MAT 364 Course Notes
An overview
1. Examples of spaces that commonly occur in low-dimensional
topology: disk, ball, surfaces, knots, graphs,
Mobius band, more exotic examples like witch's broom and devil's antenna.
2. Intuitive concept of a homeomorphism.
3. Examples of homeomorphic and non-homeomorphic spaces.
4. Examples of topological invariants: connectivity and the number of
connected components, simple-connectivity, Euler characteristic.
5. Main problem of topology: to identify and study those properties
of spaces which are invariant under homeomorphisms.
Some point-set topology in Rn
6. The n-dimensional Euclidean space Rn
and its linear structure.
7. Norm of a vector and the (Euclidean) distance between points
in Rn; basic properties; triangle inequality
and its main message ("two points that are close to a third point must
be close to each other").
8. Balls in Rn.
9. Open sets in Rn; examples.
10. Every ball is open.
11. Unions and intersections; complement of a set; De Morgan's laws.
12. The union of any family of open subsets of Rn
is open. The intersection of a finite family of open subsets of
Rn is open. The intersection of an infinite family
of open sets may or may not be open.
13. Closed sets in Rn; examples.
14. The intersection of any family of closed subsets of
Rn is closed. The union of a finite
family of closed subsets of Rn is closed.
The union of an infinite family
of closed sets may or may not be closed.
15. The middle-third Cantor set C.
16. Properties of C: it is non-empty and closed; it
contains all the endpoints of the intervals that occur in its construction but
also many other points; in fact C is uncountable; C has zero
"length".
17. Relatively open subsets of a given set; examples.
18. Recalling the notion of continuity from one-variable calculus:
the rigorous approach.
19. Definition of continuity for maps between Euclidean sets
(the "epsilon-delta" definition).
20. Examples of continuous and discontinuous maps between Euclidean
sets.
21. Topological description of continuity: preimages of open sets
must be open.
22. The image of an open set under a continuous map need
not be open (think of folding an open ball in the plane along one of its
diagonals).
23. To prove continuity, it suffices to check that preimages of
balls are open; examples showing how this works.
24. Compositions of continuous maps are continuous.
25. Definition of a homeomorphism.
26. Examples of homeomorphic subsets of Euclidean spaces: Any two closed
line segments, any two balls in Rn, a closed
segment and a closed semi-circle, an open interval and the real line,
an open ball in R2 and the plane, an open square
and an open ball in the plane.
27. The property of being homeomorphic is "transitive": If X
is homeomorphic to Y and if Y is homeomorphic to Z,
then X is homeomorphic to Z.
28. Stereographic projections from punctured circle to R
and from punctured sphere to R2; properties.
29. Paths in Euclidean sets.
30. Path-connected sets; examples: Euclideans spaces, a circle
in the plane, a sphere in the space, a circle with a point removed,
punctured plane, a sphere with a slit along the equator.
31. Examples of sets which are not path-connected: The real
line with a point removed, a circle with two points removed,
a sphere with the equator removed; the crucial point in all these
examples is the Intermediate Value Theorem of one-variable calculus.
32. The continuous image of a path-connected set is
path-connected. (This theorem is useful for proving path-connectivity,
whenever you can represent your set as the image of a "simple"
path-connected set under a continuous map. For example, the cylinder
is path-connected since it is the image of the plane under a
continuous map, e.g., wrapping the plane around the cylinder over and
over again.)
33. Path-connectivity is a topological property: If X
and Y are homeomorphic, then X is path-connected iff
Y is.
34. The (path) component of a point in a set; examples
35. Any set can be partitioned into a number (finite or infinite)
of path components. Each path component is path-connected.
36. Let f be a homeomorphism between X and Y.
Then f maps each component of X onto a component of Y
homeomorphically, and it gives a one-to-one correspondence between components
of X and those of Y.
37. The concept of an n-cut point of a set. Homeomorphic
sets have the same number of n-cut points for each n.
Knots and links
38. Definition of a knot in R3.
39. Equivalence of knots: Two knots are equivalent if one can
be continuously deformed to another.
40. Main problem of knot theory: Given two knots, decide whether or not
they are equivalent.
41. Examples of knots: The unknot (or trivial knot), the left and right
trefoil knots, the figure 8 knot, torus knots.
42. Knot diagrams; a knot may have several different looking diagrams;
each knot diagram consists of a number of "strands", a number of "crossings",
and a choice of "over/under" at each crossing.
43. Crossing number of a knot; the unknot has crossing number 0;
there are no knots of crossing number 1 or 2; the trefoil knots have
crossing number 3.
44. Oriented knots; each knot has two different orientations;
an orientation determines a notion of "left" or "right" at each crossing of the
knot diagram, and this notion is independent of the choice of the orientation.
45. Definition of Reidemeister moves on a knot diagram.
46. Theorem: Two knots K1 and K2
are equivalent iff any diagram of K1 can be obtained from
any diagram of K2 by a finite sequence of Reidemeister moves.
47. Examples of how to deform knots using Reidemeister moves.
48. Corollary of the above theorem: To verify that a certain
property of knots is invariant under deformation, it suffices to check
that it is invariant under Reidemeister moves.
49. Coloring a knot diagram; examples.
50. Let two diagrams represent equivalent knots. Then one diagram is
colorable iff the other diagram is. As a corollary, we can define a knot
to be colorable if one (hence every one) of its diagrams is colorable.
51. Corollary: A colorable knot cannot be equivalent to a non-colorable
knot. For instance, since the trefoil knot is colorable and the unknot is not,
we conclude that the trefoil knot cannot be untied.
52. If two knots are both colorable or both non-colorable,
they may be still different knots. For instance, neither the figure eight
knot nor the unknot is colorable, and they are not equivalent.
53. Definition of the connected sum
K1#K2 of two knots K1
and K2.
54. Properties of the connected sum:
K1#K2=
K2#K1
for all knots K1 and K2
(connected sum is a commutative operation); If K0 denotes
the unknot, then K#K0=K for every knot
K (the unknot acts as the identity element).
55. Theorem: If K is a non-trivial knot, then there is no knot
J such that K#J is the unknot. (In a fancier language,
this means that no non-trivial knot has an inverse in the semigroup defined
by the binary operation #.)
56. Links; examples: Hopf link, King Solomon's knot, Borromean rings,
Whitehead link.
57. Link diagrams, deformation of links, oriented links.
58. Definition of the linking number L(K,J)
of two oriented knots K and J; examples.
59. The linking number is invariant under deformation.
60. Properties of the linking number: L(K,J)=L(J,K);
L(-K,J)=L(K,-J)=-L(K,J) (here -K is the knot K
with the reverse orientation).
61. Two oriented knots with non-zero linking number are linked, i.e.,
there is no deformation which separates them from each other.
62. If two oriented knots have linking number zero, they may or
may not be linked.
63. The Conway polynomial of an oriented link; defining properties.
64. Examples of how to compute the Conway polynomials of simple oriented
knots and links.
65. If an oriented link is split (i.e., if it is equivalent to the
union of two links contained in disjoint balls), its Conway
polynomial is identically zero.
66. Examples which demonstrate the application of the above statement.
67. Two equivalent oriented links have the same Conway polynomials, but
two oriented links with the same Conway polynomials may or may not be equivalent.
Topology of Surfaces
68. Definition of a surface: A path-connected set which is locally
homeomorphic to an open disk in the plane.
69. Examples of surfaces: sphere, torus, any open
path-connected subset of the plane, a cone in R3,
the surface of a growing tree in R3 with infinitely
many branchings, Mobius band (without its boundary curve), Klein's bottle.
70. Informal discussion of orientation; sphere and torus are
orientable, Mobius band and Klein's bottle are not.
71. Definition of a bounded set in Rn;
examples.
72. Definition of a compact set in Rn;
examples.
73. Examples of compact surfaces.
74. Cutting compact surfaces into simples pieces and recovering
them by gluing back these pieces; examples: the sphere is homeomorphic
to two closed disks glued along their boundary circles, the torus is
homeomorphic to a closed square with opposite edges glued with the same
orientation.
75. Let X be the surface of a g-hole torus. Cutting
X along a standard system of 2g loops
a1, ..., ag and
b1, ..., bg will give a
topological 4g-gon whose edges are labeled with alternating
a's and b's. Going aroung the boundary of this polygon
in the counter-clockwise direction, we read the word
a1
b1
a1-1
b1-1 ...
agbg
ag-1
bg-1.
The surface X can be recovered by gluing (or "identifying") the
edges that come with the same label. When gluing, one must be careful
to match orientations.
76. The Klein Bottle can be constructed by identifying the opposite
edges of a square, one pair with matching orientations and the other
pair with reversed orientations. It is a compact non-orientable
surface which cannot be realized as a subset of
R3. Using this model, it is easy to see that the
Klein Bottle is homeomorphic to the two copies of a Mobius band, glued
along the boundary circles.
77. Definition of the connected sum
X#Y of two surfaces X and Y; examples.
78. Properties of the connected sum:
X#Y=Y#X for all surfaces X and Y
(connected sum is a commutative operation); If S2 denotes
the sphere, then X#S2=X for every surface
X (the sphere acts as the identity element).
79. Definition of a pair of pants; pant decomposition of a
g-hole torus; examples.
80. Triangulations of surfaces; examples of triangulated surfaces;
examples of decompositions which are not triangulations.
81. Theorem: Every compact surface admits a (finite) triangulation.
82. Classification Theorem: Every compact orientable surface is
homeomorphic to either the sphere or a g-hole torus for some
positive integer g.
83. Sketch of the proof of the Classification Theorem:
Step 1. Given a compact orientable surface X, choose a triangulation
{T1, ... Tk} of it. Label the triangles
in such a way that Ti has a common edge with
Ti+1 for every i=1,...,k-1
(possible since X is path-connected). Label the edges of these
triangles and orient the boundary of each Ti in the positive
sense (possible since X is orientable). Thus, each edge inherits two
opposite orientations from the two triangles sharing it.
Step 2. Choose a homeomorphic copy S1 of T1
in the plane, label and orient its edges accordingly. Choose a
homeomorphic copy S2 of T2 in the plane,
label and orient its edges accordingly, and glue it to S1
along the edge corresponding to the common edge of T1
and T2. Repeat this process until you take all the
triangles into account. End up with a topological (k+1)-gon P
(the union of the Si's) in the plane whose edges are
labeled, with each label appearing exactly twice. Gluing these edges in
pairs will produce a homeomorphic copy of X. Going around the boundary
of P in the counter-clockwise direction, one reads a word by putting
theedge labels together.
Step 3. If you ever see a label a adjacent to its inverse
a-1 in the word, glue the corresponding edges to
eliminate both. If all the edges of P get eliminated this way,
it is easy to see that X must be homeomorphic to the sphere.
Step 4. Make sure all the vertices of the polygon P correspond to
the same point q on X. If not, apply a simple cut-and-paste
surgery on P to replace all vertices other than q by q.
Step 5. For every label a, there must be a label b such that
in the counter-clockwise direction you see
... a ... b ... a-1 ...
b-1 ...
By applying a simple cut-and-paste surgery on P, replace these four
edges by four new edges which appear as
... cdc-1d-1 ...
In the course of this modification, things done in the previous steps will
not be disturbed. So this process can be continued until we get a polygon
with the word
a1
b1
a1-1
b1-1 ...
agbg
ag-1
bg-1
for some positive integer g. We know that gluing the edges of such a
polygon gives a surface homeomorphic to a g-hole torus.
84. Definition of the Euler characteristic of a compact
triangulated surface as v-e+f; examples.
85. The Euler characteristic does not depend on the choice of
the triangulation; it only depends on the surface itself.
One way of verifying this is to show that: (i) The count
v-e+f does not change by "refining" a triangulation,
(ii) Any two triangulations of the same surface have a common refinement.
86. The Euler characteristic is a topological invariant: Two
homeomorphic surfaces have the same Euler characteristic.
87. For any two compact surfaces X and Y,
E.C. (X#Y) = E.C. (X) + E.C. (Y) -2
88. The Euler characteristic of the g-hole torus is
2-2g.
89. The "genus" of a compact orientable surface X
is defined as g = (2-E.C.(X))/2.
90. Two compact orientable surfaces are homeomorphic if and only
if they have the same genera (or the same Euler characteristics).
91. Generalizations of the notion of triangulation: polyhedral
structures, cell decompositions.
92. Examples of polyhedral structures on compact surfaces.
93. The Euler's formula v-e+f = E.C. (X)
holds for any polyhedral structure on a compact surface X
(as usual, v is the number of vertices, e
the number of edges, and f the number of faces of the polyhedron).
94. Definition of a regular spherical polyhedron (aka "Platonic
solid"); There are only
five Platonic solids:
| v
| e
| f
| faces
| degree at each vertex
| name
|
| 4
| 6
| 4
| triangle
| 3
| tetrahedron
|
| 8
| 12
| 6
| square
| 3
| hexahedron (cube)
|
| 6
| 12
| 8
| triangle
| 4
| octahedron
|
| 20
| 30
| 12
| pentagon
| 3
| dodecahedron
|
| 12
| 30
| 20
| triangle
| 5
| icosahedron
|
Vector Fields on Surfaces
95. Definition of a continuous vector field in a region in
R2; trajectories; examples.
96. Singular points of vector fields; isolated singular points; examples.
97. The index of a vector field at an isolated singularity; examples.
98. The definition of index indX(p)
does not depend on the choice of the loop going around the singular point
p, as long as the loop does not pass through
any singularity and encloses only p.
99. The index of a vector field along a simple closed curve; examples.
100. Theorem: Let X be a continuous vector field in the plane and
C be a simple closed curve not passing through the singular points of
X. Suppose p1, ..., pn
are the singular points of X in the domain enclosed by
C. Then
indX(C)=indX(p1) +
... + indX(pn).
101. Corollary: If indX(C) is non-zero, there
must be at least one singular point of X in the domain
enclosed by C.
102. Vector fields on surfaces; examples.
103. Theorem (Poincare-Hopf): Let S be a compact
orientable surface and X be a continuous vector field on
S with finitely many singularities p1, ...,
pn. Then
indX(p1) +
... + indX(pn) = E.C.(S).
104. Corollary: If S is not homeomorphic to a torus
(whose Euler characteristic is zero), then every continuous vector field on
S must have at least one singular point.
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