Ideas of geometric topology in algebraic geometry

Morsels of geometric topologyAlgebro-geometric parallels

Ideas of geometric topologyin algebraic geometry

or geometric applications of A1-homotopy theory

Aravind AsokUCLA

March 6, 2009

Aravind Asok UCLA Ideas of geometric topology in algebraic geometry

Outline

1 Morsels of geometric topology

2 Algebro-geometric parallels

Outline

1 Morsels of geometric topology

2 Algebro-geometric parallels

The Poincare conjecture and how not to prove it

Conjecture (Poincare)

Any closed 3-manifold M homotopy equivalent to S3 ishomeomorphic to S3.

“Theorem” (J.H.C. Whitehead, 1934)

Proof of “Theorem”.

1 Take a homotopy equivalence f : M → S3

2 Remove a point to get a homotopy equivalence M \ ∗ → R3,i.e., M \ ∗ is open and contractible

3 Conclude that M \ ∗ is homeomorphic to R3

4 By continuity, M is homeomorphic to S3

The Whitehead manifold

Question

What’s wrong with this proof?

Example (J.H.C. Whitehead, ’35)

There is an open 3-manifold homotopy equivalent to ∗ but nothomeomorphic to R3!

Manifold is an open subset of S3

Closed complement W∞ is the Whitehead continuum

Question

Figure: Whitehead Continuum W∞ by artist Lun-Yi Tsai

Obstructions to the Poincare conjecture?

Question

Is this an isolated pathology?

No! Uncountably many such beasts in every dimension ≥ 3.

Dimension 3: McMillan (’62), Dimension ≥ 5: Curtis-Kwun(’65), Dimension 4: Glaser (’67)

Question

Are these even pathological, e.g., is there additional structure?

For any open contractible Mn, one has Mn × R ∼= Rn+1

Dimension ≥ 5: Stallings (’62) + Siebenmann (’68),Dimension 4: Freedman (’82), Dimension 3: Perelman (+Wall) (’06 + ’62)

Question

Morals of the story

In high dimensions, homotopy equivalence andhomeomorphism are very different notions

Open contractible manifolds measure the difference betweenhomotopy and homeomorphism

The theory is rich (there are many examples in highdimensions)

The theory has beautiful structure “stably”: after crossingwith R these manifolds become homeomorphic to Rn

We can construct all open contractible manifolds viaquotients of translation actions of R.

We can characterize Rn algebro-topologically (maybe moreon this later)

Morals of the story

Algebraic varieties

Definition

An algebraic variety is the locus of simultaneous solutions to asystem of polynomial equations over an algebraically closed field k .

We consider algebraic varieties up to

isomorphism (polynomial map with polynomial inverse), or

if k = C, homotopy equivalence

The simplest algebraic variety (arguably) is Ank .

Question

Is Ank distinguished in any way? Can we characterize it among

algebraic varieties of a given dimension?

Algebraic varieties

Definition

Question

Algebraic varieties

Definition

Question

Algebraic varieties

Definition

Question

Algebraic varieties

Definition

Question

Can we characterize affine space?

Question

Is AnC the only contractible algebraic variety of a given dimension?

No! There exist contractible smooth complex surfaces notisomorphic to A2

Ramanujam (’74), many many others...

Question

Can one characterize affine space in some way?

Remark

Even if we could characterize AnC “topologically,” this would be

unsatisfactory because such a characterization would notnecessarily make sense over other fields.

Question

Remark

Question

Remark

Question

Remark

A1-homotopy theory

Idea (Algebro-geometric version of homotopy theory)

Think: homotopies parameterized by A1

Morel-Voevodsky (’98) showed this could be done in areasonable way: unstable A1-homotopy category

Question

How different are A1-homotopy equivalence and isomorphism?

Measure this via A1-contractibility.

A1-homotopy theory

Question

A1-homotopy theory

Question

A1-homotopy theory

Question

A1-homotopy theory

Question

Does this new notion recover old results?

Example

Affine space Ank is A1-contractible.

Why? Use radial scaling to write down a contraction.

Question

What do low dimensional A1-contractibles look like?

Only A1-contractible smooth variety of dimension 1 is A1

Many known contractible smooth complex surfaces notisomorphic to A2 are not A1-contractible

Idea: not all pairs of points can be connected by affine lines;expect A2 is the only A1-contractible

Example

Question

Example

Question

Example

Question

Is A1-contractibility a rich notion?

Question

Is Ank the only A1-contractible variety?

No! In dimensions ≥ 4, there are many A1-contractiblevarieties over any field. (A., B. Doran ’07)

Construction

Take An, and construct free actions of A1 by translations.

Form the quotient just like topology!

Check one can produce examples not isomorphic to affinespace.

Question

Construction

Question

Construction

Question

Construction

Question

Construction

A representative example

Example (cf. Winkelmann ’90)

Take the 4-dimensional quadric Q4 defined byx1x3 + x2x4 + x2

5 = 1.

Remove E2 the locus of points where x1 = x2 = 0 and x5 = 1(isomorphic to A2).

Picture: take the tangent bundle to a sphere, and remove thetangent space at a point.

Any regular function on Q4 \ E2 extends uniquely to Q4.

This example is indicative of the general construction: allknown examples are complements of codimension ≥ 2subspaces in an affine variety.

5 = 1.

We might ask...

Question

Is An the only smooth affine A1-contractible variety?

Is A3 the only smooth affine A1-contractible 3-fold?

For n ≥ 4 unlikely if topological situation provides anyindication

We have no idea. However...

Try to use quotient constructions

Problem: hard to distinguish varieties from affine space, butA1-homotopy theory suggests new (hard to compute)invariants

We might ask...

Question

We might ask...

Question

We might ask...

Question

We might ask...

Question

We might ask...

Question

We have no idea.

However...

We might ask...

Question

Does A1-contractiblity help prove anything new?

A variety X such that X × An ∼= An+k will be called stablyisomorphic to affine space; any such variety is A1-contractible.

Question (Kaliman/Makar-Limanov ’97, Koras/Russell ’97)

Is the (contractible) smooth complex affine 3-fold defined by theequation

x + x2y + z2 + t3 = 0

stably isomorphic to affine space?

Makar-Limanov (’97) proved above hypersurface is notisomorphic to A3

C, but the stable result remains open.

Expectation (in progress): the above hypersurface is notA1-contractible.

x + x2y + z2 + t3 = 0

Thank you!

See http://www.math.ucla.edu/~asok for more information

Ideas of geometric topology in algebraic geometry

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