Post 2
Help out one of your classmates by showing what will happen if a lower
weekly salary, but a higher commission rate is used. Choose one of your
classmates posts and adjust their figures as follo
Post 3
Choose one of your classmates posts and calculate a rough estimate of the total cost of
the shed. Make sure you choose a post that does not already have an average
calculated. One way to find t
Module 5 : Seifert Van kampen Theorem & its application
Lecture 22 : Fundamental groups of certain orthogonal groups
Exercises:
1. Show that the sphere
is isomorphic (as a topological group) to
.
2. S
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 l
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
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 fin
Module 7 : Relative homology,exicism and the Jordan Brouwer separation theorem
Lecture 40 : Inductive limits
Exercises:
1. Prove lemma (40.1)
2. Show that the Prfer group (40.3) is the inductive limit
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 it
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 t
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 t
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 lectur
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
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 comp
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
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
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
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
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
Module 5 : Seifert Van kampen Theorem & its application
Lecture 25 : Adjunction Spaces
Exercises:
1. We have obtained
by attaching
constant map on the boundary of
an adjunction space.
2. Show that if
Module 4 : Theory of Covering Spaces
Lecture 20 : Orbit Spaces
Exercises:
is a finite group acting freely on a Hausdorff space then the action is
1. Suppose that
properly discontinuous and hence deduc
Module 4 : Theory of covering spaces
Lecture 19 : Deck transformations
Exercises:
1. Suppose that
and
are topological groups and
projection that is also a group homomorphism then ker
is a covering
De
Module 4 : Theory of covering spaces
Lecture 17 : Action of the fundamental group on the fibers
Exercises:
1. Describe a path in
.
2. Let
whose image under the standard map represents the generator of
Module 4 : Theory of Covering Spaces
Lecture 18 : The lifting criterion
Exercises:
1. For the map
in example (18.3) show that
is the map
given by
.
2. Suppose
is a path connected topological group wit
Module 6 : Basic Homology Theory
Lecture 36 : Maps of Spheres
Exercises:
1. Show that if
and
are two reflections (each with respect to a coordinate plane)
then they are conjugate by a homeomorphism. D
Module 4 : Theory Covering Spaces
Lecture 15 : Covering Projections
Exercises:
given by
1. Explain why the map
is not a covering projection?
2. Show that the map
given by
is a covering projection for
Module 3 : Fundamental groups & its basic properties
Lecture 10 : Brouwer's theorem and its applications
Exercises:
1. Suppose that a space
has the fixed point property, is it necessary that it be con
Module 3 : Fundamnetal groups and it's basic properties
Lecture 12 & 13 :The fundamental group of the circle.
Exercises:
1. Formulate and prove the Borsuk Ulam theorem for continuous maps from
line.
t
Module 3 : Fundamental groups & its basic properties
Lecture 11 : Homotopies of maps. Deformation retracts
Exercises:
1. Check that the map
constructed in the proof of theorem 11.3 is continuous and i
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 s
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