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
Morsels of geometric topologyAlgebro-geometric parallels
Outline
1 Morsels of geometric topology
2 Algebro-geometric parallels
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
Outline
1 Morsels of geometric topology
2 Algebro-geometric parallels
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-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)
Any closed 3-manifold M homotopy equivalent to S3 ishomeomorphic to S3.
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
The Poincare conjecture and how not to prove it
“Theorem” (J.H.C. Whitehead, 1934)
Any closed 3-manifold M homotopy equivalent to S3 ishomeomorphic to S3.
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
The Poincare conjecture and how not to prove it
“Theorem” (J.H.C. Whitehead, 1934)
Any closed 3-manifold M homotopy equivalent to S3 ishomeomorphic to S3.
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
The Poincare conjecture and how not to prove it
“Theorem” (J.H.C. Whitehead, 1934)
Any closed 3-manifold M homotopy equivalent to S3 ishomeomorphic to S3.
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
The Poincare conjecture and how not to prove it
“Theorem” (J.H.C. Whitehead, 1934)
Any closed 3-manifold M homotopy equivalent to S3 ishomeomorphic to S3.
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
The Poincare conjecture and how not to prove it
“Theorem” (J.H.C. Whitehead, 1934)
Any closed 3-manifold M homotopy equivalent to S3 ishomeomorphic to S3.
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
Figure: Whitehead Continuum W∞ by artist Lun-Yi Tsai
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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)
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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)
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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)
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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)
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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)
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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)
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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)
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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)
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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)
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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)
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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?
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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?
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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?
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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?
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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?
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
C!
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
C!
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
C!
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
C!
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
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.
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
Morsels of geometric topologyAlgebro-geometric parallels
Thank you!
See http://www.math.ucla.edu/~asok for more information
Aravind Asok UCLA Ideas of geometric topology in algebraic geometry
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