a (brief) history of homotopy theoryweb.stanford.edu/~vogti/hismathseminar_homotopytheory.pdf ·...

A (Brief) History of Homotopy Theory

Isabel Vogt

April 26, 2013

Motivation

Why I’m giving this talk:

“Dealing with ideas in the form they were firstdiscovered often shines a light on the primalmotivation for them (...) Why did anyone dream upthe notion of homotopy, and homotopy groups?”

–Mazur

Why you might be interested in listening:

Homotopy as a tool preceds homotopy as a concept

Homotopy groups were very elusive

Ushers in transition from analysis to topology

Isabel Vogt A (Brief) History of Homotopy Theory

What is Homotopy?

A continuousdeformation fromone path to theother

Red and Blue curves: the images of amaps

f, g : [0, 1] = Y → X

f and g are homotopic (as maps fromY into X) if there exists a family of ctsmaps ht for t ∈ [0, 1] such that

h0 = f, h1 = g

∀x ∈ [0, 1], t 7→ ht(x) continuous


What is Homotopy?

f, g : X → Y , are homotopic ⇔

∃H : X × [0, 1]→ Y continuous

s.t. H(x, 0) = f(x), H(x, 1) = g(x)

Homotopy is an equivalence relation.

t = 0 t = 1

+

t = 0 t = 1

=

0 1/2 1

It is very important to remember this is dependent on thetopological spaces X → Y


Homotopy Groups

Now consider based homotopy classes of maps of the sphere intothe space X:

π1(X,x0) = [S1, ∗;X,x0]

Why does this form a group:

e : S1 → x0 : the constant map is an identity

f−1 : indicates transversing f in the opposite direction

[f ][g] = [f · g] : indicates transversing first f and then g

can check associativity

all of these respect homotopy


Origins of the concept of homotopy and the fundamental group

Analysis

Cauchy and mobility of path, 1825Riemann and connectivity, 1851Jordan and deformations of curves, 1866-1882

Analysis Situs

Poincare and a definition of π1 of a manifold, 1892


Higher homotopy groups

Hopf, and a nontrivial map S3 → S2 , 1931

Cech, introduction of abstract homotopy groups, 1932

Hurewicz, higher homotopy groups and homotopyequivalence, 1935

Eilenberg and obstruction theory, 1940


A summary

Cauchy, 1825

Riemann, 1851

Jordan, 1866-1882

Poincare, 1892

Hopf, 1931

Cech, 1932

Hurewicz, 1935

Eilenberg, 1940


A summary

Cauchy, 1825

Riemann, 1851

Jordan, 1866-1882

Poincare, 1892

Hopf, 1931

Cech, 1932

Hurewicz, 1935

Eilenberg, 1940

What’s going on?

Hmmm....

Oh.


The First Uses of Homotopy

Want to integrate some complex function f between theseendpoints, does the path matter?

Gauss and Poisson both note ∼ 1815 that it can!


Cauchy’s Work on Integration

I =

∫ X+iY

x0+iy0

f(z)dz

Identify complex numbers with points in the plane

Introduce a “mobile curve” joining x0 + iy0 and X + iY

x = φ(t), y = χ(t), monotone functions of t, continuouslydifferentiable

I = A+ iB =

∫ T

t0

f(φ(t) + iχ(t))[φ′(t) + iχ′(t)

]dt

Theorem

The result is independent of the choice of φ and χ if f iscomplex differentiable for x0 ≤ x ≤ X, y0 ≤ y ≤ Y


Cauchy’s Work on Integration

“If one wants to pass from one curve to another,which is not infinitely near the first, one can imagine athird mobile curve, which is variable in its shape, andhave it coincide successively and at different instanceswith both fixed curves.”


Riemann and Connectivity

Definition

1 A surface is simply-connected if every cross-cut (interiorarc joining boundaries) on the surface divides the surface.

2 A surface has connectivity number n if n− 1 cross-cutsturn it into a simply-connected surface.

surface connectivity number

sphere 1torus 3

g-hole torus 2g + 1

Note: this is not in general equivalent to saying a space isn-connected using πn = 0!


Riemann and Connectivity

Simply ConnectedIt does coincide with our standard definition of simplyconnected (π1(X) = 0)

By classification of surfaces, we only have the plane and thesphere that are simply connected ⇒ Jordan curve theorem

The quantifier is very necessary, there are even notnull-homotopic crosscuts that divide a surface which is notsimply connected


Jordan and deformations of curves

In 1966, published a paper on closed curves on surfaces:

Definition

Any two closed contours, drawn on a given surface, are calledreducible into one another, if one can pass from one to the otherby a progressive deformation.

“Any two contours drawn inthe plane are reducible to one an-other; however this is not true onany surface: for instance, on atorus a meridian and a parallelare two irreducible contours.”


Elementary Contours

S

A

C ′0

C ′′0

C ′1

C ′′1

P

b

Γ0 Γ1

•

•

•a0 •a1


Elementary Contours

Figure: Punctured 2-hole torus

n : the maximal number of non-intersecting closed curves thatdo not divide the surface (what we call the genus)


Elementary Contours

S

C ′0C ′′0

C ′1C ′′1

•a0 •

a1

Figure: Cut at these curves

C0, ..., Cn : the maximal curves that do not divide the surfaceC ′i and C ′′i : the two “sides” after the cuta0, ..., an : points on each curve


Elementary Contours

S

C ′0C ′′0

C ′1C ′′1

Γ0 Γ1•a0 •

a1

Figure: Cut at the curves Γi as well

Γ0, ...,Γn : closed curves through the ai


Elementary Contours

S

A

C ′0C ′′0

C ′1C ′′1b

Γ0 Γ1

••a0 •

a1

Figure: Choose a point on each boundary curve

b0, ..., bm : points on the boundary curves A0...Am


Elementary Contours

S

A

C ′0C ′′0

C ′1C ′′1

P

b

Γ0 Γ1

•

••a0 •

a1

Figure: Choose a point P and connect it to the ai and bi

P : point on the surface


Elementary Contours

Three types of “elementary contours”:

•

[PaiCiaiP ][Ci]

•

[PbiAibiP ][Ai]

•

[PaiΓiaiP ][Γi]


Jordan’s Idea

Claim (Jordan)

Every closed contour on the surface S is reducible to a uniquesequence of elementary contours.

In fact, he considers free deformations, so the sequence ofelementary contours must allow cyclic permutations

In hindsight, he has obtained a set of generators of π1S forS a genus n orientable surface with m boundary curves

With the relation:

[A0]...[Am−1][C0][Γ0][C0]−1[Γ0]

−1...[Cn−1][Γn−1][Cn−1]−1[Γn−1]

−1 ' 1


Why didn’t he realize he wrote down generators for thefundamental group?


Why didn’t he realize he wrote down generators for thefundamental group?

He lacked:

Continuous deformation of based loops (only requires theelementary contours to go through the point P )

Relation between “elementary contours”

The abstract group concept had not yet been formulated!

It is not obvious how to interpret the fundamental group asa permutation group


A Classification Theorem

Theorem (Jordan, 1866)

Two orientable surfaces with boundaries are homeomorphic ifand only if they have the same genus and the same number ofboundary curves.

His terminology for homeomorphic: “applicable, one to theother without tearing or duplication”


The Transition to Analysis Situs

The theory of integration provides an abelian structure∫α·β

f =

∫αf +

∫βf =

∫β·α

f

But if we integrated a multi-valued function this might notbe the case!

The permutations of values of a multi-valued functionalong α · β need not be the same as along β · αQuestions of underlying topology of a space began to bestudied in their own right!




f =

∫αf +

∫βf =

∫β·α

f


The permutations of values of a multi-valued functionalong α · β need not be the same as along β · α

Questions of underlying topology of a space began to bestudied in their own right!




f =

∫αf +

∫βf =

∫β·α

f


The permutations of values of a multi-valued functionalong α · β need not be the same as along β · αQuestions of underlying topology of a space began to bestudied in their own right!


Poincare and the Fundamental Group

Buildup to the fundamental group :

Consider unbranched multi-valued functions Fi on amanifold

If the functions are continued around a loop, they undergoa permutation

He shows, permutations along closed paths on a manifoldform a group

Depends on the functions Fi !

Let G be group of all such functions


Poincare and the Fundamental Group

Introduces null-homotopic paths:

If the point M describes an infinitely smallcontour on the manifold, the functions F willreturn to their original values. This remains trueif M describes a loop on the manifold, that is tosay, if it varies from M0 to M1 following anarbitrary path M0BM1, then describes aninfinitely small contour and returns from M1 toM0 traversing the same path M1BM0.

Composition of paths transversed in opposite directions:

M0AM1BM0 +M0BM1CM0 ≡M0AM1CM0

Emphasizes that M0AM1CM0 6≡M0CM1AM0


But...

Even Poincare gets some stuff wrong

M0BM0 ≡ 0 if the closed contour M0BM0

constitutes the complete boundary of a 2-dimensionalmanifold contained in the the manifold.

But he latter corrects it:

K ≡ 0 mod V ⇔ there is a simply connected region in V ofwhich K is boundary


The Fundamental Group

“In this way, on can imagine a group G satisfying the followingconditions:

1 For each closed contour M0BM0 there is a correspondingsubstitution S of the group

2 S reduces to the identical substitution if and onlyM0BM0 ≡ 0

3 If S and S′ correspond to the contours C and C ′ and ifC ′′ = C + C ′, the substitution corresponding to C ′′ will beSS′

This group G will be called the fundamental group of themanifold V . ”


Why Higher Homotopy Groups?

Good question!

Vienna, 1931

Cech presents a description of higherhomotopy groups

πnX := [Sn, ∗;X,x0]

No applications for the homotopygroups

Only one theorem

They were abelian (for n ≥ 2)


Why Higher Homotopy Groups?

Persuaded by others that they could not possibly containany additional information above the already knownabelian homology groups

Cech actually withdrew his paper

π1X encompasses more information than its abeliancounterpart H1(X)

H1(X;Z) ' π1X/[π1X,π1X]


Why Higher Homotopy Groups? - In Hindsight

There’s interesting stuff going on: Hopf fibration

Compare to H∗(Sn) = 0 for ∗ > n

Action of the fundamental group on higher homotopygroups

Eilenberg’s obstruction theory


Hurewicz and a redefinition, 1935

Let Ω(X,x0) be the loop space: set of based loops

S1 → X

with basepointx1 : S1 → x0

and the compact-open topology.

Definition/Proposition

1 Ωp(X,x0) = Ω(Ωp−1 (X,x0) , xp−1

)2 πp(X,x0) = π1

(Ωp−1 (X,x0) , xp−1

)


An Action of π1 on πn

Given a path α : [0, 1]→ X with endpoints a, b ∈ X, induces anisomorphism

πn(X, a) ' πn(X, b)

Depends only on the homotopy class of the map.

π1(X) acts by automorphisms on every πn(X,x0)


Homotopy Groups of Spheres

Brouwer: 1912: degrees of maps on a sphere

[f ]∼←→ degf

Hopf: 1925-1930: maps of spheres into spheres

“...may righty be called the starting point ofhomotopy theory”

–Dieudonne

The Hopf fibration: S3 p−→ S2

Hurewicz: 1935: πmS1 = 0 for m ≥ 2

Freudenthal: 1937: homotopy suspension

πi(Sn) ' πi+1(S

n+1) : i < 2n− 1


Brouwer and Degree

Let f : Sn → Sn, then

f∗ : Hn(Sn)→ Hn(Sn)

given byα ∈ Z 7→ dα

then, deg f = d

Brower’s definition:Involves simplicial approximation andcounting

Theorem (Brouwer)

The degree of a map is invariant under homotopy.


Brouwer and Degree

At ICM 1912, conjectured that the converse was also true:

Conjecture

For M a connected, compact, orientable n-dimensionalmanifold, and f and g two continuous maps

M → Sn

of the same degree, then f ' g.

Published a proof for M = S2

Incredibly long, dense, intricate, and imprecise!

4 reductions to finally apply a result of Klein on compactRiemann surfaces of genus 0

Questionable arguments


Hopf and maps of spheres

Proved Hopf’s conjecture

⇒ [f ]∼←→ degf

⇒ πn(Sn)∼−→ Z

Introduced the Hopf invariant H(f)

Used that to prove that

π3(S2) ' Z


The Hopf Fibration, 1931

S1 −→ S3 p−→ S2

Direct construction:Identify R4 with C2 and R3 with C× R

S3 := (z0, z1) : |z0|2 + |z1|2 = 1S2 := (z, x) : |z|2 + x2 = 1p(z0, z1) = (2z0z1, |z0|2 − |z1|2)

Clearly maps to S2, can check that

p(z0, z1) = p(w0, w1)⇔ (w0, w1) = λ(z0, z1), |λ| = 1


Freudenthal and stable homotopy groups

The based (reduced) suspension of X, ΣX is defined as

ΣX = X × [0, 1]/(X × 0 ∪X × 1 ∪ x0 × [0, 1])

There is a natural bijection:

[X,x0; ΩY, y1]∼−→ [ΣX,x0;Y, y0]

(Σ and Ω are adjoint functors)



[X,x0; ΩY, y1]∼−→ [ΣX,x0;Y, y0]

We also have a natural map (Y, y0)→ (ΩΣY, y1)

[X,x0;Y, y0] −→ [X,x0; ΩΣY, y1]

Putting the two together:

E : [X,x0;Y, y0] −→ [ΣX,x0; ΣY, y0]

In particular:

E : πr(Sn)→ πr+1(S

n+1)

Freudenthal showed that this is an isomorphism for r < 2n− 1and surjective for r = 2n− 1



Which means:

πr(Sn)

E−→ πr+1(Sn+1)...

E−→ πr+k(Sn+k)...

Are isomorphisms for k > r − 2n+ 1

The homotopy groups of spheres stabilize!


Samuel Eilenberg

Figure: Eilenberg


Samuel Eilenberg

Figure: Hmmm...?


Samuel Eilenberg

Figure: Eilenberg


Eilenberg ⊆ Bourbaki

On homotopy groups and fibre spaces

Eilenberg +MacLane

Met in 1940 in Ann Arbor, wrote 15 papers together until1954

Introduced category theory, the concepts of Hom, Ext,functor, natural transformation

Cohomology of groups

The relation between homotopy and homology

Eilenberg + Steenrod

Axiomatized homology and cohomology

Eilenberg + Cartan

Homological Algebra


Eilenberg Obstruction Theory

Given a mapX1 −→ Y

If Y is simply connected this extends to a map on X2.

Null-homotopy in Y gives a way to fill in 2-skeleton.

This generalizes:

Have a map defined on Xn, and (n+ 1)-cell σ

Attaching map gσ : Sn → Xn

f extends to σ if and only if f gσ is null-homotopic

Ie if the cochain c(f) : σ → [f gσ] ∈ Cn+1(X;πn(Y )) iszero

c(f): the obstruction cocycle

The immediate result: this is always possible if πn(Y ) = 0


References

“History of Topology” edited by I.M. James

“A Brief, Subjective History of Homology and HomotopyTheory in This Century” by Peter Hilton

“A History of Algebraic and Differential Topology” byDieudonne


a (brief) history of homotopy theoryweb.stanford.edu/~vogti/hismathseminar_homotopytheory.pdf ·...

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