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Posts tagged with algebraic topology

Why is the boundary of a boundary always null? What is a coboundary?

Beware, some of the + signs in Ghrist’s Elementary Applied Topology are formal sums, which is pasting things together
http://upload.wikimedia.org/wikipedia/en/e/e8/F2_Cayley_Graph.png
rather than umming in the usual dimension-reducing sense.










An illustration I made for @michiexile’s A∞ for the layman.
 
The left side of the picture just explains the conventional algebraic topology setup: x₀ basepoint and composition-of-paths-which-are-functions-what-the-heck-are-we-talking-about-anymore. That’s the usual setup to explaining homotopy.
The right side of the picture represents @michiexile’s exploration of associativity. That’s (ab)c=a(bc). Simple to state in an algebraic formula, but it takes some pencil calisthenics to work out what that rule is saying!

My little innovation here was to replace parentheses () with \fbox{}es which I think are easier to read.

An illustration I made for @michiexile’s A for the layman.

 


The left side of the picture just explains the conventional algebraic topology setup: x₀ basepoint and composition-of-paths-which-are-functions-what-the-heck-are-we-talking-about-anymore. That’s the usual setup to explaining homotopy.

The right side of the picture represents @michiexile’s exploration of associativity. That’s (ab)c=a(bc). Simple to state in an algebraic formula, but it takes some pencil calisthenics to work out what that rule is saying!

image

My little innovation here was to replace parentheses () with \fbox{}es which I think are easier to read.


hi-res




Once you’ve accepted that Pac Man takes place on a torus

you can extend the same trick to make higher-genus manifolds.

hi-res




http://2.bp.blogspot.com/-jTsy6D2Kc-E/T9Qm0CVmOnI/AAAAAAAABfs/YHSWk-j95Kc/s1600/tomato+cam.jpg

Cylinder = line-segment × disc

C = | × ●

The “product rule” from calculus works as well with the boundary operator as with the differentiation operator .

∂C  =   ∂| × ●   +   | × ∂●

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Oops. Typo. Sorry, I did this really late at night! cos and sin need to be swapped.

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Oops. Another typo. Wrong formula for circumference.

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Minute 5. Examples of interesting 1-D topological objects:

  • hyperbola
  • Cantor set
  • line segment
  • pair of circles
  • pair of lines
  • Mercedes-Benz sign (subtract the circle or not)
  • knots / links
  • graphs

UPDATE: @bebischof corrects me:




  • Why bicontinuity is the right condition for topological equivalence (homeomorphism): if continuity of the inverse isn’t required, then a circle could be equivalent to a line (.99999 and 0 would be neighbours) — Minute 8 or so.
  • Geometric construction (no complex numbers) of the circle group.
  • Pappos’ theorem. (Minute 31)
  • Pascal’s theorem.
  • Desargues’ theorem.
Hat tip to +Ozymandias Haynes.




Calculus is topology.

The reason is that the matrix of the exterior derivative is equivalent to the transpose of the matrix of the boundary operator. That fact has been known for some time, but its practical consequences have only been understood recently.

[S]uppose you know the boundary of each k-cell in a cell complex in terms of (k−1)-cells, i.e., the boundary operator. Then you also know the exterior derivative of all discrete differential forms (i.e., cochains). So, you know calculus. Smooth or discrete.

Peter Saveliev

(Source: inperc.com)




How come you can turn a T-shirt inside out?

The automorphism group of a 3-times punctured sphere has 12 path-components (12 elements up to isotopy). There are 6 elements that preserve orientation, and 6 that reverse. In particular the orientation-reversing automorphisms reverse the orientation of all the boundary circles.

File this one under things I want to understand.

How come you can turn a T-shirt inside out?

The automorphism group of a 3-times punctured sphere has 12 path-components (12 elements up to isotopy). There are 6 elements that preserve orientation, and 6 that reverse. In particular the orientation-reversing automorphisms reverse the orientation of all the boundary circles.

File this one under things I want to understand.


hi-res