0
3476987: 8.0
3486966: 7.5
3477002: 8.5
3502309: 10.0
3564029: 7.5
3490076: 5.5
3481549: 10.0
3471543: 8.0
3354946: 3.0
3481565: 9.5
3558754: 6.0
3345653: 10.0
3471624: 10.0
3477025: 5.5
3502317: 10.0
3471675: 9.0
3245683: 5.0