*Homology of the little disks operad*

Posts tagged with **algebraic topology**

If homology classes in configuration spaces were viewed as planetary systems, cohomology classes would represent planetary alignments.

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.

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.

(Source: math.cornell.edu)

hi-res

`Cylinder = line-segment × disc`

`C = | × ●`

The “product rule” from calculus works as well with the boundary operator `∂`

as with the differentiation operator `∂`

.

`∂C = ∂| × ● + | × ∂●`

Oops. Typo. Sorry, I did this really late at night! `cos`

and `sin`

need to be swapped.

Oops. Another typo. Wrong formula for circumference.

**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:

@isomorphisms cantor set is zero dimensional blah blah #blah

— Bryan B. (@BEBischof) December 16, 2012

- 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.

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.

(Source: inperc.com)

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

The

automorphism groupof a3-times punctured spherehas 12 path-components (12 elements up to isotopy). There are 6 elements that preserve orientation, and 6 that reverse. In particularthe orientation-reversing automorphisms reverse the orientation of all the boundary circles.

File this one under things I want to understand.

hi-res