Module 3 : Fundamental groups & its basic properties
Lecture 9 : Functorial properties of the fundamental group
Exercises:
1. Show that the sphere
from
retracts onto one of its longitudes. If
is the space obtained
by taking its union with a diameter, ther
Module 7 : Relative homology,exicism and the Jordan Brouwer separation theorem
Lecture 39 : Excisim Theorem
Theorem 38.2 (Excision):
Let
be a pair and
be a subset of
interior of
such that the closure of
. Then, the homomorphism
induced by inclusion
is con
Lectures - XXIX/XXX The singular chain complex and homology groups
The program of developing a calculus of chains is now formalized in this lecture. We introduce
a new algebraic category of chain complexes and maps between them and prove the fundamental
t
Lecture - XXXII The abelianization of the fundamental group
In this lecture we shall establish a basic result relating the fundamental group 1 (X, x0 ) and the rst
homology group H1 (X). The result is elegant and states that H1 (X) is the abelianization o
Lecture - XXVII (Test IV)
1. Use the Seifert Van Kampen theorem to compute the fundamental group of the double torus.
2. Let K be a compact subset of R3 and regard S 3 as the one point compactication of R3 . Show
that 1 (R3 K) = 1 (S 3 K).
3. If C is the
Lecture - XXXIV Small simplicies
Recall that the Goursat lemma in complex analysis is proved by subdividing a triangle into four
smaller triangles determined by the midpoints of the sides of the given triangle. The integral over
the given triangle is then
Lecture - XXXI The homology groups and their functoriality
Having laid the algebraic foundations in the previous lecture we shall formally dene the homology
functors Hn , n = 0, 1, 2, . . . from the category Top to the category AbGr. We shall discuss H0 (
Module 7 : Relative homology,exicism and the Jordan Brouwer separation theorem
Lecture 38 : Relative homology
The homology groups
we have hitherto been studying are called the absolute homology
that we define here provide us a tool for
groups. The relativ
Lecture - XXXIII Homotopy invariance of homology
Homotopy of maps is one of the most important notions in topology and it is of interest to know
what is its eect on the induced maps in homology. The result is simple and direct namely, if f :
X Y and g : X
Lecture XXII - Fundamental group of SO(3, R) and SO(4, R)
For many applications, it is important to know The fundamental groups of the classical groups.
We shall discuss in detail the orthogonal groups SO(3, R) and SO(4, R) since their underlying topologi
Lecture - XXXVI Maps of spheres
We are now in a position to prove the general Brouwers xed point theorem as well as a few other
surprising results concerning maps of spheres. As demonstrated in lecture 10, these higher dimensional
analogues were inaccessi
Lecture - XXXV The Mayer Vietoris sequence and its applications
The proof of Mayer Vietoris sequence is reminiscent of the Seifert Van Kampen theorem. While
the Seifert Van Kampen theorem enables us to relate the fundamental group of a union U V in terms
Lecture - XL Inductive limits
We have frequently encountered situations where a certain space X is canonically embedded in a
larger space Y . A familiar example the sequence of orthogonal groups and the canonical inclusions
SO(2, R) SO(3, R) SO(4, R) . .
Module 4 : Theory of covering space
Lecture 16 : Lifting of paths and homotopies
Exercises:
1. Use the general results of this section to give an efficient and transparent proof that
. First show that for any loop based at , the map
given by
is well defin
Module 3 : Fundamnetal groups and it's basic properties
Lecture 8 : Categories and Functors
Exercises:
1. Recast the notion of homotopy of paths in terms of morphisms of the category
as follows
2. Define a binary operation on
Show that this defines a grou
Module 2 : General Topology
Lecture 5 : Topological groups
Exercises:
1. Show that in a topological group, the connected component of the identity is a normal
subgroup.
2. Show that the action of the group
on the sphere
given by matrix
multiplication is t
Module 2 : General Topology
Lecture 4 : Further preliminaries from general topology
Exercises
1. What happens if we omit the surjectivity hypothesis on the function
the definition of quotient topology on
induced by
?
by collapsing its
2. Show that the spa
Module 3 : Fundamnetal groups and it's basic properties
Lecture 7 : Paths, homotopies and the fundamental group
Exercises:
1. Explicitly construct a homotopy between the loop
sphere
and the constant loop based at
on the
. Note that an explicit formula is
Module 2 : General Topology
Lecture 3 : More Preliminaries from general topology
Exercises:
1. Prove that any continuous function
has a fixed point, that is
such that
to say, there exists a point
.
is connected. Is it true that if
2. Prove that the unit i
Module 2 : General Topology
Lecture 2 : Preliminaries from general topology
Exercises:
1. Prove that a topological space is compact if and only if it satisfies the following condition
known as the finite intersection property. For every family
of closed s
Module 1 : General Topology
Lecture 6 : (Test - I)
1. Prove that the intervals
that if
and
is open in
and
are non-homeomorphic subsets of
are homeomorphic subsets of , then
. Is an injective continuous map
. Prove
is open in
if and only if
a homeomorphism
Module 6 : Basic homology theory
Lecture 37 : Test V
1. Calculate the homology groups of the double torus.
2. Show that any homeomorphism of
3. Show that
4. Regard
given by
is not a retract of
onto itself must preserve the boundary.
. Use the lifting crit
Lecture - XXV Adjunction Spaces
The notion of push-outs in the category Top leads to an important class of spaces known as
adjunction spaces. We shall see that most of the important spaces encountered are adjunction spaces.
This lecture may be regarded as
Lecture - XXVIII Introductory remarks on homology theory
In the rst part of the course we focused on the fundamental group and its basic properties. We
discussed an elegant solution of the lifting problem for covering projections in terms of the fundament
Lectures - XXIII and XXIV Coproducts and Pushouts
We now discuss further categorical constructions that are essential for the formulation of the Seifert
Van Kampen theorem. We rst discuss the notion of coproducts which is a prerequisite for a proof
of the
Lecture XIX - Deck Transformations
Given a covering projection p : X X, the deck transformations are, roughly speaking, the
symmetries of the covering space. Thus it should not come as a surprise that they play a crucial part
in the theory of covering spa
Lecture XI - Homotopies of maps. Deformation retracts.
We generalize the notion of homotopy of paths to homotopy of a pair of continuous maps between
topological spaces. This would be particularly useful in the second part of the course. It also leads to
Lecture VIII - Categories and Functors
Note that one often works with several types of mathematical objects such as groups, abelian
groups, vector spaces and topological spaces. Thus one talks of the family of all groups or the family
of all topological s
Lecture II - Preliminaries from general topology:
We discuss in this lecture a few notions of general topology that are covered in earlier courses
but are of frequent use in algebraic topology. We shall prove the existence of Lebesgue number for a
coverin
Lecture V - Topological Groups
A topological group is a topological space which is also a group such that the group operations
(multiplication and inversion) are continuous. They arise naturally as continuous groups of symmetries
of topological spaces. A