André Neves – Min-max theory and applications II PART FIVE

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“…this is equal to zero if i is bigger than one and z2 if i equals one, so every homotopic group vanishes but the first one.”

The instance of i=1 is considered to be the ‘first’ one in some sequence. Although it is associated with 1, the idea of firstness is a little more complex; two is known as the first prime number, twelve is the first abundant number. Firstness suggests a sequence, a time-based progression of doing things and the selected entity being at the head. And the others, being equal to zero, vanish.

Like infinity, the conceptualisation of zero was a huge step in the development of mathematics; whole books have been written about it. Zero isn’t so much an absence as an entity, not a vacuum but a clearly delineated space into which nothing else can enter. The form of zero is a circle, an 0, a boundary; sometimes it’s crossed through in the way of a road sign, calling upon something in order to negate it. To vanish, then, the homotopic groups do not pop out of existence, it is not as though they had never been called.

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In particular, we can then compute the k singular homology, right. Because so then we get that the K singular homology of this infinite-dimensional space with coefficients mod z 2 just has one generator to the power of k. So there’s uh. Lambda bar we call lambda bar is the generator of h 1 and that generates the whole homology ring.”

The λ sign on the blackboard, the Greek lambda with a bar above it, is said to “generate” a ring, something with a circular structure. The word suggests something of automation, creation, a repetitive process with a concrete outcome. Lambda bar becomes a machine, a factory, churning out a ring of stuff that loops back on itself.

A bar above a symbol can point to something that has features in common with that which the original symbol points to, but is distinct in other ways. For example, the notation can be used to refer to the complex conjugate of a real number (the number with equal real part and imaginary part equal in magnitude but opposite in sign):

Geometric representation of z and its conjugate z̅ in the complex plane. The complex conjugate is found by reflecting z across the real axis.

“Complex conjugate picture” by Oleg Alexandrov – Vectorized version of with some tweaks. Licensed under CC BY-SA 3.0 via Commons –

The bar, then, leaves the symbol undisturbed but is a mark of alteration, of a different orientation. The left-to-right organisation of English writing supposes a baseline, a piece of ground that fills up and is then folded back on itself to create a new line; we create a temporary representation of the earth below our feet and line up our thoughts along it. The bar symbolically flips it, placing the ‘ground’ up above the symbol, leaving everything undisturbed but suddenly upside-down.

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So now I’m going to explain what the crucial issue is.”

This, quite different, bar pops up as Neves lays out his plan for the next section. The bar draws a literal line under the previous statements, embellishing it with slashes, a gesture close to framing or illuminating a text.


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