Download SAM III General Topology

SAM III
General
Topology
Lecture 5
Contents
Homotopy
Homotopy of
paths
Fundamental
groupoid
Convex subsets
of Rn
Product in a
category
SAM III
General Topology
Lecture 5
The definition
Preservation of
products by the
fundamental
group
SAM — Seminar in Abstract Mathematics [Version 20130306]
is created by Zurab Janelidze at Mathematics Division, Stellenbosch Univeristy
SAM III
General
Topology
Lecture 5
Contents
Homotopy
Homotopy of
paths
Fundamental
groupoid
Convex subsets
of Rn
Product in a
category
The definition
Preservation of
products by the
fundamental
group
1 Homotopy
Homotopy of paths
Fundamental groupoid
Convex subsets of Rn
2 Product in a category
The definition
Preservation of products by the fundamental group
Homotopy of paths
SAM III
General
Topology
Lecture 5
Contents
Definition
Consider two points x and y in a topological space (X , τ ). Let f : [0, 1] → X and
g : [0, 1] → X be two paths from x to y . A homotopy from f to g is a continuous
map h : [0, 1]2 → X such that h(x, 0) = f (x) and h(x, 1) = g (x) for all x ∈ [0, 1].
We say that f and g are homotopic, and when there is a homotopy from f to g .
Homotopy
Homotopy of
paths
Fundamental
groupoid
Convex subsets
of Rn
Product in a
category
The definition
Preservation of
products by the
fundamental
group
Homotopy classes
Show that the relation “f is homotopic to g ”, written as f ' g , is an equivalence
relation. The equivalence class of a path f under this equivalence relation is
called the homotopy class of f .
Composition of paths
If f is a path from x to y and g is a path from y to z, then we can define a path
g · f from x to y as follows:
(g · f )(x) =
f (x),
g (x),
0 6 x 6 21 ,
1
6 x 6 1.
2
Show that if f ' f 0 and g ' g 0 then g · f ' g 0 · f 0 .
Fundamental groupoid
SAM III
General
Topology
Definition
A groupoid is a category in which every morphism is an isomorphism.
Lecture 5
Contents
Homotopy
Homotopy of
paths
Fundamental
groupoid
Convex subsets
of Rn
Product in a
category
The definition
Preservation of
products by the
fundamental
group
Fundamental groupoid
To a topological space X = (X , τ ) one associates a category Π1 X defined as
follows: objects of the category are points in the topological space, while a
morphism from x to y is a homotopy class [f ] of a path f from x to y ;
composition of morphisms is defined by the formula [g ] ◦ [f ] = [g ◦ f ]; for an
object x, the identity morphism is given by the homotopy class [x] where x stands
for the path [0, 1] → X , r 7→ x. Verify that Π1 X is indeed a category, and that
moreover, it is a groupoid. This groupoid is called the fundamental groupoid of
the space.
Fundamental group
The fundamental group of a topological space X = (X , τ ), at a point x ∈ X ,
written as π1 (X , x), is the group of automorphisms of x in the fundamental
groupoid Π1 X of the space. Show that in a path-connected space, fundamental
groups at any two distinct points are isomorphic to each other. Deduce this from
a general observation about an abstract groupoid.
Convex sets in Rn
SAM III
General
Topology
Lecture 5
Contents
Homotopy
Homotopy of
paths
Fundamental
groupoid
Convex subsets
of Rn
Product in a
category
The definition
Preservation of
products by the
fundamental
group
Definition
A subset S of Rn is said to be convex when S is closed under
linear combinations c1 v1 + c2 v2 where c1 + c2 = 1 and
c1 c2 > 0.
Fundmental groupoid of a convex subset of Rn
A groupoid is codiscrete when for any two objects x and y
there is exactly one morphism x → y . Show that if a subspace
of Rn is a convex set then its fundamental groupoid is
codiscrete. In general, show that the fundamental groupoid of
a topological space is codiscrete if and only if the space is
path-connected and the fundamental group at any point is a
trivial group.
The definition
SAM III
General
Topology
Definition
Let X and Y be two objects in a category. Their product is defined as a diagram
Lecture 5
Contents
X
o
π1
π2
X ×Y
/Y
Homotopy
Homotopy of
paths
Fundamental
groupoid
Convex subsets
of Rn
Product in a
category
The definition
Preservation of
products by the
fundamental
group
such that for any other such diagram
X
o
f
C
g
/Y
there is a unique morphism C → X × Y , denoted as (f , g ), such that
π1 (f , g ) = f and π2 (f , g ) = g :
X
π1
π2
/< Y
X ×O Y
bFo F
FF
xx
x
FF (f ,g ) xx
f FFF
xx g
xx
C
Preservation of products by the fundamental group
SAM III
General
Topology
Lecture 5
Contents
Homotopy
Homotopy of
paths
Fundamental
groupoid
Convex subsets
of Rn
Product in a
category
The definition
Preservation of
products by the
fundamental
group
Products in Grpd
Define the category Grpd of groupoids and construct products in this
category.
Products in Top
Show that in the category Top of topological spaces, product of two
spaces with the canonical projection maps is the same as their
categorical product.
Functoriality of the fundamental group and preservation of products
Define a structure-preserving map between categories, and call it a
functor. Show that the process of assigning to a topological space X
its fundamental groupoid ΠX defines a functor Top → Grpd.
Formulate what does it mean for a functor to preserve products and
show that the fundamental groupoid functor is such.