Transversal homotopy theory Joint with Conor Smyth, inspired by Baez - PowerPoint PPT Presentation

Transversal homotopy theory Joint with Conor Smyth, inspired by Baez and Dolan Details in arXiv:0910.3322 March, 2010

Homotopy groups A homotopy of continuous maps f , g : X → Y is a continuous map h : X × [0 , 1] → Y such that h ( x , 0) = f ( x ) and h ( x , 1) = g ( x ).

Homotopy groups A homotopy of continuous maps f , g : X → Y is a continuous map h : X × [0 , 1] → Y such that h ( x , 0) = f ( x ) and h ( x , 1) = g ( x ). Fix basepoints ∗ . All maps and homotopies preserve basepoints.

Homotopy groups A homotopy of continuous maps f , g : X → Y is a continuous map h : X × [0 , 1] → Y such that h ( x , 0) = f ( x ) and h ( x , 1) = g ( x ). Fix basepoints ∗ . All maps and homotopies preserve basepoints. The n th homotopy group of a topological space X is π n ( X ) = { f : S n → X } / homotopy

Homotopy groups A homotopy of continuous maps f , g : X → Y is a continuous map h : X × [0 , 1] → Y such that h ( x , 0) = f ( x ) and h ( x , 1) = g ( x ). Fix basepoints ∗ . All maps and homotopies preserve basepoints. The n th homotopy group of a topological space X is π n ( X ) = { f : S n → X } / homotopy For n = 0 it is a set, for n = 1 a group, and for n ≥ 2 an abelian group where group operation arises from

Homotopy groups of spheres � 1 � 2 � 3 � 4 � 5 � 6 � 7 � 8 � 9 � 10 � 11 � 12 � 13 � 14 � 15 S 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S 1 Z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 S 2 Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 Z 2 Z 2 Z 12 � Z 2 Z 84 � Z 2 Z 2 0 Z Z 2 2 2 S 3 Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 Z 2 Z 12 � Z 2 Z 2 Z 84 � Z 2 Z 2 0 0 Z 2 2 3 5 S 4 Z � Z 12 Z 24 � Z 3 Z 15 Z 2 Z 120 � Z 12 � Z 2 Z 84 � Z 2 Z 2 Z 2 Z 2 Z 2 Z 2 0 0 0 Z 3 S 5 Z 2 Z 2 Z 24 Z 2 Z 2 Z 2 Z 30 Z 2 Z 2 Z 72 � Z 2 0 0 0 0 Z 3 S 6 Z 2 Z 2 Z 24 Z 2 Z 60 Z 24 � Z 2 Z 2 0 0 0 0 0 Z 0 Z 3 S 7 Z 2 Z 2 Z 24 Z 2 Z 120 Z 2 0 0 0 0 0 0 Z 0 0 S 8 Z 2 Z 2 Z 24 Z � Z 120 Z 2 0 0 0 0 0 0 0 Z 0 0 1 1 Table from en.wikipedia.org/wiki/Homotopy groups of spheres .

Homotopy groups of spheres � 1 � 2 � 3 � 4 � 5 � 6 � 7 � 8 � 9 � 10 � 11 � 12 � 13 � 14 � 15 S 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S 1 Z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 S 2 Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 Z 2 Z 2 Z 12 � Z 2 Z 84 � Z 2 Z 2 0 Z Z 2 2 2 S 3 Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 Z 2 Z 12 � Z 2 Z 2 Z 84 � Z 2 Z 2 0 0 Z 2 2 3 5 S 4 Z � Z 12 Z 24 � Z 3 Z 15 Z 2 Z 120 � Z 12 � Z 2 Z 84 � Z 2 Z 2 Z 2 Z 2 Z 2 Z 2 0 0 0 Z 3 S 5 Z 2 Z 2 Z 24 Z 2 Z 2 Z 2 Z 30 Z 2 Z 2 Z 72 � Z 2 0 0 0 0 Z 3 S 6 Z 2 Z 2 Z 24 Z 2 Z 60 Z 24 � Z 2 Z 2 0 0 0 0 0 Z 0 Z 3 S 7 Z 2 Z 2 Z 24 Z 2 Z 120 Z 2 0 0 0 0 0 0 Z 0 0 S 8 Z 2 Z 2 Z 24 Z � Z 120 Z 2 0 0 0 0 0 0 0 Z 0 0 1 ◮ π n ( S k ) = 0 for n < k 1 Table from en.wikipedia.org/wiki/Homotopy groups of spheres .

Homotopy groups of spheres � 1 � 2 � 3 � 4 � 5 � 6 � 7 � 8 � 9 � 10 � 11 � 12 � 13 � 14 � 15 S 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S 1 Z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 S 2 Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 Z 2 Z 2 Z 12 � Z 2 Z 84 � Z 2 Z 2 0 Z Z 2 2 2 S 3 Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 Z 2 Z 12 � Z 2 Z 2 Z 84 � Z 2 Z 2 0 0 Z 2 2 3 5 S 4 Z � Z 12 Z 24 � Z 3 Z 15 Z 2 Z 120 � Z 12 � Z 2 Z 84 � Z 2 Z 2 Z 2 Z 2 Z 2 Z 2 0 0 0 Z 3 S 5 Z 2 Z 2 Z 24 Z 2 Z 2 Z 2 Z 30 Z 2 Z 2 Z 72 � Z 2 0 0 0 0 Z 3 S 6 Z 2 Z 2 Z 24 Z 2 Z 60 Z 24 � Z 2 Z 2 0 0 0 0 0 Z 0 Z 3 S 7 Z 2 Z 2 Z 24 Z 2 Z 120 Z 2 0 0 0 0 0 0 Z 0 0 S 8 Z 2 Z 2 Z 24 Z � Z 120 Z 2 0 0 0 0 0 0 0 Z 0 0 1 ◮ π n ( S k ) = 0 for n < k ◮ π n ( S n ) ∼ = Z 1 Table from en.wikipedia.org/wiki/Homotopy groups of spheres .

Homotopy groups of spheres � 1 � 2 � 3 � 4 � 5 � 6 � 7 � 8 � 9 � 10 � 11 � 12 � 13 � 14 � 15 S 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S 1 Z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 S 2 Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 Z 2 Z 2 Z 12 � Z 2 Z 84 � Z 2 Z 2 0 Z Z 2 2 2 S 3 Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 Z 2 Z 12 � Z 2 Z 2 Z 84 � Z 2 Z 2 0 0 Z 2 2 3 5 S 4 Z � Z 12 Z 24 � Z 3 Z 15 Z 2 Z 120 � Z 12 � Z 2 Z 84 � Z 2 Z 2 Z 2 Z 2 Z 2 Z 2 0 0 0 Z 3 S 5 Z 2 Z 2 Z 24 Z 2 Z 2 Z 2 Z 30 Z 2 Z 2 Z 72 � Z 2 0 0 0 0 Z 3 S 6 Z 2 Z 2 Z 24 Z 2 Z 60 Z 24 � Z 2 Z 2 0 0 0 0 0 Z 0 Z 3 S 7 Z 2 Z 2 Z 24 Z 2 Z 120 Z 2 0 0 0 0 0 0 Z 0 0 S 8 Z 2 Z 2 Z 24 Z � Z 120 Z 2 0 0 0 0 0 0 0 Z 0 0 1 ◮ π n ( S k ) = 0 for n < k ◮ π n ( S n ) ∼ = Z ◮ π n ( S k ) ∼ = π n +1 ( S k +1 ) for 2 k ≥ n + 2 (Freudenthal, 1937) 1 Table from en.wikipedia.org/wiki/Homotopy groups of spheres .

Homotopy groups of spheres � 1 � 2 � 3 � 4 � 5 � 6 � 7 � 8 � 9 � 10 � 11 � 12 � 13 � 14 � 15 S 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S 1 Z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 S 2 Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 Z 2 Z 2 Z 12 � Z 2 Z 84 � Z 2 Z 2 0 Z Z 2 2 2 S 3 Z 2 Z 2 Z 12 Z 2 Z 2 Z 3 Z 15 Z 2 Z 12 � Z 2 Z 2 Z 84 � Z 2 Z 2 0 0 Z 2 2 3 5 S 4 Z � Z 12 Z 24 � Z 3 Z 15 Z 2 Z 120 � Z 12 � Z 2 Z 84 � Z 2 Z 2 Z 2 Z 2 Z 2 Z 2 0 0 0 Z 3 S 5 Z 2 Z 2 Z 24 Z 2 Z 2 Z 2 Z 30 Z 2 Z 2 Z 72 � Z 2 0 0 0 0 Z 3 S 6 Z 2 Z 2 Z 24 Z 2 Z 60 Z 24 � Z 2 Z 2 0 0 0 0 0 Z 0 Z 3 S 7 Z 2 Z 2 Z 24 Z 2 Z 120 Z 2 0 0 0 0 0 0 Z 0 0 S 8 Z 2 Z 2 Z 24 Z � Z 120 Z 2 0 0 0 0 0 0 0 Z 0 0 1 ◮ π n ( S k ) = 0 for n < k ◮ π n ( S n ) ∼ = Z ◮ π n ( S k ) ∼ = π n +1 ( S k +1 ) for 2 k ≥ n + 2 (Freudenthal, 1937) ◮ finite unless k = n or k = 2 m , n = 4 m − 1 (Serre, 1951) 1 Table from en.wikipedia.org/wiki/Homotopy groups of spheres .

Geometrical interpretation Lev Pontrjagin gave a geometric interpretation of the homotopy groups of spheres in terms of bordism theory of smooth manifolds (1938). Perhaps curiously for a topologist he was blind.

The Pontrjagin construction — preliminaries Theorem (Smooth approximation) A continuous map f : M → N of smooth manifolds, smooth on closed A ⊂ M, is homotopic rel A to a smooth map.

The Pontrjagin construction — preliminaries Theorem (Smooth approximation) A continuous map f : M → N of smooth manifolds, smooth on closed A ⊂ M, is homotopic rel A to a smooth map. Recall f is transverse to B if df ( T x M ) + T fx B = T fx N for all x ∈ f − 1 B . This implies f − 1 B is a submanifold and df induces Nf − 1 B ∼ = f ∗ NB . Not transverse Transverse

The Pontrjagin construction — preliminaries Theorem (Smooth approximation) A continuous map f : M → N of smooth manifolds, smooth on closed A ⊂ M, is homotopic rel A to a smooth map. Recall f is transverse to B if df ( T x M ) + T fx B = T fx N for all x ∈ f − 1 B . This implies f − 1 B is a submanifold and df induces Nf − 1 B ∼ = f ∗ NB . Not transverse Transverse Theorem (Transversal approximation) A smooth map f : M → N is homotopic to a map transverse to a compact submanifold B ⊂ N by a homotopy local to f − 1 B.

Framed submanifolds. . . Fix p ∈ S k (not the basepoint).

Framed submanifolds. . . Fix p ∈ S k (not the basepoint). ◮ f : S n → S k transverse to p ⇒ f − 1 p a codim k sbmfld

Framed submanifolds. . . Fix p ∈ S k (not the basepoint). ◮ f : S n → S k transverse to p ⇒ f − 1 p a codim k sbmfld ◮ h : S n × [0 , 1] → S k transverse to p ⇒ h − 1 p a bordism

Framed submanifolds. . . Fix p ∈ S k (not the basepoint). ◮ f : S n → S k transverse to p ⇒ f − 1 p a codim k sbmfld ◮ h : S n × [0 , 1] → S k transverse to p ⇒ h − 1 p a bordism S n × { 0 } h − 1 ( p ) S n × { 1 }

Framed submanifolds. . . Fix p ∈ S k (not the basepoint). ◮ f : S n → S k transverse to p ⇒ f − 1 p a codim k sbmfld ◮ h : S n × [0 , 1] → S k transverse to p ⇒ h − 1 p a bordism S n × { 0 } h − 1 ( p ) S n × { 1 } = f − 1 p × R k is trivial, with given Furthermore, Nf − 1 p ∼ = f ∗ Np ∼ trivialisation, i.e. f − 1 p is framed, and similarly for h .

. . . and collapse maps Given codimension k framed submanifold A ⊂ M construct ‘collapse’ map f : M → S k as follows:

. . . and collapse maps Given codimension k framed submanifold A ⊂ M construct ‘collapse’ map f : M → S k as follows: ◮ f ( a ) = p for all a ∈ A

. . . and collapse maps Given codimension k framed submanifold A ⊂ M construct ‘collapse’ map f : M → S k as follows: ◮ f ( a ) = p for all a ∈ A ◮ extend to tubular neighbourhood U of A by → R k ∼ = S k − ∗ U ∼ = NA ∼ = A × R k π 2 −

. . . and collapse maps Given codimension k framed submanifold A ⊂ M construct ‘collapse’ map f : M → S k as follows: ◮ f ( a ) = p for all a ∈ A ◮ extend to tubular neighbourhood U of A by → R k ∼ = S k − ∗ U ∼ = NA ∼ = A × R k π 2 − ◮ for x �∈ U set f ( x ) = ∗ and smooth rel nbhd of A .

. . . and collapse maps Given codimension k framed submanifold A ⊂ M construct ‘collapse’ map f : M → S k as follows: ◮ f ( a ) = p for all a ∈ A ◮ extend to tubular neighbourhood U of A by → R k ∼ = S k − ∗ U ∼ = NA ∼ = A × R k π 2 − ◮ for x �∈ U set f ( x ) = ∗ and smooth rel nbhd of A . The resulting f is transversal to p with f − 1 p = A .