nLab
partition of unity

Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

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Basic concepts

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Extra stuff, structure, properties

Examples

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Contents

Idea

A partition of unity is a partition of the unit function on a topological space into a sum of continuous functions that are each non-zero only on small parts of the space.

Definition

Let XX be a topological space. A (point finite) partition of unity on XX is a collection {u j} jJ\{u_j\}_{j \in J} of continuous functions u j:X[0,1]u_j \colon X \to [0,1], jJj\in J to the closed interval with its Euclidean metric topology such that

  1. For each xXx\in X, there is only a finite number of jJj\in J such that u j(x)0u_j(x) \neq 0 (point finiteness condition);

  2. jJu j(x)=1\sum_{j \in J} u_j(x) = 1 for all xXx\in X.

A partition of unity defines an open cover of XX, consisting of the open sets u j 1(0,1]u_j^{-1}(0,1]. Call this the induced cover.

Sometimes (rarely) the condition that {u j} J\{u_j\}_J is point finite is dropped. In this case we refer to a non-point finite partition of unity (see red herring principle). In this case for each point of XX at most countably-many of the functions u ju_j are non-zero, and we have to interpret the sum in 1. above as being a convergent infinite series.

Given a cover 𝒰={U j} jJ\mathcal{U} = \{U_j\}_{j\in J} of a topological space (open cover or closed or neither), the partition of unity {u j} J\{u_j\}_J is subordinate to 𝒰\mathcal{U} if for all jJj\in J,

u j 1(0,1]¯U j. \overline{u_j^{-1}(0,1]} \subset U_j.

What this means is that the open sets u j 1(0,1]u_j^{-1}(0,1] form an open cover refining the cover 𝒰\mathcal{U}.

Properties

Existence on paracompact topological spaces

Proposition

(paracompact Hausdorff spaces equivalently admit subordinate partitions of unity)

Assuming the axiom of choice then:

Let (X,τ)(X,\tau) be a topological space. Then the following are equivalent:

  1. (X,τ)(X,\tau) is a paracompact Hausdorff space.

  2. Every open cover of (X,τ)(X,\tau) admits a subordinate partition of unity.

Similarly normal spaces are equivalently those such that every locally finite cover has a subordinate partition of unity (reference Bourbaki, Topology Generale - find this!)

Existence on smooth manifolds

Paracompact smooth manifolds even have smooth partitions of unity subordinate to any open cover (this follows from the existence of a smooth bump function on [1,1][-1,1]). It is not true, however, that analytic manifolds have analytic partitions of unity - the aforementioned bump function is smooth but not analytic:

Proposition

(smooth manifolds admit smooth partitions of unity)

Let XX be a paracompact smooth manifold. Then every open cover {U iX} iI\{U_i \subset X\}_{i \in I} has a subordinate partition of unity by functions {f i:U i} iI\{f_i \colon U_i \to \mathbb{R}\}_{i \in I} which are smooth functions.

Proof

Since XX is paracompact, the given open cover has a refinement to a locally finite cover, and then there exists one with the same index set (this prop.):

{U iX} iI. \left\{ U'_i \subset X \right\}_{i \in I} \,.

Recall that the smooth manifold XX is a normal topological space, because it is a paracompact Hausdorff space by definition and paracompact Hausdorff spaces are normal. Therefore we may invoke the shrinking lemma to obtain yet another open cover with the same index set

{V iX} iI \left\{ V_i \subset X \right\}_{i \in I}

with the property that

iI(V iCl(V i)U iU i). \underset{i \in I}{\forall} \left( V_i \subset Cl(V_i) \subset U'_i \subset U_i \right) \,.

Now let

{ nϕ jX} jJ \{\mathbb{R}^n \underoverset{}{\phi_j }{\to} X\}_{j \in J}

be an atlas exhibiting the smooth structure on the smooth manifold. Then by definition, for each point xXx \in X there is i xIi_x \in I and j xJj_x \in J such that

xV i xIm(ϕ j x). x \in V_{i_x} \cap Im(\phi_{j_x}) \,.

By the nature of the subspace topology, this intersection is still an open subset of Im(ϕ j x) nIm(\phi_{j_x}) \simeq \mathbb{R}^n. Therefore by the definition of the metric topology there exists a positive real number ϵ x\epsilon_{x} such that the open ball of radius ϵ x\epsilon_x around xx is an open neighbourhood of xx still contained in V i xV_{i_x}:

B x (ϵ x)V i xIm(ϕ j x). B^\circ_x(\epsilon_x) \subset V_{i_x} \cap Im(\phi_{j_x}) \,.

Let then

{B x (ϵ x)X} xX \left\{ B^\circ_x(\epsilon_x) \subset X \right\}_{x \in X}

be the collection of choices of such open balls, around each point of the manifold. This is an open cover which refines the cover {U iX} iI\{U'_i \subset X\}_{i \in I}. Again by paracompactness of XX, there exists a locally finite subcover, hence a subset of points SXS \subset X such that

{B s (ϵ s)X} sSX \left\{ B^\circ_s(\epsilon_s) \subset X \right\}_{s \in S \subset X}

is a locally finite open cover of XX.

Let then

{ nAAb sAA} sSX \left\{ \mathbb{R}^n \overset{\phantom{AA}b_s\phantom{AA}}{\longrightarrow} \mathbb{R} \right\}_{s \in S \subset X}

a set of smooth bump functions whose support is the topological closure of the chosen open ball around ss, regarded now as a subspace of the corresponding jj-th copy of n\mathbb{R}^n:

Supp(b s)=Cl(B s (ϵ s)). Supp(b_s) = Cl\left( B^\circ_s(\epsilon_s) \right) \,.

Hence the smooth bump functions b sb_s vanish on n\Cl(B s (ϵ s))\mathbb{R}^n \backslash Cl(B^\circ_s(\epsilon_s)), such that their extension by zero to functions b^ s\hat b_s on all of XX

n b s ϕ i s b^ s X \array{ \mathbb{R}^n &\overset{b_s}{\longrightarrow}& \mathbb{R} \\ {}^{\mathllap{\phi_{i_s}}}\downarrow & \nearrow_{\mathrlap{\hat b_s}} \\ X }

are still smooth functions: b^ sC (X,)\hat b_s \in C^\infty(X,\mathbb{R}).

Now by local finiteness of both the cover {U iX} iI\{U'_i \subset X\}_{i \in I} and of the cover {B s (ϵ s)} sS\{B^\circ_s(\epsilon_s)\}_{s \in S} we have that the sum

iIsSU ib^ s(x) \underset{ {i \in I} \atop {s \in S \cap U'_i }}{\sum} \hat b_s(x) \;\in\; \mathbb{R}

is well defined for each xXx \in X (only finitely many of the summands are non-zero) and by the covering property each point xx is contained in at least one of the patches of the cover, hence in the interior of the support of at least one of the b^ s\hat b_s and so

sSb^ s(x)>0 \underset{s \in S}{\sum} \hat b_s(x) \;\gt\; 0

for all xXx \in X. This means that it makes sense to define

f isSU ib^ siIsSU ib^ s f_i \;\coloneqq\; \frac{ \underset{s \in S \cap U'_i}{\sum} \hat b_s } { \underset{ { i \in I} \atop { s \in S \cap U'_i} }{\sum} \hat b_s }

and these are still smooth functions: f iC (X,)f_i \in C^\infty(X,\mathbb{R}).

We claim now that these form the required partition of unity subordinate to the original cover:

  1. By construction of the various open covers we have

    sSU i(Supp(b s)=Cl(B s (ϵ s))Cl(V i)U iU i) \underset{s \in S \cap U_i}{\forall} \left(Supp(b_s) = Cl\left(B^\circ_s(\epsilon_s)\right) \subset Cl(V_i) \subset U'_i \subset U_i\right)

    and hence

    Supp(f i)U i. Supp(f_i) \subset U_i \,.
  2. By construction of the functions f if_i we have

    iIf i(x)=iIsSU ib^ siIsSU ib^ s=1. \underset{i \in I}{\sum} f_i(x) = \underset{i \in I}{\sum} \frac{ \underset{s \in S \cap U_i}{\sum} \hat b_s } { \underset{ {i \in I} \atop {s \in S \cap U_i} }{\sum} \hat b_s } = 1 \,.

From a non-point finite partition of unity to a partition of unity

Definition

A collection of functions 𝒰={u i:X[0,1]}\mathcal{U} = \{u_i : X \to [0,1]\} such that every xXx\in X is in the support of some u iu_i. Then 𝒰\mathcal{U} is called locally finite if the cover u i 1(0,1]u_i^{-1}(0,1] (i.e. the induced cover) is locally finite.

Proposition

(Mather, 1965)

Let {u i} J\{u_i\}_J be a non-point finite partition of unity. Then there is a locally finite partition of unity {v i} iJ\{v_i\}_{i\in J} such that the induced cover of the latter is a refinement of the induced cover of the former.

(For a proof, see p.354 of Dold’s Lectures on algebraic topology. Google books link to page 354, which may or may not be visible)

This implies that (loc. finite) numerable covers are cofinal in induced covers arising from collections of functions as in the definition. In particular, given the Milnor classifying space MG\mathcal{B}^M G of a topological group GG, which comes with a countable family of ‘coordinate functions’ MG[0,1]\mathcal{B}^M G \to [0,1], has a numerable cover. This is shown by Dold to be a trivialising cover for the universal bundle constructed by Milnor, and so the universal bundle is numerable?.

Applications

Maps to geometric realizations

Partitions of unity can be used in constructing maps from spaces to geometric realizations of simplicial spaces (incl. simplicial sets) - for example a classifying map for a GG-bundle where GG is a Lie group.

Coboundaries for Cech cocycles

Partitions of unity can be used to give explicit coboundaries for the cocycles of the complex of functions on a cover.

Let {U iX}\{U_i \to X\} be a open cover and {ρ iC(X,)}\{\rho_i \in C(X,\mathbb{R})\} a collection of functions with

  • (xnotU i)ρ i(x)=0(x not \in U_i) \Rightarrow \rho_i(x) = 0

  • iρ i=const 1\sum_i \rho_i = const_1.

Write C({U i}):Δ opTopC(\{U_i\}) : \Delta^{op} \to Top for the Cech nerve of the cover and C(C({U i}),)C(C(\{U_i\}), \mathbb{R}) for the cosimplicial ring of functions on this simplicial topological space; and (C (C({U i}),),δ)(C_\bullet(C(\{U_i\}), \mathbb{R}), \delta) for the corresponding (normalized) cochain complex: its differential is the alternating sum of the pullbacks of functions along the face maps, i.e. along the restriction maps

δ= k(1) kδ k *. \delta = \sum_k (-1)^k \delta_{k}^* \,.

For instance for f={f i 1,i 2,,i nC(U i 1U i n+1)}f = \{f_{i_1, i_2, \cdots, i_n} \in C(U_{i_1} \cap \cdots \cap U_{i_{n+1}})\} a collection of functions in degree nn, we have

(δf) i 0i ni n+1= k=0 n+1(1) kf i 0i k1i k+1i n+1. (\delta f)_{i_0 \cdots i_n i_{n+1}} = \sum_{k = 0}^{n+1} (-1)^k f_{i_0 \cdots i_{k-1} i_{k+1} \cdots i_{n+1}} \,.

This cochain complex has vanishing cochain cohomology in positive degree. We can explicitly construct corresponding coboundaries using the partion of unity:

assume that with the above notation ff is a cocycle in positive degree, in that δf=0\delta f = 0. Then define the (n1)(n-1)-cochain

λ i 1i n:= i 0ρ i 0f i 0i 1i n. \lambda_{i_1 \cdots i_n} := \sum_{i_0} \rho_{i_0} f_{i_0 i_1 \cdots i_n} \,.

Here in the summands on the right the product is defined on U i 0U i 1U i nU_{i_0} \cap U_{i_1} \cap \cdots \cap U_{i_n} and extended as 0 to all of U i 1U i nU_{i_1} \cap \cdots \cap U_{i_n}.

With this definition we have

δλ=f. \delta \lambda = f \,.

To see this we compute

(δλ) i 1i n+1 := i 0ρ i 0 k=1 n(1) kf i 0i 1i k1i k+1i n+1 =± i 0ρ i 0f i 1i n+1 =f i 1i n+1, \begin{aligned} (\delta \lambda)_{i_1 \cdots i_{n+1}} & := \sum_{i_0} \rho_{i_0} \sum_{k=1}^n (-1)^k f_{i_0 i_1 \cdots i_{k-1} i_{k+1} \cdots i_{n+1}} \\ & = \pm \sum_{i_0} \rho_{i_0} f_{i_1 \cdots i_{n+1}} \\ & = f_{i_1 \cdots i_{n+1}} \end{aligned} \,,

where in the second step we used the condition δf=0\delta f = 0 and in the last step we used the property of the partition of unity.

This construction is used a lot in Cech cohomology. For instance it can be used to show in Chech cocycles that every principal bundle admits a connection on a bundle (see there for the details).

References

  • Albrecht Dold, Partitions of unity in the theory of fibrations, Ann. of Math. 78. (1963), 223-255.

  • Albrecht Dold, Lectures on algebraic topology, Springer Classics in Mathematics (1980), p.354.

  • M. Mather, Paracompactness and partitions of unity, PhD thesis, Cambridge (1965).

Discussion of partitions of unity in constructive mathematics is in

Revised on May 12, 2017 13:36:55 by Urs Schreiber (185.46.137.15)