Stable Homotopy Theory and Geometry

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Paul VanKoughnett

October 24, 2014

Paul VanKoughnett Stable homotopy theory and geometry (always ‘geometric’ – Hausdorﬀ, compactly generated, . . . ) 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 . . .

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Building topological spaces

Problem: How to combinatorially describe a topological space X ?

Paul VanKoughnett Stable homotopy theory and geometry 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 . . .

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Building topological spaces

Problem: How to combinatorially describe a topological space X ? (always ‘geometric’ – Hausdorﬀ, compactly generated, . . . )

Paul VanKoughnett Stable homotopy theory and geometry 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 . . .

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Building topological spaces

Problem: How to combinatorially describe a topological space X ? (always ‘geometric’ – Hausdorﬀ, compactly generated, . . . ) Solution: Build it in stages, out of cells.

Paul VanKoughnett Stable homotopy theory and geometry 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 . . .

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Building topological spaces

Paul VanKoughnett Stable homotopy theory and geometry 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 . . .

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Building topological spaces

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Building topological spaces

Problem: How to combinatorially describe a topological space X ? (always ‘geometric’ – Hausdorﬀ, compactly generated, . . . ) 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 Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Homotopy

Two of these spaces are equivalent if the attaching maps of one can be deformed into the attaching maps of the other.

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Homotopy

Two of these spaces are equivalent if the attaching maps of one can be deformed into the attaching maps of the other.

Deﬁnition Given two maps f , g : X → Y , a homotopy f ∼ g is map H : 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 Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Homotopy

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Homotopy

Deﬁnition Given two maps f , g : X → Y , a homotopy f ∼ g is map H : 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; maps preserve basepoint; homotopies don’t move basepoint.

Paul VanKoughnett Stable homotopy theory and geometry Entirely described by an attaching map S3 → S2. 2 If this map is homotopic to a trivial one, then CP is equivalent to S4 t S2/basepoints. Is it? How many other complexes like this are there?

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

An example

2 1 2 CP is constructed by attaching a 4-cell to CP = S .

Paul VanKoughnett Stable homotopy theory and geometry 2 If this map is homotopic to a trivial one, then CP is equivalent to S4 t S2/basepoints. Is it? How many other complexes like this are there?

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

An example

2 1 2 CP is constructed by attaching a 4-cell to CP = S . Entirely described by an attaching map S3 → S2.

Paul VanKoughnett Stable homotopy theory and geometry Is it? How many other complexes like this are there?

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

An example

2 1 2 CP is constructed by attaching a 4-cell to CP = S . Entirely described by an attaching map S3 → S2. 2 If this map is homotopic to a trivial one, then CP is equivalent to S4 t S2/basepoints.

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

An example

2 1 2 CP is constructed by attaching a 4-cell to CP = S . Entirely described by an attaching map S3 → S2. 2 If this map is homotopic to a trivial one, then CP is equivalent to S4 t S2/basepoints. Is it? How many other complexes like this are there?

Paul VanKoughnett Stable homotopy theory and geometry Example

0 π0X = [S , X ] = {path components of X }

1 π1X = [S , X ] = fundamental group of X

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Homotopy groups

Deﬁnition The nth homotopy group of a space X is

n πnX := [S , X ].

Paul VanKoughnett Stable homotopy theory and geometry 1 π1X = [S , X ] = fundamental group of X

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Homotopy groups

Deﬁnition The nth homotopy group of a space X is

n πnX := [S , X ].

Example

0 π0X = [S , X ] = {path components of X }

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Homotopy groups

Deﬁnition The nth homotopy group of a space X is

n πnX := [S , X ].

Example

0 π0X = [S , X ] = {path components of X }

1 π1X = [S , X ] = fundamental group of X

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Homotopy groups

Deﬁnition The nth homotopy group of a space X is

n πnX := [S , X ].

Example

0 π0X = [S , X ] = {path components of X }

1 π1X = [S , X ] = fundamental group of X

Paul VanKoughnett Stable homotopy theory and geometry π0 is just a set. πn is abelian for n ≥ 2 (why?).

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Why are they groups?

Because we can pinch a sphere:

Paul VanKoughnett Stable homotopy theory and geometry πn is abelian for n ≥ 2 (why?).

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Why are they groups?

Because we can pinch a sphere:

π0 is just a set.

Paul VanKoughnett Stable homotopy theory and geometry 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Z

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Homotopy groups of spheres

n 9 8 7 6 5 4 3 2 1 n πk S 1 2 3 4 5 6 7 8 9 k

Paul VanKoughnett Stable homotopy theory and geometry 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Homotopy groups of spheres

n 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 n πk S 1 2 3 4 5 6 7 8 9 k

Paul VanKoughnett Stable homotopy theory and geometry 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Homotopy groups of spheres

n 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 Z n πk S 1 2 3 4 5 6 7 8 9 k

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Homotopy groups of spheres

n 9 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 n πk S 1 2 3 4 5 6 7 8 9 k

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Homotopy groups of spheres

n 9 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 0 0 0 0 0 0 0 0 n πk S 1 2 3 4 5 6 7 8 9 k

Paul VanKoughnett Stable homotopy theory and geometry 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 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 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 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

n n+1 E : πk S → πk+1S

is a surjection for k = 2n − 1 and an isomorphism for k < 2n − 1.

Paul VanKoughnett Stable homotopy theory and geometry 1 2 ∼ 3 ∼ Z = π1S π2S → π3S → · · · n πnS is cyclic. . . and must be Z, because the degree of a map is homotopy invariant.

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Using the suspension theorem

Theorem (Freudenthal suspension theorem) The suspension map

n n+1 E : πk S → πk+1S

is a surjection for k = 2n − 1 and an isomorphism for k < 2n − 1.

Paul VanKoughnett Stable homotopy theory and geometry n πnS is cyclic. . . and must be Z, because the degree of a map is homotopy invariant.

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Using the suspension theorem

Theorem (Freudenthal suspension theorem) The suspension map

n n+1 E : πk S → πk+1S

is a surjection for k = 2n − 1 and an isomorphism for k < 2n − 1.

1 2 ∼ 3 ∼ Z = π1S π2S → π3S → · · ·

Paul VanKoughnett Stable homotopy theory and geometry and must be Z, because the degree of a map is homotopy invariant.

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Using the suspension theorem

Theorem (Freudenthal suspension theorem) The suspension map

n n+1 E : πk S → πk+1S

is a surjection for k = 2n − 1 and an isomorphism for k < 2n − 1.

1 2 ∼ 3 ∼ Z = π1S π2S → π3S → · · · n πnS is cyclic. . .

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Using the suspension theorem

Theorem (Freudenthal suspension theorem) The suspension map

n n+1 E : πk S → πk+1S

is a surjection for k = 2n − 1 and an isomorphism for k < 2n − 1.

1 2 ∼ 3 ∼ Z = π1S π2S → π3S → · · · n πnS is cyclic. . . and must be Z, because the degree of a map is homotopy invariant.

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

n πnS = Z

Degree two maps:

Paul VanKoughnett Stable homotopy theory and geometry Is everything else zero? NO!

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Homotopy groups of spheres

n 9 0 0 0 0 0 0 0 0 Z 8 0 0 0 0 0 0 0 Z 7 0 0 0 0 0 0 Z 6 0 0 0 0 0 Z 5 0 0 0 0 Z 4 0 0 0 Z 3 0 0 Z 2 0 Z 1 Z 0 0 0 0 0 0 0 0 n πk S 1 2 3 4 5 6 7 8 9 k

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Homotopy groups of spheres

n 9 0 0 0 0 0 0 0 0 Z 8 0 0 0 0 0 0 0 Z 7 0 0 0 0 0 0 Z 6 0 0 0 0 0 Z 5 0 0 0 0 Z 4 0 0 0 Z 3 0 0 Z 2 0 Z 1 Z 0 0 0 0 0 0 0 0 n πk S 1 2 3 4 5 6 7 8 9 k Is everything else zero? NO!

Paul VanKoughnett Stable homotopy theory and geometry 2 S 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.

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

The Hopf ﬁbration

3 2 S is the unit sphere in C , with coordinates z, w.

Paul VanKoughnett Stable homotopy theory and geometry 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.

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

The Hopf ﬁbration

3 2 S is the unit sphere in C , with coordinates z, w. 2 S is the Riemann sphere C ∪ {∞}, with coordinate λ.

Paul VanKoughnett Stable homotopy theory and geometry The ﬁber over any point is a circle. The ﬁber over two points are two linked circles.

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

The Hopf ﬁbration

3 2 S is the unit sphere in C , with coordinates z, w. 2 S is the Riemann sphere C ∪ {∞}, with coordinate λ. η : S3 → S2 sends (z, w) 7→ z/w.

Paul VanKoughnett Stable homotopy theory and geometry The ﬁber over two points are two linked circles.

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

The Hopf ﬁbration

3 2 S is the unit sphere in C , with coordinates z, w. 2 S 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.

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

The Hopf ﬁbration

3 2 S is the unit sphere in C , with coordinates z, w. 2 S 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 Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Homotopy groups of spheres

n 9 0 0 0 0 0 0 0 0 Z 8 0 0 0 0 0 0 0 Z 2 7 0 0 0 0 0 0 Z 2 2 6 0 0 0 0 0 Z 2 2 24 5 0 0 0 0 Z 2 2 24 2 4 0 0 0 Z 2 2 Z ⊕ 12 2 ⊕ 2 2 ⊕ 2 3 0 0 Z 2 2 12 2 2 3 2 0 Z Z 2 2 12 2 2 3 1 Z 0 0 0 0 0 0 0 0 n πk S 1 2 3 4 5 6 7 8 9 k

Paul VanKoughnett Stable homotopy theory and geometry The degree of a map Sn → Sn is an integer, because its ﬁbers are signed 0-manifolds. The ‘degree’ of a map Sk → Sn should just be its ﬁber – a stably framed (n − k)-manifold.

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Generalizing the degree

We want a more geometric characterization of these homotopy elements.

Paul VanKoughnett Stable homotopy theory and geometry The ‘degree’ of a map Sk → Sn should just be its ﬁber – a stably framed (n − k)-manifold.

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Generalizing the degree

We want a more geometric characterization of these homotopy elements. The degree of a map Sn → Sn is an integer, because its ﬁbers are signed 0-manifolds.

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Generalizing the degree

We want a more geometric characterization of these homotopy elements. The degree of a map Sn → Sn is an integer, because its ﬁbers are signed 0-manifolds. The ‘degree’ of a map Sk → Sn should just be its ﬁber – a stably framed (n − k)-manifold.

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Generalizing the degree

Paul VanKoughnett Stable homotopy theory and geometry n+k n+k n+k+1 We identify M ,→ R with M ,→ R ,→ R the inclusion into the ﬁrst n + k coordinates, together with the larger framing given by adding an upward-pointing normal vector everywhere.

Deﬁnition A framed cobordism of n-dimensional stably framed manifolds M, N is an (n + 1)-manifold W with ∂W =∼ M t N, together with a stable framing on W extending those on M and N.

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Stably framed manifolds

Deﬁnition A stably framed manifold is a manifold Mn with an embedding n n+k i : M ,→ R , k 0, and a trivialization of the normal bundle ∼ k Ni M = M × R .

Paul VanKoughnett Stable homotopy theory and geometry Deﬁnition A framed cobordism of n-dimensional stably framed manifolds M, N is an (n + 1)-manifold W with ∂W =∼ M t N, together with a stable framing on W extending those on M and N.

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Stably framed manifolds

Deﬁnition A stably framed manifold is a manifold Mn with an embedding n n+k i : M ,→ R , k 0, and a trivialization of the normal bundle ∼ k Ni M = M × R . n+k n+k n+k+1 We identify M ,→ R with M ,→ R ,→ R the inclusion into the ﬁrst n + k coordinates, together with the larger framing given by adding an upward-pointing normal vector everywhere.

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Stably framed manifolds

Deﬁnition A framed cobordism of n-dimensional stably framed manifolds M, N is an (n + 1)-manifold W with ∂W =∼ M t N, together with a stable framing on W extending those on M and N.

Paul VanKoughnett Stable homotopy theory and geometry Theorem There is a canonical isomorphism

fr ∼ k Ωn = πn+k S , k 0.

Homotopy elements can be described as framed n-manifolds!

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

The Pontryagin-Thom isomorphism

fr Ωn := {stably framed n-manifolds}/framed cobordism

Paul VanKoughnett Stable homotopy theory and geometry Homotopy elements can be described as framed n-manifolds!

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

The Pontryagin-Thom isomorphism

fr Ωn := {stably framed n-manifolds}/framed cobordism

Theorem There is a canonical isomorphism

fr ∼ k Ωn = πn+k S , k 0.

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

The Pontryagin-Thom isomorphism

fr Ωn := {stably framed n-manifolds}/framed cobordism

Theorem There is a canonical isomorphism

fr ∼ k Ωn = πn+k S , k 0.

Homotopy elements can be described as framed n-manifolds!

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

n 0-manifolds: πnS

n A 0-manifold in R is a set of points. A framing is a sign attached to each point: orient their normal bundles either with or against n the orientation of R .

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

n 0-manifolds: πnS

n A 0-manifold in R is a set of points. A framing is a sign attached to each point: orient their normal bundles either with or against n the orientation of R .

Paul VanKoughnett Stable homotopy theory and geometry Framings are classiﬁed by π1SO(n) = Z/2 for n ≥ 3 (and Z for n ≥ 2). The Hopf map S3 → S2 corresponds to S1 → S3 together with a basis for its normal bundle that twists once.

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

n 1-manifolds: πn+1S

1 n+1 A 1-manifold is just a circle. A framing on S → R is n determined by how a basis for R rotates as you go around the circle.

Paul VanKoughnett Stable homotopy theory and geometry The Hopf map S3 → S2 corresponds to S1 → S3 together with a basis for its normal bundle that twists once.

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

n 1-manifolds: πn+1S

1 n+1 A 1-manifold is just a circle. A framing on S → R is n determined by how a basis for R rotates as you go around the circle.

Framings are classiﬁed by π1SO(n) = Z/2 for n ≥ 3 (and Z for n ≥ 2).

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

n 1-manifolds: πn+1S

1 n+1 A 1-manifold is just a circle. A framing on S → R is n determined by how a basis for R rotates as you go around the circle.

Framings are classiﬁed by π1SO(n) = Z/2 for n ≥ 3 (and Z for n ≥ 2). The Hopf map S3 → S2 corresponds to S1 → S3 together with a basis for its normal bundle that twists once.

Paul VanKoughnett Stable homotopy theory and geometry Framed cobordisms require framed surgery. On any surface, we can do surgery to decrease the genus. . . but not all surgeries can be made framed. Pontryagin: whether or not we can do framed surgery on a 1-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 a n sphere. ∴ πn+2S = 0, n 0.

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

n 2-manifolds: πn+2S

One way to make cobordisms is through surgery.

Paul VanKoughnett Stable homotopy theory and geometry On any surface, we can do surgery to decrease the genus. . . but not all surgeries can be made framed. Pontryagin: whether or not we can do framed surgery on a 1-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 a n sphere. ∴ πn+2S = 0, n 0.

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

n 2-manifolds: πn+2S

One way to make cobordisms is through surgery. Framed cobordisms require framed surgery.

Paul VanKoughnett Stable homotopy theory and geometry Pontryagin: whether or not we can do framed surgery on a 1-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 a n sphere. ∴ πn+2S = 0, n 0.

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

n 2-manifolds: πn+2S

One way to make cobordisms is through surgery. Framed cobordisms require framed surgery. On any surface, we can do surgery to decrease the genus. . . but not all surgeries can be made framed.

Paul VanKoughnett Stable homotopy theory and geometry 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 a n sphere. ∴ πn+2S = 0, n 0.

Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

n 2-manifolds: πn+2S

One way to make cobordisms is through surgery. Framed cobordisms require framed surgery. On any surface, we can do surgery to decrease the genus. . . but not all surgeries can be made framed. Pontryagin: whether or not we can do framed surgery on a 1-cycle is determined by a map

φ : H1(M; Z/2) → Z/2.

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

n 2-manifolds: πn+2S

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 a n sphere. ∴ πn+2S = 0, n 0.

Paul VanKoughnett Stable homotopy theory and geometry Whether or not we can do framed surgery on M depends on the nature of this quadratic map. We can conclude that

2 ∼ πn+2S = Z/2, n 0.

A representative for the nontrivial class is given by the product of two nontrivially framed circles.

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 do surgery on two-cycles, we might not be able to on their sum.

Paul VanKoughnett Stable homotopy theory and geometry 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 do surgery on two-cycles, we might not be able to on their sum. Whether or not we can do framed surgery on M depends on the nature of this quadratic map. We can conclude that

2 ∼ πn+2S = Z/2, n 0.

A representative for the nontrivial class is given by the product of two nontrivially framed circles.

Paul VanKoughnett Stable homotopy theory and geometry Homotopy Homotopy groups of spheres Pontryagin-Thom Examples

Homotopy groups of spheres

Paul VanKoughnett Stable homotopy theory and geometry