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JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Algebraic topology

I will summarize the results I developed and presented another thread in a more coherent and easier-to-follow fashion here.

1. HOMOTOPY

You have two topological spaces X and Y and two continuous functions

. Suppose for each

, there corresponds a continuous function

such that

and

for all

.

A continuous function

is callled a homotopy from f to g. We say that f and g are homotopic iff there exists a homotopy from f to g, and we write

; we can also express the fact that H is a homotopy from f to g by writing

.

The relation “is homotopic to” is an equivalence relation on

, the class of all continuous functions from X to Y.

(i) For any

, the function

is a homotopy from f to itself.

(ii) For any

, if

, then

is a homotopy from g to f.

(ii) For any

, if

and

, define

as follows.

Then

.

the equivalence classes under this equivalence relation are called homotopy classes.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Algebraic topology

2. RELATIVE HOMOTOPY

Let X, Y be topological spaces,

, and A a subset of X such that

for all

. If H is a homotopy from f to g such that

for all

, H is called a homotopy relative to A (or homotopy rel A) from f to g, and we can write

.

Like ordinary homotopy, relative homotopy is an equivalence relation on

. Indeed, if

, H is just an ordinary homotopy.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Algebraic topology

3. PATHS

Let X be a topological space and

. A path in X from a to b is a continuous function

with

and

.

If p is a path in X from a to b and q is a path in X from b to c, the product path

is the path from a to c defined by

                 

LQ

Real Member

Registered: 2006-12-04

Posts: 1,285

Re: Algebraic topology

I'm amazed, sounds important

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I see clearly now, the universe have the black dots, Thus I am on my way of inventing this remedy...

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Algebraic topology

In differential geometry, you also define the set of paths to a point to be the tangent space at that point.  The paths are of the form:

Such that

Where M is your manifold.  We then use the tangent space to define what a "derivative" means.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Algebraic topology

Thanks Ricky.

Later, when I come to n-dimensional homotopy groups, I will be defining n-dimensional paths as continuous functions from the n-dimensional hypercube to X – but for the time being I avoid jumping the gun.

Nehushtan

Member

Registered: 2013-03-09

Posts: 957

Re: Algebraic topology

4. BORSUK–ULAM THEOREM

 

 

 

 

 

 

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Nehushtan

Member

Registered: 2013-03-09

Posts: 957

Re: Algebraic topology

5. HAM-SANDWICH THEOREM

It is always possible to slice a three-layered ham sandwich with a single cut of a knife in such a way that each layer of the sandwich is divided into two exactly equal halves by the cut.

The ham-sandwich theorem can be proved using the Borsuk–Ulam theorem.

Last edited by Nehushtan (2016-04-21 00:30:28)

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