nLab partition of unity

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 $X$ be a topological space. A partition of unity on $X$ is a collection $\{u_j\}_{j \in J}$ of continuous functions $u_j \colon X \to [0,1]$, $j\in J$ to the closed interval with its Euclidean metric topology such that $\sum_{j \in J} u_j(x) = 1$ for all $x\in X$.

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

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

$$\overline{u_j^{-1}(0,1]} \subset U_j.$$

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

A partition of unity is point finite if for every $x\in X$ there is only a finite number of $j\in J$ such that $u_j(x) \neq 0$.

A partition of unity is locally finite if for every $x\in X$ there is an open neighborhood $U$ of $x$ such that for only a finite number of $j\in J$ there is $x\in U$ such that $u_j(x) \neq 0$.

Often, the property of local finiteness is included in the definition of a partition of unity. This is harmless, since a result due to Michael R. Mather (Prop. below) says that for any partition of unity we can find a locally finite partition of unity with the same indexing set and whose induced cover refines the original induced cover.

Examples

Example

Consider $\mathbb{R}$ with its Euclidean metric topology.

Let $\epsilon \in (0,\infty)$ and consider the open cover

$$\{ (n-1-\epsilon , n+1 + \epsilon) \subset \mathbb{R} \}_{n \in \mathbb{Z} \subset \mathbb{R} }.$$

Then a partition of unity $\{ f_n \colon \mathbb{R} \to [0,1] \}_{n \in \mathbb{N}}$ subordinate to this cover is given by

$$f_n(x) \coloneqq \left\{ \array{ x - (n - 1) &\vert& n - 1 \leq x \leq n \\ 1- (x-n) &\vert& n \leq x \leq n+1 \\ 0 &\vert& \text{otherwise} } \right\}.$$

Properties

Existence on paracompact topological spaces

Proposition

(paracompact Hausdorff spaces equivalently admit subordinate partitions of unity)

Assuming the axiom of choice then:

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

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

2. Every open cover of $(X,\tau)$ 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!)

The case of non-Hausdorff spaces

Slightly more generally, a topological space (not necessarily Hausdorff) is fully normal if and only every open cover admits a subordinate partition of unity.

A T1-space is fully normal if and only if it is paracompact, in which case it is also Hausdorff.

For topological spaces that are not T1-spaces, the condition of being fully normal is strictly stronger than paracompactness.

The case of locales

A regular locale is fully normal if and only if it is paracompact.

The usual proof of the existence of partitions of unity goes through for such locales since it does not make any use of points.

Existence on smooth manifolds

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

Lemma

(open cover of smooth manifold admits locally finite refinement by closed balls)

Let $X$ be a smooth manifold and let $\{U_i \subset X\}_{i \in I}$ be an open cover. Then there exists cover

$$\left\{ B_0(\epsilon_j) \underoverset{\simeq}{\psi_j}{\to} V_j \subset X \right\}_{i \in J}$$

which is a locally finite refinement of $\{U_i \subset X\}_{i \in I}$ with each patch diffeomorphic to a closed ball in Euclidean space.

Proof

First consider the special case that $X$ is compact topological space.

Let

$$\left\{ \mathbb{R}^n \underoverset{\simeq}{\phi_j}{\longrightarrow} V_j \subset X \right\}$$

be a smooth atlas representing the smooth structure on $X$. The intersections

$$\left\{ U_i \cap V_j \right\}_{i \in I, j \in J}$$

still form an open cover of $X$. Hence for each point $x \in X$ there is $i \in I$ and $j \in J$ with $x \in U_i \cap V_j$. By the nature of the Euclidean topology, there exists a closed ball $B_x$ around $\phi_j^{-1}(x)$ in $\phi_j^{-1}(U_i \cap V_j) \subset \mathbb{R}^n$. Its image $\phi_j(B_x) \subset X$ is a neighbourhood of $x \in X$ diffeomorphic to a closed ball.

The interiors of these balls form an open cover

$$\left\{ Int(B_x) \subset X \right\}_{x \in X}$$

of $X$ which, by construction, is a refinement of $\{U_i \subset X\}_{i \in I}$. By the assumption that $X$ is compact, this has a finite subcover

$$\left\{ Int(B_l) \subset X \right\}_{l \in L}$$

for $L$ a finite set. Hence

$$\left\{ B_l \subset X \right\}_{l \in L}$$

is a finite cover by closed balls, hence in particular locally finite, and by construction it is still a refinement of the orignal cover. This shows the statement for $X$ compact.

Now for general $X$, notice that without restriction we may assume that $X$ is connected, for if it is not, then we obtain the required refinement on all of $X$ by finding one on each connected component.

But if a locally Euclidean paracompact Hausdorff space $X$ is connected, then it is sigma-compact and in fact admits a countable increasing exhaustion

$$V_0 \subset V_1 \subset V_2 \subset \cdots$$

by open subsets whose topological closures

$$K_0 \subset K_1 \subset K_2 \subset \cdots$$

exhaust $X$ by compact subspaces $K_n$ (by the proof of this prop.).

For $n \in \mathbb{N}$, consider the open subspace

$$V_{n+2} \setminus K_{n-1} \;\subset\; X$$

which canonically inherits the structure of a smooth manifold (this prop.). As above we find a refinement of the restriction of $\{U_i \subset X\}_{i \in I}$ to this open subset by closed balls and since the further subspace $K_{n+1}\setminus K_n$ is still compact (by this lemma) there is a finite set $L_n$ such that

$$\{B_{l_n} \subset V_{n+2} \setminus K_{n-1} \subset X \}_{l_n \in L_n}$$

is a finite cover of $K_{n+1} \setminus K_n$ by closed balls refining the original cover.

It follows that the union of all these

$$\left\{ B_{l_n} \subset X \right\}_{n \in \mathbb{N}, l_n \in L_n}$$

is a refinement by closed balls as required. Its local finiteness follows by the fact that each $B_{l_n}$ is contained in the “strip” $V_{n+2} \setminus K_{n-1}$, each strip contains only a finite set of $B_{l_n}$-s and each strip intersects only a finite number of other strips. (Hence an open subset around a point $x$ which intersects only a finite number of elements of the refined cover is given by any one of the balls $B_{l_n}$ that contain $x$.)

Proposition

(smooth manifolds admit locally finite smooth partitions of unity)

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

Proof

By lemma the given cover has a locally finite refinement by closed subsets diffeomorphic to closed balls:

$$\left\{ B_0(\epsilon_j) \underoverset{\simeq}{\psi_j}{\to} V_j \subset X \right\}_{j \in J} \,.$$

Given this, let

$$h_j \;\colon\; X \longrightarrow \mathbb{R}$$

be the function which on $V_j$ is given by a smooth bump function

$$b_j \;\colon\; \mathbb{R} \longrightarrow \mathbb{R}$$

with support $supp(b_j) = B_0(\epsilon_j)$:

$$h_j \;\colon\; x \mapsto \left\{ \array{ b_j(\psi_j^{-1}(x)) &\vert& x \in V_j \\ 0 &\vert& \text{otherwise} } \right. \,.$$

By the nature of bump functions this is indeed a smooth function on all of $X$. By local finiteness of the cover by closed balls, the function

$$h \;\colon\; X \longrightarrow \mathbb{R}$$

given by

$$h(x) \coloneqq \underset{j \in J}{\sum} h_j(x)$$

is well defined (the sum involves only a finite number of non-vanishing contributions) and is smooth. Therefore setting

$$f_j \;\coloneqq\; \frac{h_j}{h}$$

then

$$\left\{ f_j \right\}_{j \in J}$$

is a subordinate partition of unity by smooth functions as required.

From a partition of unity to a locally finite partition of unity

Definition

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

Proposition

(Mather, 1965)

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

The proof is Proposition A.2.8 in Dold 95 and Lemma 5.1.8 on page 301 of Engelking 89.

This implies that (locally finite) numerable covers are cofinal in induced covers arising from collections of functions as in the definition. In particular, given the Milnor classifying space $\mathcal{B}^M G$ of a topological group $G$, which comes with a countable family of ‘coordinate functions’ $\mathcal{B}^M G \to [0,1]$, 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 $G$-bundle where $G$ 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 $\{U_i \to X\}$ be a open cover and $\{\rho_i \in C(X,\mathbb{R})\}$ a collection of functions with

• $(x \notin U_i) \Rightarrow \rho_i(x) = 0$

• $\sum_i \rho_i = const_1$.

Write $C(\{U_i\}) : \Delta^{op} \to Top$ for the Cech nerve of the cover and $C(C(\{U_i\}), \mathbb{R})$ for the cosimplicial ring of functions on this simplicial topological space; and $(C_\bullet(C(\{U_i\}), \mathbb{R}), \delta)$ 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

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

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

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

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 $f$ is a cocycle in positive degree, in that $\delta f = 0$. Then define the $(n-1)$-cochain

$$\lambda_{i_1 \cdots i_n} := \sum_{i_0} \rho_{i_0} f_{i_0 i_1 \cdots i_n} \,.$$

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

With this definition we have

$$\delta \lambda = f \,.$$

To see this we compute

$$\begin{aligned} (\delta \lambda)_{i_1 \cdots i_{n+1}} & := \sum_{i_0} \rho_{i_0} \sum_{k=1}^n (-1)^k f_{i_0 i_1 \cdots i_{k-1} i_{k+1} \cdots i_{n+1}} \\ & = \pm \sum_{i_0} \rho_{i_0} f_{i_1 \cdots i_{n+1}} \\ & = f_{i_1 \cdots i_{n+1}} \end{aligned} \,,$$

where in the second step we used the condition $\delta f = 0$ 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.

• Michael R. Mather, Paracompactness and partitions of unity, Mimeographed notes, Cambridge (1964).

• Michael R. Mather, Products in spectral sequences and other topics, PhD dissertation, Cambridge (1966).

• Albrecht Dold, p. 354 in: Lectures on Algebraic Topology, Springer 1995 (doi:10.1007/978-3-642-67821-9, pdf, GoogleBooks)

• Ryszard Engelking, p. 301 in: General Topology, Sigma series in pure mathematics 6, Heldermann 1989 (ISBN 388538-006-4)

Discussion of partitions of unity in constructive mathematics is in

• Frank Waaldijk, section 3.1 of modern intuitionistic topology, 1996 (pdf)