Download Algebraic Topology Introduction

1
Point Set Topology
Partitions of unity, some common topologies, connectedness, compactness
A good overflow thread discussing the definition of topology in terms of open sets http://mathoverflow.
net/questions/19152/why-is-a-topology-made-up-of-open-sets/19156#19156
1.1
Basics
Proposition 1.1. Let f : X → Y , with A ⊆ X and B ⊆ Y .
• A ⊆ f −1 ◦ f (A) with equality if and only if f is injective.
• f ◦ f −1 (B) ⊆ B with equality if and only if f is surjective.
• A bijective continuous map is a homeomorphism if and only if it is open (or closed.)
Include more properties about the inverse of a map e.g. inverse of union, intersection, etc.
Proposition 1.2. A topological space X is Hausdorff if and only if the diagonal D = {(x, x) : x ∈ X} is a
closed subset of X × X.
Some results about compact spaces:
See Terry Tao’s intuitive notes on compact spaces, http://www.math.ucla.edu/~tao/preprints/compactness.
pdf
Compact spaces can be roughly characterized as having tamed infinity. For example, points can’t quite
run off to infinity like on a non-compact space e.g. R,
For a metric space, compactness is equivalent to sequential compactness, that is, every sequence has a
convergent sub-sequence. See also, Konig’s lemma.
Theorem 1.3.
1. Every closed subset of a compact space is compact.
2. Every compact subset of a Hausdorff space is closed.
3. The image of a compact space under a continuous map is compact.
4. Let Y be Hausdorff with f : X → Y a continuous map. The inverse image of a compact set is closed.
Further if X is compact then the inverse image of a compact set is compact.
5. Let f : X → Y be a bijective continuous function. If X is compact and Y is Hausdorff, then f is a
homeomorphism.
Proof. (2) Let X be a topological space with C ⊆ X a compact subspace. Let x ∈ X − C, we aim to show
that there is an open set disjoint from C that contains X (i.e. C is closed.)
Take any y ∈ C, and there exists an open neighborhood about y, Vy , and an open neighborhood about
x, Uy , which are disjoint (definition of Hausdorff.)
Now let the point y move around in C, so the set {Vy } makes an open cover of C. There exists a finite
subset F ⊆ C, such that {Vy : y ∈ F } forms a finite open cover of C (definition of compact.)
Now for each open set Vy with y ∈ F , take its partner Uy defined above. As y runs over F , the intersection
.
U = ∩y∈F Uy is a set of points containing x and disjoint from all of the Vy , and hence disjoint from C.
Lastly, U must be open since the intersection of a finite number of open sets is open. We really did need
the finite subcover from the compact property!
(3) Let f : X → Y be continuous, with X compact. Let B be an open cover of f (X). Consider
A = {f −1 (B) : B ∈ B}. First note that A is a collection of open sets in X, since f is continuous and each
B is open in Y . Second, A is an open cover of X, since if x ∈ X there exists a B such that f (x) ∈ B i.e.
x ∈ f −1 (B). Since X is compact, there exists a finite subcover of A, call this finite collection Ā.
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Every open set in Ā has the form f −1 (B) for some open B in Y . Then, apply f to each open set in Ā,
f ◦ f −1 (B) = B since f surjects onto f (X). This finite collection of B’s forms a finite open cover of f (X),
and therefore f (X) is compact.
(4) Since Y is Hausdorff, a compact subset C of Y must be a closed subset. Then, f −1 (C) is closed in
X, by definition of continuity. If X happens to be compact, then f −1 (C) is compact, since closed subsets of
compact spaces are compact.
(5) Since f is bijective and continuous, to prove that it is a homeomorphism, all that remains to show
that f −1 is continuous. We will show that f is a closed map, and hence f −1 is continuous.
Let A be a closed subset of X. Since X is compact, A is compact (part 1). Then, f (A) is compact in Y
(part 3). Finally by part 2, since Y is Hausdorff and f (A) ⊆ Y is compact, f (A) is also closed. Therefore f
is a closed mapping, and f is a homeomorphism.
1.2
Quotient Space
Let X and Y be topological spaces. A surjective, continuous map p : X → Y is called a quotient map if
V ∈ TY if and only if p−1 (V ) ∈ TX .
If X is a topological space and Y is a set, with p : X → Y surjective, then we can topologize Y with the
quotient topology by defining V open in Y if and only if p−1 (V ) is open in X.
If we suppose that X is a topological space with subspace Z, we’d like to construct the quotient space
X/Z. However there is a problem in saying “the” quotient space, as topologizing X/Z depends crucially on
the quotient map p : X → Z. I think I was initially confused because there is not a canonical “quotient” in
the category of topological spaces, as there is for say, the category of groups or modules.
Note that if Z ⊆ X, most(?) authors define X/Z to be the space obtained by identifying Z with a point
i.e. p(x) = x if x ∈ X − Z and p(z) = ∗ if z ∈ Z.
Example 1.4. Take X to be a rectangle and glue together the boundary in some particular way, for example
we may form a torus or a Klein bottle. Both spaces are quotient spaces formed from a rectangle, taking the
quotient by the boundary, but observe that the quotient map is the tool carrying all the information.
In some circumstances there should be no confusion by what is meant by /. For example if G is a
topological group with H a normal subgroup, then G/H denotes the usual quotient group, with the quotient
topology.
Frequently quotient spaces arise as equivalence relations, as any surjective map of sets p : X → Y defines
an equivalence relation on X by x1 ∼ x2 if and only if p(x1 ) = p(x2 ). In general, if R is an equivalence
relation on X, then the quotient space X/R is the set of all equivalence classes of X under R, with topology
defined by the quotient map p : X → X/R, which assigns to each x its equivalence class.
Example 1.5. Let
1.3
Homeomorphisms/ Homotopy
Definition 1.6. Two spaces X and Y are homotopy equivlent, or of the same homotopy type, if there
exist continuous maps f : X → Y and h : Y → X such that f ◦ h ∼ idY and h ◦ f ∼ idX . We may call f
and h “homotopy inverses”.
Proposition 1.7. Homotopy equivalence of spaces defines an equivalence relation.
Example 1.8. A deformation retract from a space X to a subspace A is a homotopy between a retraction
and the identity map on X. That is, there exists a continuous map F : X × [0, 1] → X such that for every
x ∈ X and every a ∈ A, F (x, 0) = id(x) = x, F (x, 1) ∈ A (the retraction map on x) with F (a, 1) = a.
If we add the condition that F (a, t) = a for all t ∈ [0, 1], so that the homotopy fixes A at all times t, we
may call it a strong deformation retract. Hatcher takes this as the definition of deformation retract.
n
n
For example,
there is a “strong” deformation retract from R − {0} → S . Use the homotopy F (x, t) =
(1 − t) +
t
kxk
x.
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Example 1.9. Every homeomorphism defines a homotopy equivilance, but the converse isn’t
true in general.
Let f : X → Y be a homeomorphism. By definition of homeomorphism, f −1 is a continuous map taking
Y to X, so take h = f −1 as in the definition of homotopy equivalence. Then, not only are f ◦ f −1 and
f −1 ◦ f homotopic to the respective identity maps, they are on the nose equal.
The converse is certainly not true. That is, if X and Y are homotopy equivalent spaces, they are not
necessarily homeomorphic. For example take the letter X and the letter Y. These spaces are homotopy
equivalent but not homeomorphic.
Certainly Y is a deformation retract of X so that X and Y are homotopy equivalent. To show this more
rigorously, take the map f : X → Y taking three of the prongs of X to the three of Y and the fourth to the
center of Y , and the map g : Y → X taking those three prongs back to X, define a homotopy equivalence
between X and Y . On the other hand, removing the center point from X gives a space with 4 connected
components, but removing any point from Y gives at most 3. Thus, the spaces are not homeomorphic.
Another example of homotopy equivalent spaces which are not homeomorphic is the mobius band and the
cylinder (some interesting answers on the math stack exchange question), or a point and the disk (another
deformation retract.)
1.4
Connectedness
A separation of a topological space X is a pair of disjoint, nonempty open subsets of X whose union is X.
X is said to be connected if no separation exists (which is sort of a lame definition.)
Proposition 1.10. Path connected implies connected, but the converse is not true. For example, the
topologist’s sine curve is defined T = {(x, sin( x1 ) : x ∈ (0, 1)} ∪ {(0, 0)}, and is connected but not pathconnected. Take the origin and some other point on the curve, since the sine part of the curve is not
continuous at the origin there is no path (continous image of the unit interval into the space) connecting the
points.
Example 1.11. For a manifold, path connected and connected are equivalent. A quick proof: Let X be a
manifold with x ∈ X. If X is path-connected, then it’s connected as the above proposition states. Suppose on
the other hand that X is connected and define the set U to be the set of all points in X that may be connected
by a path to X.
We will show that U is open. Take u ∈ U , by definition of manifold there is a neighborhood, Bu ⊆ X
which contains u and is homeomorphic to an open ball in Rn . Since path-connectedness is an invariant under
homeomorphism, Bu is path-connected. But x ∈ Bu , so the definition of U dictates that Bu ⊆ U . Therefore,
U is open.
.
By the same reasoning, V = X − U is open in X. Clearly X = U ∪ V , and since x ∈ U , V must be
empty, otherwise we would have a separation of the connected space X.
Definition 1.12. Define a relation ∼ on a topological space X by x ∼ y if there exists a connected subspace
C of X containing x and y. Prove that this is an equivalence relation. The equivalence classes are called
connected components of X.
We see that each connected component (equivalence class) is in fact connected, and is thus a maximal
connected subset of X. In general if A is a connected set, with A ⊂ B ⊂ Ā, then B must be connected.
So by maximality, every connected component C satisfies C = C̄. Therefore, connected components are
closed.
Proposition 1.13. A locally constant function on a connected component of a topological space X is constant.
Example 1.14. Let p : E → X be a vector bundle (with finite dimensional fibers)[3]. Local triviality of
vector bundles says that for any open U ⊆ X, p−1 (U ) is homeomorphic to U × V for some vector space V .
Then, the function on X which maps x → vdim(Ex ) is locally constant. It is therefore constant on the
connected components of X. We see that the fibers over a connected component are all vector spaces of the
same dimension, and hence isomorphic.
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Example 1.15. [2] pg. 3
Let G be a topological group with G0 the component of the identity e ∈ G. Then, G0 is a closed normal
subgroup of G. Normality buys us that G/G0 is a group, and further it is also a topological group.
As discussed above, any connected component is closed, so G0 is closed. Now we show that G0 is normal.
Recall that left and right multiplication are homeomorphism in a topological group.
Definition 1.16. A space X is called n-connected if πi (X) is trivial for 1 ≤ i ≤ n. A 1-connected space
(trivial fundamental group) is also called simply connected.
1.5
Operations on Spaces
e.g. suspension and loop
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Homology Theory
Topological spaces are often huge and complicated objects, one strategy to simplify their study is to convert
topological questions into algebraic questions via the (co)homology functors from the category of topological
spaces to an algebraic category.
Different homology theories apply to different classes of topological spaces. Simplicial, singular, and
cellular homology are three of the most common theories.
Simplicial theory deals with polyhedron (or spaces homeomorphic/homotopic to polyhedron). Simplicial
theory is combinatorial in nature, as we can count vertices, edges, faces, etc., and they are glued together
linearly so everything is nice. The combinatorial aspect of simplicial theory is exploited more generally in
abstract homotopy theory using simplicial sets. It is also exploited via computational topology software (e.g.
Javaplex.)
Singular theory is a generalization of simplicial theory which applies to all spaces, and was fundamental
to rigorously establish the theory. Cellular theory
2.1
Simplicial Homology
Basic definitions as in Munkres [1]. Given a set of vertices {a0 , a1 , . . . , an } ⊆ Rm , geometric independence
is the condition that a1 − a0 , . . . , an − a0 forms a linearly independent set of vectors. An n-simplexP
is built
m
from
a
geometrically
independent
set
of
vertices
by
taking
the
set
of
all
points
x
∈
R
where
x
=
i ti ai ,
P
ti = 1, and ti ≥ 0. Geometrically, a 0-simplex is a point, a 1-simplex a line, a 2-simplex a triangle, a
3-simplex a solid tetrahedron, and so-on for the higher dimensional analogoues.
A polyhedron is a space formed by joining together simplices in a way that satisfies a couple of axioms.
This is captured in the definition of simplicial complex. The algebraic topology
Definition 2.1. A simplicial complex K is a collection of simplices in Rm such that
1. Every face of a simplex of K is in K.
2. The intersection of any two simplexes of k is a face of each of them.
The underlying space of a simplicial complex K is denoted |K| and signifies the union of all simplices in
K. A space that is the underlying space of a simplicial complex is called a polyhedron.
Lemma 2.2. A collection K of simplies is a simplicial complex if and only if the following hold:
1. Every face of a simplex of K is in K.
2. Every pair of distinct simplices of K have disjoint interiors.
Specifying a map between two simplicial complexes K and L is fairly straight-forward. Let K and L
be simplicial complexes with f : K 0 → L0 a map on the 0-simplices. Suppose f has the property that
whenever v0 , . . . , vn are the vertices of a simplex in K, the points f (v0 ), . . . , f (vn ) are the vertices for a
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simplex
P in L. Then, we
P can extend f to a continuous map on the underlying spaces g : |K| → |L| defined:
x=
ti vi ⇒ g(x) = ti f (vi ). This map is linear on the vertex set and is called a simplicial map.
In the case that f is a bijection, and a set of vertices forms a simplex in K if and only if the image of
the vertices form a simplex in L, then it can be shown that g : |K| → |L| is a homeomorphism, and is called
a simplicial homeomorphism.
As Munkres aptly observes, “In practice, specifying a polyhedron X by giving a collection of simplices
whose union is X is not a very convenient way of dealing with specific polyhedra...it is messy to specify all
of the simplices and to make sure they intersect only when they should.” An abstract simplicial complex
simplifies this issue. It is defined as a collection S of finite nonempty sets, such that if A ∈ S, so is every
nonempty subset of A.
A convenient way of expressing an abstract simplicial complex
Theorem 2.3. (Classification theorem of closed surfaces)
Every connected closed surface is homeomorphic to a connected sum of real projective planes and g-tori.
(The sphere is considered the connected sum of zero tori and zero projective planes.)
Both the torus and the projective plane can be triangulated, and so any connected closed surface is
triangulable. As Hatcher puts it (pg. 102), “A polygon with any number of sides can be cut along the
diagonals into triangles, so in fact all closed surfaces can be constructed from triangles by identifying edges.”
Proposition 2.4. To what extent do the homology groups classify the closed, connected surfaces? General
formula for homology of connected sum of manifolds of higher dimension?
Which manifolds can be triangulated? May be hard to answer in general: http://mathoverflow.net/
questions/44021/which-manifolds-are-homeomorphic-to-simplicial-complexes.
2.2
Singular Homology
Proposition 2.5. ([4] pg. 109) Decompose a space X into its path-components Xα . There is an isomorphism
of Hn (X) with the direct sum ⊕α Hn (Xα ).
Proof. Let σ : ∆n → X be a singular n-simplex. The image of a path-connected space under a continuous
map is path-connected, thus σ(∆n ) ⊆ Xα for some α. Then, the singular chain group Cn (X) (which is
made of finite formal linear combinations of n-simplices) adds over the path-connected components of X i.e.
Cn (X) = ⊕α Cn (Xα ). The kernel and image of the boundary map necessarily preserve this decomposition,
and so the homology groups do too.
Lemma 2.6. Let f : X → Y be a continuous map. Then, f induces a chain map f# on the singular chain
groups. That is, for each n, f# : Cn (X) → Cn (Y ) is a homomorphism, and ∂f# = f# ∂
Moreover, f induces the map f∗ , with f∗ : Hn (X) → Hn (Y ), a homomorphism for all n.
Proof.
Lemma 2.7. Suppose that f and g are continuous maps taking X to Y with f ' g. Then, f∗ = g∗ as maps
on homology.
Proof. By definition there exists a continuous map F : X × [0, 1] → Y with the property that F (x, 0) = f (x)
and F (x, 1) = g(x).
Corollary 2.8. Homotopy equivalent spaces have isomorphic homology
Proof. This is a simple application of functoriality of the homology functor. Let f : X → Y and g : Y → X
be continuous maps such that f ◦ g ' IdY and g ◦ f ' IdX . Then as maps on homology,(f ◦ g)∗ ' (IdY )∗
and (g ◦ f )∗ ' (IdX )∗ . Since the homology functor takes identities to identities and splits over composition,
we have that f∗ : H∗ (X) → H∗ (Y ) and g∗ : H∗ (Y ) → H∗ (X) are inverses as group homomorphisms, and
therefore, H∗ (X) ∼
= H∗ (Y ).
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Example 2.9. The converse to the above corollary is not true. Topological spaces with the same
homology are not necessarily homotopic!
For example, take X = S 1 ∨ S 1 ∨ S 2 and Y = S 1 × S 1 . Since homology is additive over the wedge, we see
that these two spaces have the same homology groups. However, a computation of the fundamental groups
shows that they cannot be homeomorphic.
Poincare’s conjecture states that every every simply connected, closed 3-manifold is homeomorphic to the
3-sphere. The Poincare homology sphere is an example of a closed 3-manifold which has the same homology
as the 3-sphere but has a non-trivial fundamental group (and thus is not homotopic to the 3-sphere.)
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3.1
Cohomology Theory
Differentiable manifolds and the de Rham cohomology
It is typical that students of mathematics begin their education by passing through a sequence of calculus
classes, so from a pedagogical perspective, it may be best to motivate the study of algebraic topology by
thinking about topological spaces which are differentiable manifolds and the associated cohomology groups
called de Rham cohomology groups.
These objects generalize many of the concepts which students encounter in calculus. In particular the
fundamental theorems of multivariable calculus: Stokes’ theorem, Green’s theorem, the Divergence theorem,
and the fundamental theorem of line integrals all have a simple explanation in the language of differential
topology.
3.2
Interpreting cup/cap product in differential manifolds
Cup product should correspond to wedge product. Cap product is a pairing of a homology class and a
cohomology class, should have to do with an integral. Poincare duality.
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Homotopy Theory
While (co)homology theory uses functors to address topological problems in an algebraic category, homotopy
theory is more concerned with the intrinsic algebraic properties of the category T OP (topological spaces
with continuous maps.) For example, while the algebraic notions of kernel or cokernel do not apply to a
continuous map between topological spaces, there do exist the related ideas fibration and (co)fibration.
I’ve often read that the point of attaching algebraic invariants to topological spaces is to
Let X and Y be CW-complexes, with f : X → Y a continuous map. If f∗ : πn (X) → πn (Y )
Include Hurewicz and Whitehead theorems. Then define eilenberg maclane spaces
Proposition 4.1. A CW-complex X is K(G, 1) if and only π1 (X) ∼
= G, and the universal cover of X is
contractible.
4.1
Fibrations and Fiber Bundles
pg. 19 Benson
A Serre fibration is a map p : E → B that satisfies the homotopy lifting property with respect to all
simplicial complexes. Serre fibrations give rise to a long exact sequence of homotopy groups. On homology
and cohomology, a Serre fibration gives rise to a spectral sequence.
Note that all fibers in a path component of B are homotopy equivalent.
A covering space p : E → B is a fiber bundle such that the fibers are discrete sets. The fundamental
group π1 (B, b0 ) acts on F = p−1 (b0 ). If E and B are both path-connected
Thm: Every fiber bundle is a Serre Fibration.
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5
Common spaces and their topological properties
5.1
The Sphere
Topological Properties
1. Path connected for all n > 0
2. Simply Connected for n > 1
3. Orientable for all n.
Proof: Hn (S n ; Z) = Z for all n. (See computation below)
4. S n is compact.
It’s clearly closed and bounded, so apply the Heini-Borel theorem.
5. Smooth manifold for all n
Computations
Homology Groups
Cohomology
Homotopy Groups
Special Cases/Misc.
1. S n is a Lie Group for n = 0, 1, 3
2. Poincare Conjecture - Every simply-connected closed 3-manifold is homeomorphic to the 3-sphere.
3. Exotic Spheres, Milnor’s paper on the 7-sphere. Constructing differentiable manifolds which are homeomorphic but not diffeomorphic to the 7-sphere. http://www.scientificamerican.com/article/
hypersphere-exotica/
5.2
Projective Space
Topological Properties
1. Projective space can be realized as an orbit space.
The antipodal action of Z/2Z on S n has as orbit space RP n . Similarly, let S 1 act on S 2n+1 by
multiplication in each factor. The orbit space S 2n+1 /S 1 can be identified with CP n .
2. Projective space is compact.
Consider its realization as an orbit space. e.g. The quotient map p : S n → RP n /(Z/2Z) is a
continuous map on a compact space, and since the continuous image of compact is compact, we
see that RP n is compact.
Computations
Homology Groups
Cohomology
Homotopy Groups
Special Cases/Misc.
1. The Riemann Sphere CP 1 . Mobius transformations...
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5.3
Grassmann and Stiefel Manifolds
Let W be a vector space. As a set, the Stiefel variety V n (W ) is the collection of all ordered, orthonormal
n-frames, and the Grassmanian Gn (W ) is the collection of all n-dimensional subspaces of W .
Topological Properties
1. How to topologize the sets...
2. The Grassmanian Gn (Rm ) is a homogeneous space.
By basic linear algebra, any n-dimensional subspace of Rm can be transformed to any other ndimensional subspace. Thus O(n) acts on Gn (Rm ) transitively. If x ∈ Gn (Rm ), then since O(n)
is a compact Lie group, we have the relationship between orbits and isotropy: O(n)/O(n)x ∼
= O(n)x.
Now, transitivity buys us that O(n)x ∼
= Gn (Rm ). Similarly, if we replace R by C, and O(n) by U (n),
the same argument holds.
3. The Grassmanian is a compact manifold.
In the last item, we saw that the quotient O(n)/O(n)x is homeomorphic to Gn (Rm ). Then, the quotient
map q : O(n) → O(n)/O(n)x is a continuous, surjective map on a compact space. Thus, its image is
compact.
4. The Grassmanian Gn (V ) is a smooth manifold when V = R or C.
Computations
Homology Groups
Cohomology
Homotopy Groups
Special Cases/Misc.
1.
5.4
The Unitary Group
Topological Properties
1.
Computations
Homology Groups
Cohomology
Homotopy Groups
Group Cohomology
Special Cases/Misc.
1.
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5.5
The Special Unitary Group
Topological Properties
1.
Computations
Homology Groups
Cohomology
Homotopy Groups
Group Cohomology
Special Cases/Misc.
1.
5.6
The Orthogonal Group
Topological Properties
1.
Computations
Homology Groups
Cohomology
Homotopy Groups
Group Cohomology
Special Cases/Misc.
1.
6
Abstract Homotopy Theory
Eilenberg Maclane Spaces, Model Categories, K-Theory...
7
Things that split
References
[1] Munkres Intro to Alg Top
[2] Bredon Transformation Groups
[3] Atiyah K-Theory
[4] Hatcher
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