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

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

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

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

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

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

Solution to Math256a section IV.3 (H.Zhu, 1994)
3.1) One direction follows easily from 3.3.4. We show that D is not very
ample when deg(D) < 5. Now suppose D is very ample, then l(D) = l(D
P Q) + 2 2. Futhermore, if l(D) = 2, dim|D| = 1, thus |D| denes a

Hartshorne, Chapter 1.3 Answers to exercises.
REB 1994
3.1a Follows from exercise 1.1 as 2 ane varieties are isomorphic if and only if their coordinate rings are.
3.1b The coordinate ring of any proper subset of A1 has invertible elements not in k and o i

Solution to Math256a section IV.1
1.1) Choose a positive integer n larger than deg(K) = 2g 2, and g. By
Riemann-Roch theorem, l(nP ) = n+1g > 1. Thus there exists a non-constant
rational function f over X which has a pole at P of order n > 0, and regular