Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups...
Transcript of Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups...
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Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Stable homotopy theory and geometry
Paul VanKoughnett
October 24, 2014
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 2: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/2.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Building topological spaces
Problem: How to combinatorially describe a topologicalspace X?
(always ‘geometric’ – Hausdorff, compactlygenerated, . . . )
Solution: Build it in stages, out of cells.
n-cell = Dn = unit n-disk, attached by its boundary Sn−1.
X is described entirely by attaching maps Sn−1 → X (n−1),where X (n−1) is the n-dimensional part of X .
Even simpler: Sn−1 → X (n−1)/X (n−2), a bouquet of(n − 1)-spheres; or Sn−1 → X (n−2)/X (n−3), a bouquet of(n − 2)-spheres; or . . .
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 3: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/3.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Building topological spaces
Problem: How to combinatorially describe a topologicalspace X? (always ‘geometric’ – Hausdorff, compactlygenerated, . . . )
Solution: Build it in stages, out of cells.
n-cell = Dn = unit n-disk, attached by its boundary Sn−1.
X is described entirely by attaching maps Sn−1 → X (n−1),where X (n−1) is the n-dimensional part of X .
Even simpler: Sn−1 → X (n−1)/X (n−2), a bouquet of(n − 1)-spheres; or Sn−1 → X (n−2)/X (n−3), a bouquet of(n − 2)-spheres; or . . .
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 4: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/4.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Building topological spaces
Problem: How to combinatorially describe a topologicalspace X? (always ‘geometric’ – Hausdorff, compactlygenerated, . . . )
Solution: Build it in stages, out of cells.
n-cell = Dn = unit n-disk, attached by its boundary Sn−1.
X is described entirely by attaching maps Sn−1 → X (n−1),where X (n−1) is the n-dimensional part of X .
Even simpler: Sn−1 → X (n−1)/X (n−2), a bouquet of(n − 1)-spheres; or Sn−1 → X (n−2)/X (n−3), a bouquet of(n − 2)-spheres; or . . .
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 5: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/5.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Building topological spaces
Problem: How to combinatorially describe a topologicalspace X? (always ‘geometric’ – Hausdorff, compactlygenerated, . . . )
Solution: Build it in stages, out of cells.
n-cell = Dn = unit n-disk, attached by its boundary Sn−1.
X is described entirely by attaching maps Sn−1 → X (n−1),where X (n−1) is the n-dimensional part of X .
Even simpler: Sn−1 → X (n−1)/X (n−2), a bouquet of(n − 1)-spheres; or Sn−1 → X (n−2)/X (n−3), a bouquet of(n − 2)-spheres; or . . .
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 6: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/6.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Building topological spaces
Problem: How to combinatorially describe a topologicalspace X? (always ‘geometric’ – Hausdorff, compactlygenerated, . . . )
Solution: Build it in stages, out of cells.
n-cell = Dn = unit n-disk, attached by its boundary Sn−1.
X is described entirely by attaching maps Sn−1 → X (n−1),where X (n−1) is the n-dimensional part of X .
Even simpler: Sn−1 → X (n−1)/X (n−2), a bouquet of(n − 1)-spheres; or Sn−1 → X (n−2)/X (n−3), a bouquet of(n − 2)-spheres; or . . .
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 7: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/7.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Building topological spaces
Problem: How to combinatorially describe a topologicalspace X? (always ‘geometric’ – Hausdorff, compactlygenerated, . . . )
Solution: Build it in stages, out of cells.
n-cell = Dn = unit n-disk, attached by its boundary Sn−1.
X is described entirely by attaching maps Sn−1 → X (n−1),where X (n−1) is the n-dimensional part of X .
Even simpler: Sn−1 → X (n−1)/X (n−2), a bouquet of(n − 1)-spheres; or Sn−1 → X (n−2)/X (n−3), a bouquet of(n − 2)-spheres; or . . .
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 8: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/8.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Homotopy
Two of these spaces are equivalent if the attaching maps ofone can be deformed into the attaching maps of the other.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 9: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/9.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Homotopy
Two of these spaces are equivalent if the attaching maps ofone can be deformed into the attaching maps of the other.
Definition
Given two maps f , g : X → Y , a homotopy f ∼ g is mapH : X × [0, 1]→ Y with H|X×0 = f , and H|X×1 = Y .
[X ,Y ] = maps X → Y /homotopy
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 10: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/10.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Homotopy
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 11: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/11.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Homotopy
Definition
Given two maps f , g : X → Y , a homotopy f ∼ g is mapH : X × [0, 1]→ Y with H|X×0 = f , and H|X×1 = Y .
[X ,Y ] = maps X → Y /homotopy
Remark
Always take spaces to come equipped with a fixed basepoint; mapspreserve basepoint; homotopies don’t move basepoint.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 12: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/12.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
An example
CP2 is constructed by attaching a 4-cell to CP1 = S2.
Entirely described by an attaching map S3 → S2.
If this map is homotopic to a trivial one, then CP2 isequivalent to S4 t S2/basepoints.
Is it? How many other complexes like this are there?
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 13: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/13.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
An example
CP2 is constructed by attaching a 4-cell to CP1 = S2.
Entirely described by an attaching map S3 → S2.
If this map is homotopic to a trivial one, then CP2 isequivalent to S4 t S2/basepoints.
Is it? How many other complexes like this are there?
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 14: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/14.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
An example
CP2 is constructed by attaching a 4-cell to CP1 = S2.
Entirely described by an attaching map S3 → S2.
If this map is homotopic to a trivial one, then CP2 isequivalent to S4 t S2/basepoints.
Is it? How many other complexes like this are there?
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 15: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/15.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
An example
CP2 is constructed by attaching a 4-cell to CP1 = S2.
Entirely described by an attaching map S3 → S2.
If this map is homotopic to a trivial one, then CP2 isequivalent to S4 t S2/basepoints.
Is it? How many other complexes like this are there?
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 16: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/16.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Homotopy groups
Definition
The nth homotopy group of a space X is
πnX := [Sn,X ].
Example
π0X = [S0,X ] = path components of X
π1X = [S1,X ] = fundamental group of X
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 17: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/17.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Homotopy groups
Definition
The nth homotopy group of a space X is
πnX := [Sn,X ].
Example
π0X = [S0,X ] = path components of X
π1X = [S1,X ] = fundamental group of X
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 18: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/18.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Homotopy groups
Definition
The nth homotopy group of a space X is
πnX := [Sn,X ].
Example
π0X = [S0,X ] = path components of X
π1X = [S1,X ] = fundamental group of X
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 19: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/19.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Homotopy groups
Definition
The nth homotopy group of a space X is
πnX := [Sn,X ].
Example
π0X = [S0,X ] = path components of X
π1X = [S1,X ] = fundamental group of X
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 20: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/20.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Why are they groups?
Because we can pinch a sphere:
π0 is just a set. πn is abelian for n ≥ 2 (why?).
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 21: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/21.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Why are they groups?
Because we can pinch a sphere:
π0 is just a set.
πn is abelian for n ≥ 2 (why?).
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 22: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/22.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Why are they groups?
Because we can pinch a sphere:
π0 is just a set. πn is abelian for n ≥ 2 (why?).
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 23: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/23.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Homotopy groups of spheres
n9
0 0 0 0 0 0 0 0
8
0 0 0 0 0 0 0
7
0 0 0 0 0 0
6
0 0 0 0 0
5
0 0 0 0
4
0 0 0
3
0 0
2
0
1
Z
πkSn 1 2 3 4 5 6 7 8 9 k
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 24: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/24.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Homotopy groups of spheres
n9 0
0 0 0 0 0 0 0
8 0
0 0 0 0 0 0
7 0
0 0 0 0 0
6 0
0 0 0 0
5 0
0 0 0
4 0
0 0
3 0
0
2 01
Z
πkSn 1 2 3 4 5 6 7 8 9 k
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 25: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/25.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Homotopy groups of spheres
n9 0
0 0 0 0 0 0 0
8 0
0 0 0 0 0 0
7 0
0 0 0 0 0
6 0
0 0 0 0
5 0
0 0 0
4 0
0 0
3 0
0
2 01 Z
πkSn 1 2 3 4 5 6 7 8 9 k
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 26: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/26.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Homotopy groups of spheres
n9 0 0 0 0 0 0 0 08 0 0 0 0 0 0 07 0 0 0 0 0 06 0 0 0 0 05 0 0 0 04 0 0 03 0 02 01 Z
πkSn 1 2 3 4 5 6 7 8 9 k
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 27: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/27.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Homotopy groups of spheres
n9 0 0 0 0 0 0 0 08 0 0 0 0 0 0 07 0 0 0 0 0 06 0 0 0 0 05 0 0 0 04 0 0 03 0 02 01 Z 0 0 0 0 0 0 0 0
πkSn 1 2 3 4 5 6 7 8 9 k
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 28: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/28.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
A tool: suspension
Definition
The suspension of a space X is
ΣX = X × [0, 1]/(X × 0 ∪ X × 1 ∪ ∗ × [0, 1]).
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 29: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/29.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
A tool: suspension
Definition
The suspension of a space X is
ΣX = X × [0, 1]/(X × 0 ∪ X × 1 ∪ ∗ × [0, 1]).
There are suspension maps
E : [X ,Y ]→ [ΣX ,ΣY ]
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 30: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/30.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
A tool: suspension
Definition
The suspension of a space X is
ΣX = X × [0, 1]/(X × 0 ∪ X × 1 ∪ ∗ × [0, 1]).
There are suspension maps
E : πnY → πn+1ΣY
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 31: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/31.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
A tool: suspension
Definition
The suspension of a space X is
ΣX = X × [0, 1]/(X × 0 ∪ X × 1 ∪ ∗ × [0, 1]).
Theorem (Freudenthal suspension theorem)
The suspension map
E : πkSn → πk+1S
n+1
is a surjection for k = 2n − 1 and an isomorphism for k < 2n − 1.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 32: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/32.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Using the suspension theorem
Theorem (Freudenthal suspension theorem)
The suspension map
E : πkSn → πk+1S
n+1
is a surjection for k = 2n − 1 and an isomorphism for k < 2n − 1.
Z = π1S1 π2S
2 ∼→ π3S3 ∼→ · · ·
πnSn is cyclic. . . and must be Z, because the degree of a map is
homotopy invariant.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 33: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/33.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Using the suspension theorem
Theorem (Freudenthal suspension theorem)
The suspension map
E : πkSn → πk+1S
n+1
is a surjection for k = 2n − 1 and an isomorphism for k < 2n − 1.
Z = π1S1 π2S
2 ∼→ π3S3 ∼→ · · ·
πnSn is cyclic. . . and must be Z, because the degree of a map is
homotopy invariant.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 34: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/34.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Using the suspension theorem
Theorem (Freudenthal suspension theorem)
The suspension map
E : πkSn → πk+1S
n+1
is a surjection for k = 2n − 1 and an isomorphism for k < 2n − 1.
Z = π1S1 π2S
2 ∼→ π3S3 ∼→ · · ·
πnSn is cyclic. . .
and must be Z, because the degree of a map ishomotopy invariant.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 35: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/35.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Using the suspension theorem
Theorem (Freudenthal suspension theorem)
The suspension map
E : πkSn → πk+1S
n+1
is a surjection for k = 2n − 1 and an isomorphism for k < 2n − 1.
Z = π1S1 π2S
2 ∼→ π3S3 ∼→ · · ·
πnSn is cyclic. . . and must be Z, because the degree of a map is
homotopy invariant.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 36: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/36.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
πnSn = Z
Degree two maps:
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 37: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/37.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Homotopy groups of spheres
n9 0 0 0 0 0 0 0 0 Z8 0 0 0 0 0 0 0 Z7 0 0 0 0 0 0 Z6 0 0 0 0 0 Z5 0 0 0 0 Z4 0 0 0 Z3 0 0 Z2 0 Z1 Z 0 0 0 0 0 0 0 0
πkSn 1 2 3 4 5 6 7 8 9 k
Is everything else zero? NO!
Paul VanKoughnett
Stable homotopy theory and geometry
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Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Homotopy groups of spheres
n9 0 0 0 0 0 0 0 0 Z8 0 0 0 0 0 0 0 Z7 0 0 0 0 0 0 Z6 0 0 0 0 0 Z5 0 0 0 0 Z4 0 0 0 Z3 0 0 Z2 0 Z1 Z 0 0 0 0 0 0 0 0
πkSn 1 2 3 4 5 6 7 8 9 k
Is everything else zero? NO!
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 39: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/39.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
The Hopf fibration
S3 is the unit sphere in C2, with coordinates z , w .
S2 is the Riemann sphere C ∪ ∞, with coordinate λ.
η : S3 → S2 sends (z ,w) 7→ z/w .
The fiber over any point is a circle.
The fiber over two points are two linked circles.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 40: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/40.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
The Hopf fibration
S3 is the unit sphere in C2, with coordinates z , w .
S2 is the Riemann sphere C ∪ ∞, with coordinate λ.
η : S3 → S2 sends (z ,w) 7→ z/w .
The fiber over any point is a circle.
The fiber over two points are two linked circles.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 41: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/41.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
The Hopf fibration
S3 is the unit sphere in C2, with coordinates z , w .
S2 is the Riemann sphere C ∪ ∞, with coordinate λ.
η : S3 → S2 sends (z ,w) 7→ z/w .
The fiber over any point is a circle.
The fiber over two points are two linked circles.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 42: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/42.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
The Hopf fibration
S3 is the unit sphere in C2, with coordinates z , w .
S2 is the Riemann sphere C ∪ ∞, with coordinate λ.
η : S3 → S2 sends (z ,w) 7→ z/w .
The fiber over any point is a circle.
The fiber over two points are two linked circles.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 43: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/43.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
The Hopf fibration
S3 is the unit sphere in C2, with coordinates z , w .
S2 is the Riemann sphere C ∪ ∞, with coordinate λ.
η : S3 → S2 sends (z ,w) 7→ z/w .
The fiber over any point is a circle.
The fiber over two points are two linked circles.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 44: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/44.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Homotopy groups of spheres
n9 0 0 0 0 0 0 0 0 Z8 0 0 0 0 0 0 0 Z 27 0 0 0 0 0 0 Z 2 26 0 0 0 0 0 Z 2 2 245 0 0 0 0 Z 2 2 24 24 0 0 0 Z 2 2 Z⊕ 12 2⊕ 2 2⊕ 23 0 0 Z 2 2 12 2 2 32 0 Z Z 2 2 12 2 2 31 Z 0 0 0 0 0 0 0 0
πkSn 1 2 3 4 5 6 7 8 9 k
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 45: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/45.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Generalizing the degree
We want a more geometric characterization of thesehomotopy elements.
The degree of a map Sn → Sn is an integer, because its fibersare signed 0-manifolds.
The ‘degree’ of a map Sk → Sn should just be its fiber – astably framed (n − k)-manifold.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 46: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/46.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Generalizing the degree
We want a more geometric characterization of thesehomotopy elements.
The degree of a map Sn → Sn is an integer, because its fibersare signed 0-manifolds.
The ‘degree’ of a map Sk → Sn should just be its fiber – astably framed (n − k)-manifold.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 47: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/47.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Generalizing the degree
We want a more geometric characterization of thesehomotopy elements.
The degree of a map Sn → Sn is an integer, because its fibersare signed 0-manifolds.
The ‘degree’ of a map Sk → Sn should just be its fiber – astably framed (n − k)-manifold.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 48: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/48.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Generalizing the degree
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 49: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/49.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Stably framed manifolds
Definition
A stably framed manifold is a manifold Mn with an embeddingi : Mn → Rn+k , k 0, and a trivialization of the normal bundleNiM ∼= M × Rk .
We identify M → Rn+k with M → Rn+k → Rn+k+1 the inclusioninto the first n + k coordinates, together with the larger framinggiven by adding an upward-pointing normal vector everywhere.
Definition
A framed cobordism of n-dimensional stably framed manifoldsM, N is an (n + 1)-manifold W with ∂W ∼= M t N, together witha stable framing on W extending those on M and N.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 50: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/50.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Stably framed manifolds
Definition
A stably framed manifold is a manifold Mn with an embeddingi : Mn → Rn+k , k 0, and a trivialization of the normal bundleNiM ∼= M × Rk .We identify M → Rn+k with M → Rn+k → Rn+k+1 the inclusioninto the first n + k coordinates, together with the larger framinggiven by adding an upward-pointing normal vector everywhere.
Definition
A framed cobordism of n-dimensional stably framed manifoldsM, N is an (n + 1)-manifold W with ∂W ∼= M t N, together witha stable framing on W extending those on M and N.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 51: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/51.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Stably framed manifolds
Definition
A stably framed manifold is a manifold Mn with an embeddingi : Mn → Rn+k , k 0, and a trivialization of the normal bundleNiM ∼= M × Rk .We identify M → Rn+k with M → Rn+k → Rn+k+1 the inclusioninto the first n + k coordinates, together with the larger framinggiven by adding an upward-pointing normal vector everywhere.
Definition
A framed cobordism of n-dimensional stably framed manifoldsM, N is an (n + 1)-manifold W with ∂W ∼= M t N, together witha stable framing on W extending those on M and N.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 52: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/52.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
The Pontryagin-Thom isomorphism
Ωfrn := stably framed n-manifolds/framed cobordism
Theorem
There is a canonical isomorphism
Ωfrn∼= πn+kS
k , k 0.
Homotopy elements can be described as framed n-manifolds!
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 53: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/53.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
The Pontryagin-Thom isomorphism
Ωfrn := stably framed n-manifolds/framed cobordism
Theorem
There is a canonical isomorphism
Ωfrn∼= πn+kS
k , k 0.
Homotopy elements can be described as framed n-manifolds!
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 54: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/54.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
The Pontryagin-Thom isomorphism
Ωfrn := stably framed n-manifolds/framed cobordism
Theorem
There is a canonical isomorphism
Ωfrn∼= πn+kS
k , k 0.
Homotopy elements can be described as framed n-manifolds!
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 55: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/55.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
0-manifolds: πnSn
A 0-manifold in Rn is a set of points. A framing is a sign attachedto each point: orient their normal bundles either with or againstthe orientation of Rn.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 56: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/56.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
0-manifolds: πnSn
A 0-manifold in Rn is a set of points. A framing is a sign attachedto each point: orient their normal bundles either with or againstthe orientation of Rn.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 57: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/57.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
1-manifolds: πn+1Sn
A 1-manifold is just a circle. A framing on S1 → Rn+1 isdetermined by how a basis for Rn rotates as you go aroundthe circle.
Framings are classified by π1SO(n) = Z/2 for n ≥ 3 (and Zfor n ≥ 2).
The Hopf map S3 → S2 corresponds to S1 → S3 togetherwith a basis for its normal bundle that twists once.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 58: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/58.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
1-manifolds: πn+1Sn
A 1-manifold is just a circle. A framing on S1 → Rn+1 isdetermined by how a basis for Rn rotates as you go aroundthe circle.
Framings are classified by π1SO(n) = Z/2 for n ≥ 3 (and Zfor n ≥ 2).
The Hopf map S3 → S2 corresponds to S1 → S3 togetherwith a basis for its normal bundle that twists once.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 59: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/59.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
1-manifolds: πn+1Sn
A 1-manifold is just a circle. A framing on S1 → Rn+1 isdetermined by how a basis for Rn rotates as you go aroundthe circle.
Framings are classified by π1SO(n) = Z/2 for n ≥ 3 (and Zfor n ≥ 2).
The Hopf map S3 → S2 corresponds to S1 → S3 togetherwith a basis for its normal bundle that twists once.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 60: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/60.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
2-manifolds: πn+2Sn
One way to make cobordisms is through surgery.
Framedcobordisms require framed surgery.
On any surface, we can do surgery to decrease thegenus. . . but not all surgeries can be made framed.
Pontryagin: whether or not we can do framed surgery on a1-cycle is determined by a map
φ : H1(M;Z/2)→ Z/2.
But H1(M;Z/2) is positive-dimensional if M has genus ≥ 1,so ker φ is always nonzero, so any M is framed-cobordant to asphere. ∴ πn+2S
n = 0, n 0.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 61: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/61.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
2-manifolds: πn+2Sn
One way to make cobordisms is through surgery. Framedcobordisms require framed surgery.
On any surface, we can do surgery to decrease thegenus. . . but not all surgeries can be made framed.
Pontryagin: whether or not we can do framed surgery on a1-cycle is determined by a map
φ : H1(M;Z/2)→ Z/2.
But H1(M;Z/2) is positive-dimensional if M has genus ≥ 1,so ker φ is always nonzero, so any M is framed-cobordant to asphere. ∴ πn+2S
n = 0, n 0.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 62: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/62.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
2-manifolds: πn+2Sn
One way to make cobordisms is through surgery. Framedcobordisms require framed surgery.
On any surface, we can do surgery to decrease thegenus. . . but not all surgeries can be made framed.
Pontryagin: whether or not we can do framed surgery on a1-cycle is determined by a map
φ : H1(M;Z/2)→ Z/2.
But H1(M;Z/2) is positive-dimensional if M has genus ≥ 1,so ker φ is always nonzero, so any M is framed-cobordant to asphere. ∴ πn+2S
n = 0, n 0.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 63: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/63.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
2-manifolds: πn+2Sn
One way to make cobordisms is through surgery. Framedcobordisms require framed surgery.
On any surface, we can do surgery to decrease thegenus. . . but not all surgeries can be made framed.
Pontryagin: whether or not we can do framed surgery on a1-cycle is determined by a map
φ : H1(M;Z/2)→ Z/2.
But H1(M;Z/2) is positive-dimensional if M has genus ≥ 1,so ker φ is always nonzero, so any M is framed-cobordant to asphere. ∴ πn+2S
n = 0, n 0.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 64: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/64.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
2-manifolds: πn+2Sn
One way to make cobordisms is through surgery. Framedcobordisms require framed surgery.
On any surface, we can do surgery to decrease thegenus. . . but not all surgeries can be made framed.
Pontryagin: whether or not we can do framed surgery on a1-cycle is determined by a map
φ : H1(M;Z/2)→ Z/2.
But H1(M;Z/2) is positive-dimensional if M has genus ≥ 1,so ker φ is always nonzero, so any M is framed-cobordant to asphere. ∴ πn+2S
n = 0, n 0.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 65: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/65.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Pontryagin’s mistake
φ : H1(M;Z/2)→ Z/2
is not a linear map, but a quadratic map. Even if we can dosurgery on two-cycles, we might not be able to on their sum.
Whether or not we can do framed surgery on M depends onthe nature of this quadratic map. We can conclude that
πn+2S2 ∼= Z/2, n 0.
A representative for the nontrivial class is given by theproduct of two nontrivially framed circles.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 66: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/66.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Pontryagin’s mistake
φ : H1(M;Z/2)→ Z/2
is not a linear map, but a quadratic map. Even if we can dosurgery on two-cycles, we might not be able to on their sum.
Whether or not we can do framed surgery on M depends onthe nature of this quadratic map. We can conclude that
πn+2S2 ∼= Z/2, n 0.
A representative for the nontrivial class is given by theproduct of two nontrivially framed circles.
Paul VanKoughnett
Stable homotopy theory and geometry
![Page 67: Stable homotopy theory and geometry · Stable homotopy theory and geometry. HomotopyHomotopy groups of spheresPontryagin-ThomExamples Homotopy Two of these spaces are equivalent if](https://reader034.fdocuments.net/reader034/viewer/2022050605/5fac18a5175d14214a0dffa8/html5/thumbnails/67.jpg)
Homotopy Homotopy groups of spheres Pontryagin-Thom Examples
Homotopy groups of spheres
n9 0 0 0 0 0 0 0 0 Z8 0 0 0 0 0 0 0 Z 27 0 0 0 0 0 0 Z 2 26 0 0 0 0 0 Z 2 2 245 0 0 0 0 Z 2 2 24 24 0 0 0 Z 2 2 Z⊕ 12 2⊕ 2 2⊕ 23 0 0 Z 2 2 12 2 2 32 0 Z Z 2 2 12 2 2 31 Z 0 0 0 0 0 0 0 0
πkSn 1 2 3 4 5 6 7 8 9 k
Paul VanKoughnett
Stable homotopy theory and geometry