TENSOR PRODUCTS II
KEITH CONRAD
1. Introduction
Continuing our study of tensor products, we will see how to combine two linear maps
M M and N N into a linear map M R N M R N . This leads to at modules
and linear maps between base extensions. Then we will
Problem Set #1
Math 453 Dierentiable Manifolds
Assignment: Chapter 1 #2, 5, 6, 7, 8
Chapter 2 #1, 3, 5
Clayton J. Lungstrum
January 22, 2013
Exercise 1.2
Let
f (x) =
e1/x
0
: x>0
.
: x0
(a) Show by induction that for x > 0 and k 0, the k th derivative f (
Problem Set #4
Math 453 Dierentiable Manifolds
Assignment: Chapter 5 #1
Chapter 6 #1, 2, 4
Clayton J. Lungstrum
February 13, 2013
Exercise 5.1
Let A and B be two points not on the real line R. Consider the set S = (Rcfw_0)cfw_A, B.
For any two positive re
Problem Set #5
Math 453 Dierentiable Manifolds
Assignment: Chapter 7 #1, 5, 7, 8, 9, 11
Clayton J. Lungstrum
February 20, 2013
Exercise 7.1
Let f : X Y be a map of sets, and let B Y . Prove that f (f 1 (B) = B f (X).
Therefore, if f is surjective, then f
Problem Set #2
Math 453 Dierentiable Manifolds
Assignment: Chapter 3 #1, 2, 4, 5, 7, 10, 11
Clayton J. Lungstrum
January 30, 2013
Exercise 3.1
Let e1 , . . . , en be a basis for a vector space V and let 1 , . . . , n be its dual basis in V .
Suppose [gij
Problem Set #7
Math 453 Dierentiable Manifolds
Assignment: Chapter 9 #1, 2, 3, 5
Clayton J. Lungstrum
March 6, 2013
Exercise 9.1
Dene f : R2 R by
f (x, y) = x3 6xy + y 2 .
Find all values c R for which the level set f 1 (c) is a regular submanifold of R2
Problem Set #3
Math 453 Dierentiable Manifolds
Assignment: Chapter 4 #1, 2, 3, 4, 5, 6
Chapter 5 #3
Clayton J. Lungstrum
February 1, 2013
Exercise 4.1
Let be the 1-form zdx dz and let X be the vector y/x + x/y on R3 . Compute
(X) and d.
Solution.
By denit
FUNDAMENTAL GROUPS AND THE VAN KAMPENS
THEOREM
SAMUEL BLOOM
Abstract. In this paper, we dene the fundamental group of a topological
space and explore its structure, and we proceed to prove Van-Kampens Theorem, a powerful result useful for calculating the
1
SMSTC Geometry and Topology 20112012
Lecture 5
The Seifert van Kampen Theorem
Andrew Ranicki (Edinburgh)
10th November, 2011
2
Introduction
Topology and groups are closely related via the fundamental
group construction
1 : cfw_spaces cfw_groups ; X 1 (X
MATH 215B HOMEWORK 4 SOLUTIONS
1. (8 marks) Compute the homology groups of the space X obtained from n by
identifying all faces of the same dimension in the following way: [v0 , . . . , vj , . . . , vn ]
is identied with [v0 , . . . , vk , . . . , vn ] by
FUNDAMENTAL GROUPS AND COVERING SPACES
ETHAN JERZAK
Abstract. In this paper, I will briey develop the theory of fundamental
groups and covering spaces of topological spaces. Then I will point toward
the manner in which covering spaces can be used prove so
Algebraic Topology, Fall 2013
Selected Solutions for HW 6.
Hatcher 1.3.24.
Suppose we have a covering space action of a group G on a path-connected, locally pathconnected space X. Show:
a) Every path-connected covering space between X and X/G is isomorphi
CUP PRODUCTS IN COMPUTATIONAL TOPOLOGY
JONATHAN HUANG
Abstract. Topological persistence methods provide a robust framework for
analyzing large point cloud datasets topologically, and have been applied with
great success towards homology computations on si
CHAPTER 6
Singular Homology
1. Homology, Introduction
In the beginning, we suggested the idea of attaching algebraic objects to topological spaces in order to discern their properties. In language introduced later, we want functors from the category of to
Cup Products
March 7, 2012
Example
Consider the spaces X = CP 2 and Y = S 2 S 4 . The natural question
whether X and Y are homotopy equivalent arises. We know that as CW
Complexes, both X and Y have one 0-cell, one 2-cell and one 4-cell. Hence
the cellula
HW 10 SOLUTIONS, MA525
Hatcher 2.2 Problem 29
The Mayer-Vietoris sequence gives
Hn (Mg ) Hn (R) Hn (R) Hn (X) Hn1 (Mg )
The compact space R deformation retracts on a wedge of g circles. So H0 (R) = Z, H1 (R) =
g Z, Hn (R) = 0 for n > 1. Also we know H0
72
CHAPTER 5
Manifolds and Surfaces
1. Manifolds and Surfaces
Recall that an n-manifold is a Hausdor space in which every point
has a neighborhood homeomorphic to a open ball in Rn .
Here are some examples of manifolds. Rn is certainly an n-manifold.
Also