nLab embedding of smooth manifolds into formal duals of R-algebras

Redirected from "Milnor duality".
Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Algebra

Contents

Idea

The functor which sends

  1. smooth manifolds to their algebras of smooth functions with values in the real numbers, regarded just as \mathbb{R}-algebras (instead of, say smooth algebras aka “C C^\infty-rings”),

  2. smooth functions between these manifolds to the corresponding pullback/precomposition algebra homomorphisms between these function algebras

turns out to be fully faithful, hence a full subcategory embedding of SmthMfd into the opposite of Alg \mathbb{R}Alg .

This is remarkable, because such a relation between spaces and their plain algebras of functions is (more) manifest only for affine varieties in algebraic geometry, where however it holds essentially by construction. In contrast, nothing in the usual definition of smooth manifolds manifestly suggests that they behave to some extent similarly to affine varieties with respect to \mathbb{R}-algebras of smooth functions. (See also at duality between algebra and geometry.)

Related “miracles” in differential geometry, revealing a maybe surprising algebro-geometric-nature, are the facts that:

Accordingly, the embedding of smooth manifolds into formal duals of \mathbb{R}-algebras allows to import some constructions and tools from algebraic geometry into differential geometry without strengthening the notion of “algebra” to something like smooth algebras.

This is useful and is used particularly for discussions in synthetic differential geometry — cf. e.g. the emphasis on “Weil algebras”, hence of would-be function algebras on infinitesimally thickened points, in Kolář, Michor & Slovák 1993 §35.

Statement

Lemma

(Milnor’s exercise)
For a smooth manifold XSmthMfdX \in SmthMfd, the evaluation map (from its underlying set to the hom-set)

X ev Hom Alg (C (X),) x (ϕϕ(x)) \begin{array}{l} X &\overset{ev}{\longrightarrow}& Hom_{Alg_{\mathbb{R}}} \big( C^\infty(X) ,\, \mathbb{R} \big) \\ x &\mapsto& \big( \phi \,\mapsto\, \phi(x) \big) \end{array}

is a bijection.

(due to Milnor & Stasheff (1974) Prob. 1-C (p. 11), detailed proof in Kolář, Michor & Slovák 1993 Lem. 35.8 & Cor. 35.9)

Proof

We provide a simplified proof using Hadamard's lemma. Suppose MM is a smooth manifold and ϕ:C (M)R\phi\colon C^\infty(M)\to\mathbf{R} is a homomorphism of real algebras.

If M=R nM=\mathbf{R}^n for some n0n\ge0, then set y i=ϕ(x i)y_i=\phi(x_i), where x i:R nRx_i\colon\mathbf{R}^n\to\mathbf{R} is the iith coordinate function. We have

ϕ(f)=ϕ(f(y)+ i(x iy i)g i)=f(y)+ i(ϕ(x i)y i)ϕ(g i)=f(y),\phi(f)=\phi(f(y)+\sum_i (x_i-y_i)\cdot g_i)=f(y)+\sum_i (\phi(x_i)-y_i)\cdot \phi(g_i)=f(y),

where the functions g ig_i are provided by Hadamard's lemma.

For a general MM, use Whitney's embedding theorem to embed MM into some R n\mathbf{R}^n. Without loss of generality we can assume the embedding to be proper so that the subset MR nM\subset\mathbf{R}^n is closed.

Consider the composition

C (R n)C (M)R,C^\infty(\mathbf{R}^n)\to C^\infty(M)\to \mathbf{R},

where the first homomorphism ρ\rho is given by restricting along the embedding. The composition is given by evaluating at some point pR np\in\mathbf{R}^n.

If pMp\notin M, we can construct a smooth function bb that vanishes on MM and does not vanish at pp. For example, we can take the smooth function with zero locus MM constructed by the smooth Tietze extension theorem, or simply use smooth bump functions. The function bb vanishes on MM and therefore belongs to the kernel of the map ρ\rho. The evaluation homomorphism at pMp\notin M does not vanish on the element bb, hence cannot factor through the map ρ\rho, a contradiction. Therefore, we must have pMp\in M.

Milnor’s exercise (Lem. ) implies — and also is the special case for X=*X = \ast (the point) of:

Theorem

The functor

SmthMfd Alg op X C (X) f f * Y C (Y) \begin{array}{ccc} SmthMfd &\longrightarrow& Alg_{\mathbb{R}}^{op} \\ X &\mapsto& C^\infty(X) \\ \mathllap{{}^f}\Big\downarrow && \Big\uparrow\mathrlap{{}^{{f^\ast}}} \\ Y &\mapsto& C^\infty(Y) \end{array}

which sends a smooth manifold (finite dimensional, paracompact, second countable) to (the formal dual of) its \mathbb{R}-algebra of smooth functions, C (X)C (X,)C^\infty(X) \coloneqq C^\infty(X, \mathbb{R}), is a full and faithful functor.

In other words, given a pair of smooth manifolds X,YX,Y then the operation of precomposition (pullback of functions) f *(g)gff^\ast(g) \coloneqq g \circ f constitutes a natural bijection

C (X,Y) Hom Alg (C (Y),C (X)) f f * \begin{array}{l} C^\infty(X,Y) &\overset{\sim}{\longrightarrow}& Hom_{Alg_{\mathbb{R}}}\big( C^\infty(Y) ,\, C^\infty(X) \big) \\ f &\mapsto& f^\ast \end{array}

between

  1. the smooth functions XYX \to Y,

  2. the \mathbb{R}-algebra homomorphisms C (X)C (Y)C^\infty(X)\leftarrow C^\infty(Y).

In particular, the diffeomorphisms between smooth manifolds are in natural bijection to the isomorphisms between their algebras of functions.

(Kolář, Slovák & Michor 1993 Cor. 35.10)
Proof

It is clear that the functor is faithful; we need to show that it is full, hence that for any \mathbb{R}-algebra homomorphism

ϕ:C (Y,)C (X,) \phi \;\colon\; C^\infty(Y, \mathbb{R}) \longrightarrow C^\infty(X, \mathbb{R})

there exists a smooth function f:XYf \colon X \to Y such that ϕ=f *\phi = f^\ast.

To that end, observe that given a point xXx \in X, the postcomposition of ϕ\phi with the evaluation map at xx

C (Y,)ϕC (X,)ev x C^\infty(Y, \mathbb{R}) \overset{ \phi }{\longrightarrow} C^\infty(X, \mathbb{R}) \overset{ ev_x }{\longrightarrow} \mathbb{R}

is an algebra homomorphism of the form assumed in Milnor’s exercise (Lem. ) and thus given uniquely by evaluation at some point f(x)Yf(x) \in Y

ev xϕ=ev f(x). ev_x \circ \phi \;=\; ev_{f(x)} \,.

This implies that ϕ\phi acts on any gC (Y)g \in C^\infty(Y) by

g ϕ (xϕ(g)(x)) = (xev x(ϕ(g))) = (xev f(x)(g)) = (xg(f(x))) = (x(gf)(x)), \begin{array}{rcl} g &\overset{\phi}{\mapsto}& \big( x \mapsto \phi(g)(x) \big) \\ &=& \Big( x \mapsto ev_x(\phi(g)) \Big) \\ &=& \Big( x \mapsto ev_{f(x)}(g) \Big) \\ &=& \Big( x \mapsto g\big(f(x)\big) \Big) \\ &=& \Big( x \mapsto (g \circ f)(x) \Big) \mathrlap{\,,} \end{array}

hence that it acts by precomposition with the assignment ff:

ϕ(g)=gf. \phi(g) \;=\; g \circ f \,.

It just remains to observe that this ff is necessarily smooth. But since ϕ\phi is given on all gC (Y)g \in C^\infty(Y) this way, to see that ff is smooth at some xXx \in X, choose any coordinate chart UYU \subset Y around f(x)Yf(x) \in Y and consider a gg which restricts to one of the coordinate functions x ix^i on UU. Then (gf) |f 1(U)=f |f 1(U) i(g \circ f)_{\vert f^{-1}(U)} = f^i_{\vert f^{-1}(U)} is the iith coordinate component of ff restricted to a neighbourhood of xx, and this being smooth for all ii means that ff is smooth around xx.

Remark

(attribution)
For the case of diffeomorphisms, Thm. was proven by Pursell (1952), following an announcement by Shanks (1951). This is the case that most reviews focus on, e.g. Grabowski (1978), Marsden, Ratiu & Abraham (2002), Grabowski (2005).

For the case that the domain is a point the statement is left as an exercise (without reference to Pursell) in Milnor & Stasheff (1974) Problem 1-C (p. 11), sometimes now referred to as “Milnor’s exercise” (Lem. ), with a detailed proof given in Kolář, Slovák & Michor (1993) 35.8-9

The general statement of Thm. appears as Kolář, Slovák & Michor (1993) 35.10.

Remark

The statement of theorem serves as the stepping-stone for generalizations of differential geometry such as to supergeometry. On the other hand, for transporting various applications familiar from algebraic geometry to differential geometry (such as Kähler differentials, see there) the above embedding is insufficient, and instead of just remembering the associative algebra structure, one needs to remember the smooth algebra-structure on algebras of smooth functions. See also at synthetic differential geometry.

Remark

If one drops standard regularity assumptions on manifolds then theorem may break. For instance allowing uncountably many connected components, then there are counterexamples (MO discussion).

The analogous statement in topology is:

duality between \;algebra and geometry

A\phantom{A}geometryA\phantom{A}A\phantom{A}categoryA\phantom{A}A\phantom{A}dual categoryA\phantom{A}A\phantom{A}algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand-KolmogorovAlg op\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C *,comm op\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}A\phantom{A}A\phantom{A}comm. C-star-algebraA\phantom{A}
A\phantom{A}noncomm. topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cptNCTopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C * op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}A\phantom{A}A\phantom{A}general C-star-algebraA\phantom{A}
A\phantom{A}algebraic geometryA\phantom{A}A\phantom{A}NCSchemes Aff\phantom{NC}Schemes_{Aff}A\phantom{A}A\phantom{A}almost by def.TopAlg op\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op} A\phantom{A}AA\phantom{A} \phantom{A}
A\phantom{A}commutative ringA\phantom{A}
A\phantom{A}noncomm. algebraicA\phantom{A}
A\phantom{A}geometryA\phantom{A}
A\phantom{A}NCSchemes AffNCSchemes_{Aff}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg fin,red op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}A\phantom{A}A\phantom{A}fin. gen.
A\phantom{A}associative algebraA\phantom{A}A\phantom{A}
A\phantom{A}differential geometryA\phantom{A}A\phantom{A}SmoothManifoldsSmoothManifoldsA\phantom{A}A\phantom{A}Milnor's exerciseTopAlg comm op\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}A\phantom{A}SuperSpaces Cart n|q\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}A\phantom{A}A\phantom{A}Milnor's exercise Alg 2AAAA op C ( n) q\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }A\phantom{A}A\phantom{A}supercommutativeA\phantom{A}
A\phantom{A}superalgebraA\phantom{A}
A\phantom{A}formal higherA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}
A\phantom{A}(super Lie theory)A\phantom{A}
ASuperL Alg fin 𝔤A\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}AALada-MarklA sdgcAlg op CE(𝔤)A\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}A\phantom{A}differential graded-commutativeA\phantom{A}
A\phantom{A}superalgebra
A\phantom{A} (“FDAs”)

in physics:

A\phantom{A}algebraA\phantom{A}A\phantom{A}geometryA\phantom{A}
A\phantom{A}Poisson algebraA\phantom{A}A\phantom{A}Poisson manifoldA\phantom{A}
A\phantom{A}deformation quantizationA\phantom{A}A\phantom{A}geometric quantizationA\phantom{A}
A\phantom{A}algebra of observablesA\phantom{A}space of statesA\phantom{A}
A\phantom{A}Heisenberg pictureA\phantom{A}Schrödinger pictureA\phantom{A}
A\phantom{A}AQFTA\phantom{A}A\phantom{A}FQFTA\phantom{A}
A\phantom{A}higher algebraA\phantom{A}A\phantom{A}higher geometryA\phantom{A}
A\phantom{A}Poisson n-algebraA\phantom{A}A\phantom{A}n-plectic manifoldA\phantom{A}
A\phantom{A}En-algebrasA\phantom{A}A\phantom{A}higher symplectic geometryA\phantom{A}
A\phantom{A}BD-BV quantizationA\phantom{A}A\phantom{A}higher geometric quantizationA\phantom{A}
A\phantom{A}factorization algebra of observablesA\phantom{A}A\phantom{A}extended quantum field theoryA\phantom{A}
A\phantom{A}factorization homologyA\phantom{A}A\phantom{A}cobordism representationA\phantom{A}

References

The case of the category of smooth manifolds with (just) diffeomorphisms between them is proved in

following an announcement in

  • M. E. Shanks, Rings of functions on locally compact spaces, 469th meeting of the AMS (1951) [pdf]

The statement for domain a point is due to:

with a proof spelled out in:

Expository accounts for the case of isomorphisms are in

The general statement and its proof is discussed in:

Discussion that takes the dual algebraic formulation as the very definition of smooth functions is in

The analog of the statement for real algebras refined to smooth algebras is theorem 2.8 in

Last revised on September 24, 2024 at 07:51:51. See the history of this page for a list of all contributions to it.