 Forum index » Science and Technology » Math
Author Message
melanie
science forum beginner

Joined: 15 Jul 2005
Posts: 13 Posted: Thu Jul 13, 2006 1:38 am    Post subject: Question from an old algebraic topology qualifying exam Hello,

I am struggling on an old qual, it seems difficult.
I am not sure how to solve these.
Any help would be appreciated.

Problem;

Let Y be a subset of 3 dimentinal Euclid space such that

Y = {(x,y,z) in R^3 : 9 =< x^2 + y^2 + z^2 =< 16,
and x^2 + z^2 >= 1, y^2 + z^2 >= 1 }

Let the boundary of Y be X and consider the continuous
function

f: X -> X ; (x, y, z) -> (-y, x, z)

1)Find the integer coefficient homology group H_* (X; Z).

2)Suppose f induces self-homomorphism on the first
dimentional integer coefficient homology group

f_* : H_1 (X; Z) -> H_1 (X;Z).

Find the characterisic polynomial of its matrix. Lee Rudolph
science forum Guru

Joined: 28 Apr 2005
Posts: 566 Posted: Thu Jul 13, 2006 11:07 am    Post subject: Re: Question from an old algebraic topology qualifying exam melanie <melanie@yahoo.com> writes:

 Quote: Let Y be a subset of 3 dimentinal Euclid space such that Y = {(x,y,z) in R^3 : 9 =< x^2 + y^2 + z^2 =< 16, and x^2 + z^2 >= 1, y^2 + z^2 >= 1 } Let the boundary of Y be X

Okay. So can you describe X in some other way, as an
example of a space you are (or should be) familiar with?
One (natural) way to get this description is to look
at the description above and take it bit by bit. Thus:
{(x,y,z) in R^3 : 9 =< x^2 + y^2 + z^2 =< 16} describes
the closed region between two spheres (of radii 3 and 4),
so *its* boundary is precisely those two spheres.
The inequalities x^2 + z^2 < 1, y^2 + z^2 < 1
(complementary to the other two inequalities in the
description of Y) each describe an open region that
is a tubular (product) neighborhood of a straight
line in R^3, namely, the y axis and the x axis
respectively. Y is the region between those two
spheres with those open regions both *removed*,
and (because the radius of the tubes is 1 which
is less than 3) its boundary X is the union of
the following parts: (1) the two spheres, each
with four (round) disks removed; (2) four cylindrical
surfaces, each bounded by one (round) circle on the
outer sphere and one on the inner sphere. So X is
a surface (2-dimensional manifold). *What* surface?
Count and find out.

 Quote: and consider the continuous function f: X -> X ; (x, y, z) -> (-y, x, z)

This function, being the restriction to X of an orthogonal
linear transformation A of R^3, and the spheres, cylinders,
etc., mentioned above all behaving nicely with respect to
A, can now be described--EASILY--in terms of how it
permutes the two 4-punctured 2-spheres in (1) and the
four cylinders in (2)
..
 Quote: 1)Find the integer coefficient homology group H_* (X; Z).

As soon as you have answered "*What* surface?", you will know
this *abstractly*. If you think a bit more, you should know it
*concretely*, in the sense that you will actually be able to
find, first, a geometrically described set of 1-cycles on X whose
homology classes generated the homology group, and, second,
the (fairly obvious!) relations among them--so you'll have the
group given by generators and relations.

 Quote: 2)Suppose f induces self-homomorphism on the first dimentional integer coefficient homology group f_* : H_1 (X; Z) -> H_1 (X;Z).

Why "suppose"? It does, and that's an end on't.

 Quote: Find the characterisic polynomial of its matrix.

Continue as indicated.

Lee Rudolph melanie
science forum beginner

Joined: 15 Jul 2005
Posts: 13 Posted: Thu Jul 13, 2006 4:22 pm    Post subject: Re: Question from an old algebraic topology qualifying exam Quote: melanie writes: Let Y be a subset of 3 dimentinal Euclid space such that Y = {(x,y,z) in R^3 : 9 =< x^2 + y^2 + z^2 =< 16, and x^2 + z^2 >= 1, y^2 + z^2 >= 1 } Let the boundary of Y be X Okay. So can you describe X in some other way, as an example of a space you are (or should be) familiar with? One (natural) way to get this description is to look at the description above and take it bit by bit. Thus: {(x,y,z) in R^3 : 9 =< x^2 + y^2 + z^2 =< 16} describes the closed region between two spheres (of radii 3 and 4), so *its* boundary is precisely those two spheres. The inequalities x^2 + z^2 < 1, y^2 + z^2 < 1 (complementary to the other two inequalities in the description of Y) each describe an open region that is a tubular (product) neighborhood of a straight line in R^3, namely, the y axis and the x axis respectively. Y is the region between those two spheres with those open regions both *removed*, and (because the radius of the tubes is 1 which is less than 3) its boundary X is the union of the following parts: (1) the two spheres, each with four (round) disks removed; (2) four cylindrical surfaces, each bounded by one (round) circle on the outer sphere and one on the inner sphere. So X is a surface (2-dimensional manifold). *What* surface? Count and find out.

I thimk it is the torus.

 Quote: and consider the continuous function f: X -> X ; (x, y, z) -> (-y, x, z) This function, being the restriction to X of an orthogonal linear transformation A of R^3, and the spheres, cylinders, etc., mentioned above all behaving nicely with respect to A, can now be described--EASILY--in terms of how it permutes the two 4-punctured 2-spheres in (1) and the four cylinders in (2) . 1)Find the integer coefficient homology group H_* (X; Z). As soon as you have answered "*What* surface?", you will know this *abstractly*. If you think a bit more, you should know it *concretely*, in the sense that you will actually be able to find, first, a geometrically described set of 1-cycles on X whose homology classes generated the homology group, and, second, the (fairly obvious!) relations among them--so you'll have the group given by generators and relations.

If X is torus then,
H_0 (X; Z) = Z
H_1 (X; Z) = Z (+) Z
H_q (X; Z) = 0 ( q > 1 )

 Quote: 2)Suppose f induces self-homomorphism on the first dimentional integer coefficient homology group f_* : H_1 (X; Z) -> H_1 (X;Z). Why "suppose"? It does, and that's an end on't. Find the characterisic polynomial of its matrix. Continue as indicated.

I trnaslated wrong.

 Quote: Find characrteristic equation of its matrix is right.

Is this just a characteristic equation of next matrix?
[0, -1, 0]
[1, 0, 0]
[0, 0, 1]

i.e. its characteristic equation is

(lamda)^2*(1-lamda) + 1 = 0

 Quote: Lee Rudolph Lee Rudolph
science forum Guru

Joined: 28 Apr 2005
Posts: 566 Posted: Thu Jul 13, 2006 6:51 pm    Post subject: Re: Question from an old algebraic topology qualifying exam melanie <melanie@yahoo.com> writes:

....
 Quote: Y is the region between those two spheres with those open regions both *removed*, and (because the radius of the tubes is 1 which is less than 3) its boundary X is the union of the following parts: (1) the two spheres, each with four (round) disks removed; (2) four cylindrical surfaces, each bounded by one (round) circle on the outer sphere and one on the inner sphere. So X is a surface (2-dimensional manifold). *What* surface? Count and find out. I thimk it is the torus.

I think you should draw a picture.

Lee Rudolph melanie
science forum beginner

Joined: 15 Jul 2005
Posts: 13 Posted: Fri Jul 14, 2006 5:04 pm    Post subject: Re: Question from an old algebraic topology qualifying exam Quote: melanie writes: ... Y is the region between those two spheres with those open regions both *removed*, and (because the radius of the tubes is 1 which is less than 3) its boundary X is the union of the following parts: (1) the two spheres, each with four (round) disks removed; (2) four cylindrical surfaces, each bounded by one (round) circle on the outer sphere and one on the inner sphere. So X is a surface (2-dimensional manifold). *What* surface? Count and find out. I thimk it is the torus. I think you should draw a picture. Lee Rudolph

I drew a picture, it looks like two spherical shell
with four round holes. Does this look like a
familiar surface?
I have a hard time to figure out.
Could you explain in details?
Thanks; Lee Rudolph
science forum Guru

Joined: 28 Apr 2005
Posts: 566 Posted: Sat Jul 15, 2006 7:43 pm    Post subject: Re: Question from an old algebraic topology qualifying exam melanie <melanie@yahoo.com> quoted a description of a surface,
which I rephrased thus:

 Quote: Y is the region between those two spheres with those open regions both *removed*, and (because the radius of the tubes is 1 which is less than 3) its boundary X is the union of the following parts: (1) the two spheres, each with four (round) disks removed; (2) four cylindrical surfaces, each bounded by one (round) circle on the outer sphere and one on the inner sphere. So X is a surface (2-dimensional manifold). *What* surface?

Later I recommended that melanie draw a picture. Now:

 Quote: I drew a picture, it looks like two spherical shell with four round holes

in each of the two.

No, it doesn't: it looks like that PLUS the lateral
surfaces of four right circular cylinders; each of
these four pieces of cylinder has one boundary
circle on the outer sphere (where it concides
with the boundary of one of the round holes on
that sphere) and one on the inner sphere (where
it coincides with the boundary of one of the round
holes on *that* sphere).

 Quote: Does this look like a familiar surface?

Do you know the "classification of surfaces"? If you
do, then you have enough information to know which
surface it is (and it is familiar to you); if not,
then it may not be familiar to you. Even if it
is unfamiliar, you do have enough information to
(in theory) compute the homological information you
originally asked about, but you may not have enough
sophistication (in practice).

Lee Rudolph  Display posts from previous: All Posts1 Day7 Days2 Weeks1 Month3 Months6 Months1 Year Oldest FirstNewest First
 The time now is Wed Mar 27, 2019 2:13 am | All times are GMT Forum index » Science and Technology » Math
 Jump to: Select a forum-------------------Forum index|___Science and Technology    |___Math    |   |___Research    |   |___num-analysis    |   |___Symbolic    |   |___Combinatorics    |   |___Probability    |   |   |___Prediction    |   |       |   |___Undergraduate    |   |___Recreational    |       |___Physics    |   |___Research    |   |___New Theories    |   |___Acoustics    |   |___Electromagnetics    |   |___Strings    |   |___Particle    |   |___Fusion    |   |___Relativity    |       |___Chem    |   |___Analytical    |   |___Electrochem    |   |   |___Battery    |   |       |   |___Coatings    |       |___Engineering        |___Control        |___Mechanics        |___Chemical

 Topic Author Forum Replies Last Post Similar Topics Question about Life. socratus Probability 0 Sun Jan 06, 2008 10:01 pm Probability Question dumont Probability 0 Mon Oct 23, 2006 3:38 pm Question about exponention WingDragon@gmail.com Math 2 Fri Jul 21, 2006 8:13 am question on solartron 1260 carrie_yao@hotmail.com Electrochem 0 Fri Jul 21, 2006 7:11 am A Combinatorics/Graph Theory Question mathlover Undergraduate 1 Wed Jul 19, 2006 11:30 pm