# Topology

## Definitions

### General

A basis for a topology is a subset of sets s.t. every element can be written as a union of elements of , and s.t.

• Intersections can be expressed using the basis:

A space is said to be first-countable if each point has a countable neighbourhood basis (local base).

That is, for each point there exists a sequence of neighborhoods of such that for any there exists an integer with . Since every neighborhood of any point contains an open neighborhood of that point, the neigbhourhood basis can be chosen wlog to consist of open neighborhoods.

We say a topological space is is second-countable if and only if there exists a countable basis for the topology of .

### Homotopy

Two continuous functions from one topological space to another are called homotopic if one can be continuously deformed into the other, with such a deformation called a homotopy between the two functions.

Formally, a homotopy between two continuous functions and from a topological space to a topological space is defined to be a continuous function such that,

### de Morgan's laws

Let be a set.

A family of subsets of is a set together with a subset for each .

De Morgan's laws state that

### Homeomorphism

A homeomorphism or topological isomorphism is a continuous function between topological spaces that has a continuous inverse function.

### Covering space

A map between connected manifolds is a covering map if every point of has a connected neighborhood such that the restriction of to each component of is a homeomorphism onto .

Let be a topological space.

A covering space of is a topological space together with a continuous surjective map

such that for every , there exists an open neighborhood of such that is a union of disjoint open sets in , each of which is mapped homeomorphically into by .

• Such is said to be evenly covered
• The disjoint open sets in that project homeomorphically into by are called sheets
• is called the covering map
• is called the base space of the covering
• is called the total space of the covering

For any point in the base, the inverse image of in is necessarily a discrete space called the fiber of .

Every manifold has a unique (up to homeomorphism) simply connected covering space , known as its universal cover.

Equivalently, it's a connected covering space with trivial 1st fundamental group, i.e. .

### Wedge-sum / one-point union

The wedge sum is a "one-point union" of a family of topological spaces.

Specifically, if and are point speces (i.e. topological space with distinguished basepoints and , resp.) the wedge sum of and is the quotient space of the disjoint union of and by the identification :

## Topological spaces

### Notation

• represents a topology, using the weird "O" since topologies consists of open sets

### Definitions

A topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

Or more rigorously, let be a set. A topology on is a collection of subsets of , called open subsets, satisfying:

• and are open (i.e. in )
• The union of any family of open subsets is open (i.e. in )
• The intersection of any finite family of open subsets is open (i.e. in )

A topological space is then a pair consisting of a set together with a topology on .

The definition of a topological space relies only upon set theory and is the most general notion of mathematical space that allows for the definition of concepts such as:

• continuity
• connectedness
• convergence

A topology is a way of constructing a set of subsets of such that theese subsets are open and satisfy the properties described above.

Let be a topological space. Then let .

Then

is a topology on , called the induced (subset) topology.

Let and be topological spaces.

Equip with the so-called product topology, implicitly defined by:

where .

A product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology.

The initial topology (or weak topology or limit topology or projective topology) on a set , wrt. a family of functions on , is the coarsest topology on that makes those functions continuous.

More formally,given a set and an indexed family of topological spaces with functions

the initial topology on is the coarsest topology on such that

is continuous.

Explicitly, the initial topology may be described as the topology generated by sets of the form

where is an open set . These sets are often called cylinder sets. If contains exactly one element, all the open sets of are cylinder sets.

Let be a set.

A subset is cofinite if is finite.

There is a topology on consisting of all the cofinite subsets together with called the cofinite topology:

Let be a set. The discrete topology is

i.e. has as a basis.

Let be a set. The indiscrete topology is

### Quotient topology

Let be a topological space, and let be an equivalence relation on .

The quotient space, , is defined to be the set of equivalence classes of elements on :

equipped with the topology where the open sets are defined to be those sets of equivalence classes whose unions are open sets in :

Equivalently, we can define them to be open sets with an open preimage under the surjective map , which sends a point in to the equivalence class containing it:

In this way, the quotient topology is the final topology on the quotient space wrt. the map .

Quotient topology is in fact a topology.

If is a function, then is continuous if and only if is continuous.

#### Examples

##### Natural topology on the circle

Consider . Since we're considering and as the same, this is equivalent to the circle .

Hence, the natural topology on is then just taking the quotient topology of the std. topology on !

##### Equivalence classes, surjective functions

Surjections and . Say if

if and only if there exists a bijection s.t.. .

: Suppose , then

: Suppose for , choose s.t. and set

Then we check:

• well-defined
• bijection
##### open in for open in

Consider then open in , but

Theorem thm:continuous-on-quotient-topology-iff-composition-with-can-is-continuous then tells us that any function which has integer periods, is also continuous on .

### Examples

#### Standard topology

we define the open ball as

i.e. the "open ball of radius around the point " as we recognize from Analysis.

We say that if and only if

and call the resulting topology the standard topology on .

We have the set

And construct the topology as follows:

1. we define:

i.e. the "open ball of radius around the point " as we recognize from Analysis.

2. We say that if and only if

3. Then we prove that this is in fact a topology:
• Is the empty set in ? Yes, since we clearly have since there are no points in the emptyset
• Is the set in ? Yes, is open, thus by definition we have for all
• Is the union of any collection of open subsets open? Suppose such that is closed.

### Topology of metric spaces

Let be a set, and let and be metrics on .

We say that and are Lipschitz equivalent if there existreal numbers s.t.

Lipschitz equivalence implies topological equivalence, but the reverse does NOT hold.

If a topological space is metrizable, then it is also Hausdorff.

All metrics on finite space are topological equivalent.

Metrisable implies Hausdorff, and the only Haussdorf topology is the discrete topology, hence all metrics on the finite space must include the Hausdorff topology, i.e. the discrete topology.

#### Regular and normal

A topological space is regular if for all closed sets and all with , there exist disjoint open sets such that and .

A topological space is normal if for all disjoint closed sets , there exist disjoint open sets such that and .

### Continuity

Let and be topological spaces.

Further, let be a map. Then is said to be continuous if and only if

where the preimage of the map over the set is defined as

Given any set and subset there is an inclusion function defined

and for , we have

Let be a topological space, and let be a subspace of .

Then the inclusion function is continuous.

Subspace topology is defined in such a way that all preimages of open sets are open, but nothing else is open.

In other words, the subspace topology is the smallest (coarsest) topology on such that is continuous.

#### Gluing lemma for continuous maps

Suppose one of the following conditions hold:

1. are finitely many closed subsets of
2. is a collection of open subsets of

such that

Suppose that for all we have a continuous map that agree on overlaps, i.e.

Then there exists a unique such that

#### TFAE for topological continuity

Let and be topological spaces. Let be a function.

The following statements are equivalent:

• is continuous
• open, is open
• closed, is closed
• Closures

### Convergence

Let be a topological space, and let be a sequence in , and .

Then converges to if for all open sets containing , there exists such that for all .

More concisely, if and only if

Let be a Hausdorff topological space.

Then each sequence in converges to at most one point.

#### Nets

Let be a directed set with order relation and be a topological space with topology .

A function is said to be a net.

If is a directed set, we often write a net from to in the form which expresses the fact that the element of is mapped to the element in .

Why do we need these? Well, for a metric spaces , the following statements are equivalent:

1. The map is continuous.
2. Given any point , and any sequence s.t. , then .

BUT! This is not true for general topological spaces!!! It is true that , but the difficulty encountered in attempting to prove (the untrue) statement is the fact that topological spaces are, in general, not first-countable!

Nets are introduced to try an have a similar equivalence-statement as a above (but now we need to replace "sequence" with "net" in (2)).

Let be a Haussdorff topological space and be a convergent net.

Then the limit is unique.

##### Convergence of series

Let be a Hausdorff topological space.

Given an index set , and a function from into , we define the sums sums as follows:

Let be a collection of all finite subsets of , directed by inclusion. Then

This is called unconditonal convergence, since it does not depend on the ordering of .

So basically, we're saying that if all finite series converge to if and only if the uncountable sequences converge equals .

In ,

means that for any , there is a finite set such that for any finite set with ,

In , if for all , then

Suppose .

Let and take a finite with

Then for any finite with , we have

Now suppose , then for any there exists a finite such that

and the same holds for any finite in place of , so

by definition.

### Closure and interior

The closure of is the intersection of all closed subsets of that contain .

One can prove the following equivalent statements:

1. (just rephrasing of 1.)
2. If closed, then

A limit point of is a point such that every neighborhood of contains some point of not equal to , i.e.

An interior point of is a point s.t. some neighborhood of is contained in , i.e.

## Connectedness

A space is connected if it is nonempty and cannot be written as a disjoint union with and open.

A space is disconnected if it is not empty or connected.

Given spaces and , define

with opens for and open.

The and is just notation to indicate they are separate.

TFAE

1. is connected
2. If for disjoint closed subsets then or is empty
3. The only clopen subsets of are and
4. Every cnts. map from to a discrete space is constant
5. Same as 4. but with instead of discrete space
• (This is useful)

: connected, and discrete with .

Let , then , both of which are clopen, and since , then so .

: If disconnected, then , with open disjoint and nonempty.

Let

Then cnts. but not constant.

with \$B \$ and connected, then so is .

If , then is constant, but and so on .

If and connected, then is connected.

Idea is just to take , which would imply that is disconnected; contradiction!

Any quotient of a connected space is connected.

The induced map is continuous, hence by Prop. proposition:connected-then-image-is-connected we have this.

Products of connected spaces are connected.

Then connectedness implies and are constant for all . Therefore,

is nonempty with subspace covering , with each connected and we have .

Then is connected.

### Path-connected

We say a topological space is path-connected iff for all there exists a path such that and .

With path we mean a continuous function .

#### Example: connected but not path-connected (Topologist's since curve)

Not closed in . To get the closure we need to add , i.e. .

• Not path-connected since no paths from to
• Connected since it's the closure of (which is itself connected)
• TODO: Why is connected?
• is closed in and is open in

## Topological invariants

A toplogical space is T1 if for any two distinct points ,

or equivalently,

A topological space is T2 if and only if for any two distinct points

That is, for any two distinct points we can find "neighborhoods" of and such that they are non-intersecting.

This is also called a (completely) Hausdorff topology.

An example of a T2 topology is the , hence it is also T1. T2 is a "stronger" notion of "neighborhood".

### Compactness & Paracompactness

A topological space is called paracompact if and only if every open cover has an open refinement that is locally finite.

Let be a topological space, and be an open cover on this space.

We then say that is an open refinement if and only if

i.e. for all points in the manifold there exists some open cover which contains , such that

only for finitely many , where is an open refinement. We then say is locally finite.

The definition of paracompact does actually make a bit of sense! We're saying it's paracompact if

• for every open cover on we can construct some refinement where only a finite number of the sets making up this "refined" open cover does actually intersect with the open cover covering some point

Every metrizable space is a paracompact.

Let be a Hausdorff space.

Then it is paracompact if and only if every open cover admits a partition of unity subordinate to that cover.

A set of continuous functions such that:

1. for every then exists a such that

2. for all there exists an open neighborhood which contains such that only finitely many are non-zero on and

#### Compactness

Let be a collection of subsets of a metric space and suppose that is a subset of .

is said to cover if and only if

is said to be an open covering of iff covers and each is open.

Let be a covering of .

is said to a finite (respectively, countable ) subcovering iff there is a finite (respectively, countable) subset of s.t. covers .

Let be a metric space.

A subset is compact iff for every open cover of , there is a finite subcover of .

I often find myself wondering "what's so cool about this compactness?! It shows up everywhere, but why?"

Well, mainly it's just a smallest-denominator of a lot of nice properties we can deduce about a metric space. Also, one could imagine compactness being important since the basis building blocks of Topology is in fact open sets, and so by saying that any open cover has a finite open subcover, we're saying it can be described using "finite topological constructs". But honestly, I'm still not sure about all of this :)

One very interesting theorem which relies on compactness is Stone-Weierstrass theorem, which allows us to show that for example polynomials are dense in the space of continous functions! Suuuuper-important when we want to create an approximating function.

Let be a metric space and let . Then is said to be dense in if for every and for every we have that i.e., every open ball in contains a point of .

Or, alternatively, as described in thm:dense-iff-closure-eq-superset:

A metric space is said to be separable iff it contains a countable dense subset.

Where with countable dense subset we simply mean a dense subset which is countable.

We say the metric space is a precompact metric space if for every there is a cover of by finitely many closed balls of the form

Let be a complete metric space and precompact, then is compact.

Let be a metric space. Then is said to be sequentially compact if and only if every sequence in has a convergent subsequence.

Let be a topological space, and a subspace.

Then is compact (as a topological space with subspace topology) if and only if every cover of by open subsets of has a finite subcover.

If for open in (subspace topology), then open in s.t. .

Therefore

: Choose finite subcover . Then is a finite subcover of .

: Let , then open in . So we let

so there exists finite with

Every space with cofinite topology is compact.

Let . Take some so that is finite.

Then , there exists in cover with . Therefore

is a finite cover.

Idea: take away one cover â†’ left with finitely many points â†’ we good.

##### Motivation

Let . For which must be bounded?

• finite
• If of opens with bounded then is bounded.
• Any continuous is locally bounded

with bounded, e.g.

• If there exists finitely many as above, with

then is bounded.

##### Compactness NOT equivalent to:
1. is a cover of
2. covers but not finite.
• cover â†’ clearly not finite mate
3. Same as 2, but take finitely many
4. Follows from 2 by taking subcover to be the whole cover.
5. always covers and has finite subcover (e.g. )
##### Examples
• Non-compact
1. , so has no finite subcover, so not compact
2. Infinite discrete space is not compact. Consider which is an open cover, but has no finite subcover.
• Compact
1. indiscrete so . Only open covers are and , and is a finite subcover, hence is compact.
2. Any finite space is compact (for any topology)

#### Compactness and Subspaces

Every closed subspace of a compact space is compact.

Suppose compact and closed.

Take open cover of , then there exists open in with

and we know that .

Therefore is an open cover of for which we have a finite subcover (since we can cover using a finite subcover, we can simply cover the union of the using the intersection, and from this obtain a finite subcover). We then add the complement of , which is open since is closed, providing us with a finite subcover of of .

Let be HausdÃ¶rff.

If be compact, then is closed.

Take (we want to show that is open).

there exists and open in s.t.

Then there exists a finite subcover of and open in .

And for all , therefore

#### Heine-Borel

A subspace of is compact if and only if closed and bounded.

• Product of compact is compact
• is compact by weak Heine-Borel
• Closed subspace of compact space are compact
• Compact subspaces of Hausdorff spaces are closed
• Compact metric spaces are bounded
##### Examples of metric spaces s.t. closed bounded subspaces not all compact
• with discrete metric, infinite\$ then infinite subspaces are not compact
• any non-compact metric space, and let

(which is topological equivalent to ). Then closed and is bounded, but clearly not compact.

#### Compactness of images and quotients

If is continuous and is compact, then is compact.

Eevery continuous from is bounded and attains its bounds.

Every quotient space of a compact space is a compact.

Every continuous map from compact to HaussdÃ¶rff is closed.

Every continuous bijection form a compact space to a Hausdorff space is a homeomorphism.

If is a continuous surjection form compact to Hausdorff, then

#### Compact metric spaces

A compact metric space is sequentially compact.

Let be a cover of a metric space .

A Lebesgue number for is a real number with the property that for all there exists such that .

Basically, it's the smallest radius such that all the contain balls of such radius centered at every point.

Let be a sequentially compact metric space.

Then every open cover of has a Lebesgue number.

1. Suppose s.t. .
2. Sequentially compact for some sub-sequence
3. Compactness allows us to choose some for which we know that
4. "Follow" this subsequence until we get close to , i.e.
5. Let , then

6. But AND , which is a contradiction

Let be a metric space. For , an on is a subset such that covers .

We say that is totally bounded if for all , there exists a finite on .

The following are equivalent for a metric space :

1. is compact
2. is sequentially compact
3. is complete and totally bounded

### Betti numbers

Let be a topological space.

The Betti numbers are

"number of dimensional holes".

Let be compact smooth manifold (without a boundary), then dimension of vector space of dimensional harmonic forms (deRham cohomology):

1. connected .

## Topological Manifolds & Bundles

### Notation

• is a topological manifold called the total space
• is a surjective and continuous map from to the base space , called the projection map (or bundle projection)
• denotes a fibre
• is the projection onto the first factor, e.g.

### Toplogical manifolds

A paracompact Hausdorff topogical space is called a d-dimensional topological manifold if

and exists a homeomorphism .

Equivalently, given an atlas of , denoted , we can say that if and only if open in for all .

Let and be manifolds.

Then is a topological manifold of dimension called the product manifold.

#### Examples

##### Product bundle, i.e. "functions"

for some topological manifolds and , and then in this very specific case we have a section

where is a map.

The use of as notation for a some "random" topological manifold might seem a bit confusing.

Notice that we don't start out by assuming that is a fibre of , but when we then let , then clearly is a fibre for any point . Thus, using the notation for some "arbitrary" manifold ends up being "convenient".

To drive the point home, is simply defined by the map

Hence, any product manifold is a fibre bundle.

### Bundles

A bundle (of topological manifolds) is a triple where

• is a topological manifold called the total space
• is a continuous surjective / onto map from the total space to the base space, we call this the projection
• is a topological manifold called the base space

We say a bundle is a smooth bundle if and only if the projection is smooth.

Let and be a bundle.

Then is called the fibre at .

A fibre is a generalization of the notion a product toplogy.

If be a bundle s.t.

for some manifold , then is called a fibre bundle with (typical) fibre .

We can say that a fibre bundle is locally a product space between the base space and the fibre . E.g. drawing a straight line through each point a manifold can be viewed as being locally a product space but not globally!

For every , there is an open neighborhood of such that there is a homeomorphism s.t.

where is the projection onto the first factor, and is a homeomorphism.

Or equivalently, the following diagram commutes

Thes set of all is called a local trivialization of the bundle.

For example we could construct a fibre bundle by taking the complex line and drawing it "straight" through a straight line of reals at each point on this real line.

Let be a bundle.

A map is called a section of the bundle if

In the particular case of a tangent-bundle of we have

We then observe that we obtain the following "hierarchy":

Let be a bundle.

Then is a sub-bundle if

• (sub-manifold)
• (sub-manifold)
• says that if we restrict the domain of the projection to , then we recover

Let be bundle.

Consider the submanifold , then

is called the restricted bundle.

We say the bundles

and maps

is called a bundle morphism if

Two bundles and are called isomorphic as bundles if there exists bundle morphisms and , where and .

Such are called bundle isomorphisms and they clearly are the relevant structure-preserving maps for fundles.

We say a bundle is called trivial if it is isomorphic to a product bundle .

Further, we say a bundle is locally trivial if it's locally isomorphic to some product bundle.

Consider the bundle , and also some function where is also a topological manifold.

Then we can construct the so-called pull-back bundle as

A topological group, G, is a topological space which is also a group such that the group operations of product:

and taking inverses:

are continuous (in a topological sense). Here is viewed as a topological space with the product topology.

Although not part of this definition, many authors require that the topology on be Hausdorff; it is equivalent to assuming that the singleton contianing the identity element 1 is a closed subset of .

A discrete subgroup of a topological group is a subgroup s.t. there exists an open cover of in which every open subsets contains exactly one element of .

In other words, the subspace / induced topology of in is the discrete topology.

Keep in mind that is a subgroup of itself, so this also defines a discrete group.

A principal G-bundle, where denotes any topological group, is a fibre bundle together with a continuous right action such that preserves the fibres of , i.e.:

and acts freely and transistively on them.

This implies that each fibre of the bundle is homeomorphic to the group itself.

Frequently, one requires the base space to be Hausdorff and possibly paracompact.

In the same way as with the Cartesian product, a principal bundle is equipped with

1. An action of on , analogous to for a product space
2. A projection onto . For a product space, this is just the projection onto the first factor:

Unlike a product space, principal bundles lack a preferred choice of identity cross-secion; they have no preferred analog of .

## Homotopy and the Fundamental Group

### Definitions

If and are topological spaces and are continuous maps, a homotopy from to is a continuous map satisfying

for all .

If there exists a homotopy from to , we say that and are homotopic, and write .

If the homotopy satisfies for all and all , the maps and are said to be homotopic relative to .

Both "homotopic" and "homotopic relative to " are equivalence relations on the set of all continuous maps from to .

Homotopy defines the notion of "continuously deforming one path into another path ".

Two paths are said to be path-homotopic, denoted symbolically by , if they are homotopic relative to .

Explicitly, this means that there is a continuous map satisfying

For any given points , path homotopy is an equivalence relation on the set of all paths from to . The equivalence class of a path is called its path class and is denoted .

Given two paths such that , their product is the path defined by

If and , it is not hard to show that .

Therefore it makes sense to define the product of the path classes and by .

Although multiplication of paths is not associative, it is associative up to path homotopy:

When considering products of three or more actual paths (as opposed ot the path classes), the convention is to evaluate from left to right.

If is a topological space and , a loop in based at is a path in from to , i.e. a continuous map such that

The set of path-classes of loops based at is denoted by .

Equipped with the product, it is a group, called the fundamental group of based at .

The idenity element of this group is the path class of the constant path , and the inverse of is the path class of the reverse path .

It can be shown that for path-connected spaces, the fundamental groups based at different points are isomorphic.

One can roughly think about the fundamental group as the "number of loops" the path has; and the curves are then mapped to the equivalence class corresponding to the number of loops.

If is path-connected and for some (hence every) , the fundamental group is the trivial group consisting of alone (i.e. the equivalence class of the constant path), we say that is simply connected.

If and are topological spaces, and is a continuous map, then is a group homomorphism, known as the homomorphism induced by .

1. Let and be continuous maps. Then for each ,

2. For each space and each , the homomorphism induced by the identity map is the identity map of
3. If is a homomorphism, then is an isomorphism. Thus, homeomorphic spaces have isomorphic fundamental groups.

A continuous map between topological spaces is said to be a homotopy equivalence if there is a continuous map such that

Such a map is called a homotopy inverse for .

If there exists a homotopy equivalence between and , the two spaces are said to be homotopy equivalent.

If is a homotopy euqivalence, then for each , is an isomorphism.

Where

### Examples

• If and , then a homotopy is a path from to since and for some .
• Let

then this is homotopy between the paths s.t. and .

For any , with . Then are homotopic.

Let

then is continuous in both of the it's components (by cont. of and and the projection-maps) and thus, by componentswise cont. implying cont., we have that is cont.

Hence it's a homotopy, and we write

## Algebraic topology

### Notation

• i.e. closed unit ball
• map understood as continuous map, unless otherwise stated
• means homeomorphism
• homotopy equivalence
• denotes the concatenation of two paths

### Stuff

where defined by

For maps with convex, then and are homotopic.

Let , then we're good!

#### Contractible

is contractible if for some .

A homotopy equivalence induces an isomorphism

A rectraction of onto is a map

Observe then that and . In this way retractions are topological analogs of projection operators in other parts of mathematics.

A deformation retraction is a retraction with a homotopy for inclusion .

Not all retractions are deformation retractions. For example, a space always retracts onto any point via the constant map for all , but a space that defromation retracts onto a point must be path-connected since a deformation retraction of to gives a path joining each to .

is star-shaped at if contains all straight lines from to points

E.g. convex points are star-shaped.

Let , for all , is . Then we simply use the straight line homotopy

which we can since by definition of star-shaped, it contains all straight lines starting at .

The cone on a space is the quotient .

E.g. the cone on , denoted , where is a cylinder, so the cone comes from turning the bottom circle of the cylinder into a point. The resulting space is something which is topologically equivalent to a cone.

1. is a deformation retraction
2. , i.e. cone of without it's cone-point , is a deformation retraction with map .
1. Let

which is the straight line to

2. and is contractible using

##### Example

unit interval is contractible.

Let , which is homotopy from const. map to identity map since

1. Prove any map is homotopic to a constant map.
2. Prove two constant maps are homotopic if and only if path from to
3. When are two maps homotopic?
4. Construct smallest examples of space and and s.t. NOT homotopic to .
1. Let then we're good.
2. Straight forward use of definition.
3. and so , therefore if and only if and are in same path-component of

### Cell complexes

A space is called a cell complex or CW complex if constructed in the following way:

1. Start with a discrete set , whose points are regarded as 0-cells.
2. Inductively, construct the n-skeleton from by attaching n-cells via maps . This menas that is the quotient space of the disjoint union of with a collection of n-disks under idenitifications for . Thus set

where each is an open n-disk.

3. Then, either
1. stop process for finite , and let
2. continue indefinitely, setting , in which case is equipped with the weak / initial topology.

### Homotopy theory

#### Notation

• is the n-dimensional unit cube
• of denotes the boundary of , consisting of points with at least one coordinate equal to 0 or 1
• is the set of homotopy classes of maps where homotopies are required to satisfy

• For , a sum or concatenate operation in defined by

• Sometimes a homotopy is denoted rather than
• is often used to denote a covering space of

#### Fundamental Group of the Circle

is an infinite cyclic group generated by the homotopy class of the loop

based at .

• Note

where

• Thus thm:fundamental-group-of-the-circle is equivalent to saying "every loop in based at is homotopic to for a unique "
• I.e. goes around a circle some times
• Procedure of the proof will compare paths in with paths in via the map

• Can visualize by embedding to as the helix
• Then is the restriction of to the helix of the projection of to ,
• Observe that

where

thus and , winding around the helix times (upward if , downward if )

• is said to be a lift of

I find to be defined as a lift of to be confusing.

Intuition says that we are "lifting" from to by using to perform the "lift". Not "lifting" from to .

But okay. Maybe there is some obvious connection I'm missing…

To prove thm:fundamental-group-of-the-circle we need the following two facts about covering spaces .

1. For each path , with and each there is a unique lift

2. For each homotopy of starting at (i.e. the paths satisfy ) and each there is a unique lifted homotopy

of paths starting at (i.e. paths satisfying )

Let be a loop at the basepoint , representing a given element of .

By (1) there exists a lift with . Then

Another path in from to is , and

via the linear homotopy . Composing with , i.e. we get a homotopy for and , thus

Hence every loop in based at is homotopic to . Now we just need to show that is uniquely determined by .

Suppose and , thus by transitivity of , we have

We now want to show that this implies . Let be the homotopy from to . By (2), this homotopy lifts to a homotopy of paths starting at (i.e. homotopy of paths s.t. ). The uniqueness part of (1) implies that

Since is a homotopy of paths, the endpoint is independent of . For , this endpoint is and for it is , hence

as wanted.

Given maps and (which is "lifting" ), then there is a unique map which lifts and restricting to the given on .

Here denotes a covering space of .

The two properties used in proof:fundamental-group-of-the-circle are special cases of thm:existence-of-unique-lift-covering-spaces.

1. Let be a singleton set in the theorem. Then is equivalent to , which is in the statement. Then there is a unique map , which is just which "lifts" to the given on , which in this case is , as used in the proof.
2. Let . Then the homotopy in (2) gives a map by setting

A unique lift is obtained by an application of (1). Then, by thm:existence-of-unique-lift-covering-spaces, this gives a unique lift . The restrictions and are paths which "lift" the constant paths, hence they must also be constant by the uniqueness part of (1)\$. Thus, is a homotopy of paths, and lifts :

Let , where open and for basepoint for all . If each intersection is path-connected, then every loop in at is homotopic to a product of loops each of which is contained in a single .

is isomorphic to if and are path-connected.

if

Let and , where , i.e. only the n-th component is non-zero. Then each are homeomorphic to .

Choose a basepoint . If , then is path-connected. Lemma 1.15 says that every loop in based at is homotopic to a product of loops in or . Both and are zero since and are homeomorphic to .

Hence every loop in is nullhomotopic.

is NOT homeomorphic to for .

Suppose is a homeomorphism.

The case is disposed since is path-connected but the homeomorphic space is NOT path-connected.

When we cannot distinguish from by the number of path-components, but we can distinguish them by their fundamental groups. Namely, for a point , the complement is homeomorphic to , so by Prop. 1.12

(where the last is due to trivial fundamental group of ).

Hence is for and trivial for , using Prop 1.14 in the latter case.

If is a homotopy equivalence, then the induced homomorphism is an isomorphism for all .

#### Homotopy groups

is the set of homotopy classes of maps where homotopies are required to satisfy

This extends to the case of by taking to be a point and , so is just the set of path-components of .

For , a sum operation in defined by

generalizes the composition operation in !

This defines a group.

##### Intuition

The homotopy begins by shrinking the domains of and to smaller subcubes of , with the region outside these subcubes mapping to the basepoint .

Afterwards, there is room to slide the two subcubes around anywhere in \$In4 as long as they stay disjoint, so if they can be slide past each other, interchanging their positions.

Then, to finish the homotopy, the domains of and are enlarged back to their original size.

If one likes, the whole process can be done just using the coordinates and , keeping the other coordinates fixed.

##### path-connected choice of always produce isomorphic groups

If is path-connected, then any choice of basepoint always produce isomorphic groups , thus we can use the notation . That is,

Given path s.t.

we may associate to each map a new map

by shrinking the domain of to be a smaller concentric cube in , then inserting the path on each radial segment in the shell between this smaller cube and .

A homotopy of or through maps fixing or , respectively, yields a homotopy of through maps . Here are three basic properties:

1. where denotes the constant path (

The homotopies in (2) and (3) are:

1. is homotopy since both sides result in splitting into 3 paths which are successivly tranversed (both sides in same order)
2. is homotopy

For (1), the homotopy is given by

Thus we have

This can intuitively seen as first deforming to be constant on the right half of , and to be constant on the left half of , which produces maps we may call and . Then we excise a progressively wider symmetric slab of until it becomes .

Now, if we define a change-of-basepoint transformation

Then (1) shows that \$βγ is a homomorphism, while (2) and (3) imply that is an isomorphism with inverse where is reversed (often called inverse-path of ), i.e.

Thus if is path-connected, different choices of basepoint yield isomorphic groups \$πn(X, x0), which may then be written simply as .

##### Action of on

Now let us restrict attention to loops at a basepoint . Since

where, to remind ourselves, for some path is defined

The association defines a homomorphism from . We call this the action of on . Each element of acting as an automorphism of .

• then acts on itself by inner automorphisms
• then the action of makes the abelian group into a module over the group ring :

i.e. finite sums of scaled paths, with the multiplication operation defined by distributivity and the multiplication of .

• Module structure on is give by

For brevity one sometimes says is a rather than .

We say a space is abelian if it has trivial action of on all homotopy groups , since when this is the condition that be abelian.

Observe that is a functor, that is, a map induces

From the definition of we see that it's well-defined and a homomorphism for . Furthermore, the functor properties

1. , i.e. the identity map induces the identity map

are also satisfied. Finally, if

is a homotopy, then

A homotopy equivalence (in the basepointed sense) induces isomorphisms on all homotopy groups .

induces isomorphisms

for all .