Computing persistent Stiefel-Whitney classes
of line bundles

A vector bundle $\xi$ of dimension $d$ over $X$ consists of a topological space $A=A(\xi)$, the total space, a continuous map $\pi=\pi(\xi)\colon A\rightarrow X$, the projection map, and for every $x\in X$, a structure of $d$-dimensional vector space on the fiber $\pi^{-1}(\{x\})$. Moreover, $\xi$ must satisfy the local triviality condition: for every $x\in X$, there exists a neighborhood $U\subseteq X$ of $x$ and a homeomorphism $h\colon U\times\mathbb{R}^{d}\rightarrow\pi^{-1}(U)$ such that for every $y\in U$, the map $z\mapsto h(y,z)$ defines an isomorphism between the vector spaces $\mathbb{R}^{d}$ and $\pi^{-1}(\{y\})$.

Let $m\geq 0$. The Grassmann manifold $\mathcal{G}_{d}(\mathbb{R}^{m})$ is a set which consists of all $d$-dimensional linear subspaces of $\mathbb{R}^{m}$. It can be given a smooth manifold structure. When $d=1$, $\mathcal{G}_{1}(\mathbb{R}^{m})$ corresponds to the real projective space $\operatorname{\mathbb{P}}_{m-1}(\mathbb{R})$. In order to avoid mentioning $m$, it is convenient to consider the infinite Grassmannian. The infinite Grassmann manifold $\mathcal{G}_{d}(\mathbb{R}^{\infty})$ is the set of all $d$-dimensional linear subspaces of $\mathbb{R}^{\infty}$, where $\mathbb{R}^{\infty}$ is the vector space of sequences with a finite number of nonzero terms.

Let $X$ be a paracompact space. There exists a correspondence between the vector bundles over $X$ (up to isomorphism) and the continuous maps $X\rightarrow\mathcal{G}_{d}(\mathbb{R}^{\infty})$ (up to homotopy). Such a map is called a classifying map. When $X$ is compact, there exist an integer $m\geq 1$ such that a classifying map factorizes through

Consequently, in the rest of this paper, we shall consider that vector bundles are given as a continuous maps $X\rightarrow\mathcal{G}_{d}(\mathbb{R}^{m})$ or $X\rightarrow\mathcal{G}_{d}(\mathbb{R}^{\infty})$.

To each vector bundle $\xi$ over a paracompact base space $X$, one associates a sequence of cohomology classes

$\displaystyle w_{i}(\xi)\in H^{i}(X,\mathbb{Z}_{2}),~{}~{}~{}~{}~{}i\in\mathbb{N},$

called the Stiefel-Whitney classes of $\xi$. These classes satisfy:

• •

  Axiom 1: $w_{0}$ is equal to $1\in H^{0}(X,\mathbb{Z}_{2})$, and if $\xi$ is of dimension $d$, then $w_{i}(\xi)=0$ for $i>d$.

• •

  Axiom 2: if $f\colon\xi\rightarrow\eta$ is a bundle map, then $w_{i}(\xi)=\overline{f}^{*}w_{i}(\eta)$, where $\overline{f}^{*}$ is the map in cohomology induced by the underlying map $\overline{f}\colon X\rightarrow Y$ between base spaces.

• •

  Axiom 3: if $\xi,\eta$ are bundles over the same base space $X$, then for all $k\in\mathbb{N}$, $w_{k}(\xi\oplus\eta)=\sum_{i=0}^{k}w_{i}(\xi)\smile w_{k-i}(\eta)$, where $\oplus$ denotes the Withney sum, and $\smile$ denotes the cup product.

• •

  Axiom 4: if $\gamma_{1}^{1}$ denotes the tautological bundle of the projective line $\mathcal{G}_{1}(\mathbb{R}^{2})$, then $w_{1}(\gamma_{1}^{1})\neq 0$.

If such classes exists, then one proves that they are unique. A way to show that they actually exist relies on the cohomology of the Grassmannians.

The cohomology rings of the Grassmann manifolds admit a simple description: $H^{*}(\mathcal{G}_{d}(\mathbb{R}^{\infty}),\mathbb{Z}_{2})$ is the free abelian ring generated by $d$ elements $w_{1},...,w_{d}$. As a graded algebra, the degree of these elements are $|w_{1}|=1,...,|w_{d}|=d$. Hence we can write

$\displaystyle H^{*}(\mathcal{G}_{d}(\mathbb{R}^{\infty}),\mathbb{Z}_{2})\cong\mathbb{Z}_{2}[w_{1},...,w_{d}].$

The generators $w_{1},...,w_{d}$ can be seen as the Stiefel-Whitney classes of a particular vector bundle on $\mathcal{G}_{d}(\mathbb{R}^{\infty})$, called the tautological bundle. Now, for any vector bundle $\xi$, define

$\displaystyle w_{i}(\xi)=f_{\xi}^{*}(w_{i}),$

where $f_{\xi}\colon X\rightarrow\mathcal{G}_{d}(\mathbb{R}^{\infty})$ is a classifying map for $\xi$, and $f_{\xi}^{*}\colon H^{*}(X)\leftarrow H^{*}(\mathcal{G}_{d}(\mathbb{R}^{\infty}))$ the induced map in cohomology. This construction yields the Stiefel-Whitney classes.

The Stiefel-Whitney classes are invariants of isomorphism classes of vector bundles, and carry topological information. Their main interpretation is the following: the Stiefel-Whitney classes are obstructions to the existence of nowhere vanishing sections of vector bundles. Let us explain this result. A section of a vector bundle $\pi(\xi)\colon A\rightarrow X$ is a continuous map $s\colon X\rightarrow A$ such that $s(x)\in\pi^{-1}(\{x\})$ for all $x\in X$. It is nowhere vanishing if $s(x)\neq 0$ for all $x\in X$, where $0$ denotes the origin of the vector space $\pi^{-1}(\{x\})$. Given $k$ sections $s_{1},\dots,s_{k}$, we say that they are independent if the family $\left(s_{1}(x),\dots,s_{k}(x)\right)$ is free for all $x\in X$. Then the following result holds: if a vector bundle $\xi$ of dimension $d$ admits $k$ independent and nowhere vanishing sections, then the top $k$ Stiefel-Whitney classes $w_{d}(\xi),\dots,w_{d-k+1}(\xi)$ are zero.

Another property that we will use in this paper is the following: the first Stiefel-Whitney class detects orientability. More precisely, the first Stiefel-Whitney class $w_{1}(\xi)$ is zero if and only if the vector bundle $\xi$ is orientable. In the same vein, if $X$ is a compact manifold and $\tau$ its tangent bundle, then the manifold $X$ is orientable if and only if $w_{1}(\tau)=0$.

In this paper, we will particularly study line bundles, that is, vector bundles of dimension $d=1$. As a consequence of being an obstruction to nowhere vanishing sections, a line bundle $\xi$ on any topological space $X$ is trivial if and only if $w_{1}(\xi)=0$. More generally, the first Stiefel-Whitney class establishes a bijection between the isomorphism classes of line bundles over $X$ and its first cohomology group $H^{1}(X)$ over $\mathbb{Z}_{2}$. As an example, the circle $\operatorname{\mathbb{S}}_{1}$ has cohomology group $H^{1}(\operatorname{\mathbb{S}}_{1})=\mathbb{Z}_{2}$, hence admits only two isomorphism classes of line bundles. As another example, the sphere $\operatorname{\mathbb{S}}_{2}$ has trivial cohomology group $H^{1}(\operatorname{\mathbb{S}}_{2})=0$, hence only admits trivial line bundles.

A simplicial complex is a set $K$ such that there exists a set $V$, the set of vertices, with $K$ a collection of finite and non-empty subsets of $V$, and such that $K$ satisfies the following condition: for every $\sigma\in K$ and every non-empty subset $\nu\subseteq\sigma$, $\nu$ is in $K$. The elements of $K$ are called faces or simplices of the simplicial complex $K$.

For every simplex $\sigma\in K$, we define its dimension $\dim(\sigma)=\mathrm{card}(\sigma)-1$. The dimension of $K$, denoted $\dim(K)$, is the maximal dimension of its simplices. For every $i\geq 0$, the $i$-skeleton ${K}^{i}$ is defined as the subset of $K$ consisting of simplices of dimension at most $i$. Note that ${K}^{0}$ corresponds to the underlying vertex set $V$, and that ${K}^{1}$ is a graph. Given a graph $G$, the corresponding clique complex is the simplicial complex whose simplices are the sets of vertices of the cliques of $G$. We say that a simplicial complex $K$ is a flag complex if it is the clique complex of its 1-skeleton ${K}^{1}$.

Given a simplex $\sigma\in K$, its (open) star $\mathrm{St}\left(\sigma\right)$ is the set of all the simplices $\nu\in K$ that contain $\sigma$. The open star is not a simplicial complex in general. We also define its closed star $\overline{\mathrm{St}}\left(\sigma\right)$ as the smallest simplicial subcomplex of $K$ which contains $\mathrm{St}\left(\sigma\right)$.

For every $p\geq 0$, the standard $p$-simplex $\Delta^{p}$ is the topological space defined as the convex hull of the canonical basis vectors $e_{1},...,e_{p+1}$ of $\mathbb{R}^{p+1}$, endowed with the subspace topology. To each simplicial complex $K$ is attached a geometric realization. It is a topological space, denoted $\left|K\right|$, obtained by gluing the simplices of $K$ together. According to this construction, each simplex $\sigma\in K$ admits a geometric realization $\left|\sigma\right|$ which is a subset of $\left|K\right|$. The following set is a partition of $\left|K\right|$:

$$\left\{\left|\sigma\right|,\sigma\in K\right\}.$$

This allows to define the face map of $K$. It is the unique map $\mathcal{F}_{K}\colon\left|K\right|\rightarrow K$ that satisfies $x\in\left|\mathcal{F}_{K}(x)\right|$ for every $x\in\left|K\right|$.

If $\sigma$ is a face of $K$ of dimension at least 1, the subset $\left|\sigma\right|$ is canonically homeomorphic to the interior of the standard $p$-simplex $\Delta^{p}$, where $p=\dim(\sigma)$. This allows to define on $\left|K\right|$ the barycentric coordinates: for every face $\sigma=[v_{0},...,v_{p}]\in K$, the points $x\in\left|\sigma\right|$ can be written as

$$x=\sum_{i=0}^{p}\lambda_{i}v_{i}$$

with $\lambda_{0},...,\lambda_{p}>0$ and $\sum_{i=0}^{p}\lambda_{i}=1$.

If $X$ is any a topological space, a triangulation of $X$ consists of a simplicial complex $K$ together with a homeomorphism $h\colon X\rightarrow\left|K\right|$.

A simplicial map between simplicial complexes $K$ and $L$ is a map between geometric realizations $g\colon\left|K\right|\rightarrow\left|L\right|$ which sends each simplex of $K$ to a simplex of $L$ by a linear maps that sends vertices to vertices. In other words, for every $\sigma=[v_{0},...,v_{p}]\in K$, the subset $[g(v_{0}),...,g(v_{p})]$ is a simplex of $L$, and the map $g$ restricted to $\left|\sigma\right|\subset\left|K\right|$ can be written in barycentric coordinates as

$$\sum_{i=0}^{p}\lambda_{i}v_{i}~{}\longmapsto~{}\sum_{i=0}^{p}\lambda_{i}g(v_{i}).$$

(2)

A simplicial map $g\colon\left|K\right|\rightarrow\left|L\right|$ is uniquely determined by its restriction to the vertex sets $g_{|{K}^{0}}\colon{K}^{0}\rightarrow{L}^{0}$. Reciprocally, let $f\colon{K}^{0}\rightarrow{L}^{0}$ be a map between vertex sets which satisfies the following condition:

$$\forall\sigma\in K,~{}f(\sigma)\in L.$$

(3)

Then $f$ induces a simplicial map via barycentric coordinates, denoted $|f|\colon\left|K\right|\rightarrow\left|L\right|$. In the rest of this paper, a simplicial map shall either refer to a map $g\colon\left|K\right|\rightarrow\left|L\right|$ which satisfies Equation (2), to a map $f\colon{K}^{0}\rightarrow{L}^{0}$ which satisfies Equation (3), or to the induced map $f\colon K\rightarrow L$.

In general, a map $g$ may not satisfy the star condition. However, there is always a way to subdivise the simplicial complex $K$ in order to obtain an induced map which does. We describe this construction in the following paragraph.

We point out that, for some authors, such as Munkres (1984), the star condition is defined by the property $g\left(\left|\mathrm{St}\left(v\right)\right|\right)\subseteq\left|\mathrm{St}\left(w\right)\right|$. The defintion we used above, although harder to satisfy than this one, will be enough for our purposes.

Let $\Delta^{p}$ denote the standard $p$-simplex, with vertices denoted $v_{0},...,v_{p}$. The barycentric subdivision of $\Delta^{p}$ consists in decomposing $\Delta^{p}$ into $(p+1)!$ simplices of dimension $p$. It is a simplicial complex, whose vertex set corresponds to the points $\sum_{i=0}^{p}\lambda_{i}v_{i}$ for which some $\lambda_{i}$ are zero and the other ones are equal. Equivalently, one can see this new set of vertices as a the power set of the set of vertices of $\Delta^{p}$.

More generally, if $K$ is a simplicial complex, its barycentric subdivision $\text{sub}(K)$ is the simplicial complex obtained by subdivising each of its faces. The set of vertices of $\text{sub}(K)$ can be seen as a subset of the power set of the set of vertices of $K$.

If $g\colon\left|K\right|\rightarrow\left|L\right|$ is any map, there exists a canonical extended map $\left|\text{sub}(K)\right|\rightarrow\left|L\right|$, still denoted $g$. The simplicial approximation theorem states that for any two simplicial complexes $K,L$ with $K$ finite, and $g\colon\left|K\right|\rightarrow\left|L\right|$ any a continuous map, there exists $n\geq 0$ such that $g\colon\left|\mathrm{sub}^{n}\left(K\right)\right|\rightarrow\left|L\right|$ satisfies the star condition. As a consequence, such a map $g\colon\left|\mathrm{sub}^{n}\left(K\right)\right|\rightarrow\left|L\right|$ admits a simplicial approximation.

A persistence module over $T$ is a pair $(\mathbb{V},\mathbbm{v})$ where $\mathbb{V}=(V^{t})_{t\in T}$ is a family of $k$-vector spaces, and $\mathbbm{v}=(v_{s}^{t})_{s\leq t\in T}$ is a family of linear maps $v_{s}^{t}\colon V^{s}\leftarrow V^{t}$ such that:

• •

  for every $t\in T$, $v_{t}^{t}\colon V^{t}\leftarrow V^{t}$ is the identity map,

• •

  for every $r,s,t\in T$ such that $r\leq s\leq t$, $v_{r}^{s}\circ v_{s}^{t}=v_{r}^{t}$.

When there is no risk of confusion, we may denote a persistence module by $\mathbb{V}$ instead of $(\mathbb{V},\mathbbm{v})$. Given $\epsilon\geq 0$, an $\epsilon$-morphism between two persistence modules $(\mathbb{V},\mathbbm{v})$ and $(\mathbb{W},\mathbbm{w})$ is a family of linear maps $(\phi_{t}\colon V^{t}\rightarrow W^{t-\epsilon})_{t\geq\epsilon}$ such that the following diagram commutes for every $\epsilon\leq s\leq t$:

If $\epsilon=0$ and each $\phi_{t}$ is an isomorphism, the family $(\phi_{t})_{t\in T}$ is an isomorphism of persistence modules. An $\epsilon$-interleaving between two persistence modules $(\mathbb{V},\mathbbm{v})$ and $(\mathbb{W},\mathbbm{w})$ is a pair of $\epsilon$-morphisms $(\phi_{t}\colon V^{t}\rightarrow W^{t-\epsilon})_{t\geq\epsilon}$ and $(\psi_{t}\colon W^{t}\rightarrow V^{t-\epsilon})_{t\geq\epsilon}$ such that the following diagrams commute for every $t\geq 2\epsilon$:

The interleaving pseudo-distance between $(\mathbb{V},\mathbbm{v})$ and $(\mathbb{W},\mathbbm{w})$ is defined as

$$\mathrm{d}_{\mathrm{i}}\left(\mathbb{V},\mathbb{W}\right)=\inf\{\epsilon\geq 0,~{}\mathbb{V}\text{ and }\mathbb{W}\text{ are }\epsilon\text{-interleaved}\}.$$

A persistence module $(\mathbb{V},\mathbbm{v})$ is said to be pointwise finite-dimensional if for every $t\in T$, $V^{t}$ is finite-dimensional. This implies that we can define a notion of persistence barcode (Botnan and Crawley-Boevey, 2020). It comes from the algebraic decomposition of the persistence module into interval modules. Moreover, given two pointwise finite-dimensional persistence modules $\mathbb{V},\mathbb{W}$ with persistence barcodes $\mathrm{Barcode}\left(\mathbb{V}\right)$ and $\mathrm{Barcode}\left(\mathbb{W}\right)$, the so-called isometry theorem states that

$\displaystyle\mathrm{d}_{\mathrm{b}}\left(\mathrm{Barcode}\left(\mathbb{V}\right),\mathrm{Barcode}\left(\mathbb{W}\right)\right)=\mathrm{d}_{\mathrm{i}}\left(\mathbb{V},\mathbb{W}\right),$

where $\mathrm{d}_{\mathrm{i}}\left(\cdot,\cdot\right)$ denotes the interleaving distance between persistence modules, and $\mathrm{d}_{\mathrm{b}}\left(\cdot,\cdot\right)$ denotes the bottleneck distance between barcodes.

More generally, the persistence module $(\mathbb{V},\mathbbm{v})$ is said to be $q$-tame if for every $s,t\in T$ such that $s<t$, the map $v_{s}^{t}$ is of finite rank. The $q$-tameness of a persistence module ensures that we can still define a notion of persistence barcode, even though the module may not be decomposable into interval modules. Moreover, the isometry theorem still holds (Chazal et al., 2016).

A family of subsets $\mathbb{X}=(X^{t})_{t\in T}$ of $E$ is a filtration if it is non-decreasing for the inclusion, i.e. for any $s,t\in T$, if $s\leq t$ then $X^{s}\subseteq X^{t}$. Given $\epsilon\geq 0$, two filtrations $\mathbb{X}=(X^{t})_{t\in T}$ and $\mathbb{Y}=(Y^{t})_{t\in T}$ of $E$ are $\epsilon$-interleaved if, for every $t\in T$, $X^{t}\subseteq Y^{t+\epsilon}$ and $Y^{t}\subseteq X^{t+\epsilon}$. The interleaving pseudo-distance between $\mathbb{X}$ and $\mathbb{Y}$ is defined as the infimum of such $\epsilon$:

$$\mathrm{d}_{\mathrm{i}}\left(\mathbb{X},\mathbb{Y}\right)=\inf\{\epsilon,~{}\mathbb{X}\text{ and }\mathbb{Y}\text{ are }\epsilon\text{-interleaved}\}.$$

Filtrations of simplicial complexes and their interleaving distance are similarly defined: given a simplicial complex $S$, a filtration of $S$ is a non-decreasing family $\operatorname{\mathbb{S}}=(S^{t})_{t\in T}$ of subcomplexes of $S$. The interleaving pseudo-distance between two filtrations $(S_{1}^{t})_{t\in T}$ and $(S_{2}^{t})_{t\in T}$ of $S$ is the infimum of the $\epsilon\geq 0$ such that they are $\epsilon$-interleaved, i.e., for any $t\in T$, we have $S_{1}^{t}\subseteq S_{2}^{t+\epsilon}$ and $S_{2}^{t}\subseteq S_{1}^{t+\epsilon}$.

Applying the singular cohomology functor to a set filtration gives rise to a persistence module whose linear maps between cohomology groups are induced by the inclusion maps between sets. As a consequence, if two filtrations are $\epsilon$-interleaved, then their associated persistence modules are also $\epsilon$-interleaved, the interleaving homomorphisms being induced by the interleaving inclusion maps. As a consequence of the isometry theorem, if the modules are $q$-tame, then the bottleneck distance between their persistence barcodes is upperbounded by $\epsilon$. The same remarks hold when applying the simplicial cohomology functor to simplicial filtrations.

In order to visualize the evolution of a persistent Stiefel-Whitney class through the persistence module $(\mathbb{V},\mathbbm{v})$, we propose the following bar representation: the lifebar of $w_{i}(\mathbbm{p})$ is the set

$\displaystyle\left\{t\in T,w_{i}^{t}(\mathbbm{p})\neq 0\right\}.$

According to the last paragraph, the lifebar of a persistent class is an interval of $T$, of the form $[t^{\dagger},\sup(T))$ or $(t^{\dagger},\sup(T))$, where

$\displaystyle t^{\dagger}=\inf\left\{t\in T,w_{i}^{t}(\mathbbm{p})\neq 0\right\},$

with the convention $\inf(\emptyset)=\inf(T)$. In order to distinguish the lifebar of a persistent Stiefel-Whitney class from the bars of the persistence barcodes, we draw the rest of the interval hatched (see Figure 1).

We embed the Grassmannian $\mathcal{G}_{d}(\mathbb{R}^{m})$ into $M(\mathbb{R}^{m})$ via the application which sends a $d$-dimensional subspace $T\subset\mathbb{R}^{m}$ to its orthogonal projection matrix $P_{T}$. We can now see $\mathcal{G}_{d}(\mathbb{R}^{m})$ as a submanifold of $M(\mathbb{R}^{m})$. Recall that $M(\mathbb{R}^{m})$ is endowed with the Frobenius norm. According to this metric, $\mathcal{G}_{d}(\mathbb{R}^{m})$ is included in the sphere of center 0 and radius $\sqrt{d}$ of $M(\mathbb{R}^{m})$.

In the metric space $(M(\mathbb{R}^{m}),\left\|\cdot\right\|_{\mathrm{F}})$, consider the distance function to $\mathcal{G}_{d}(\mathbb{R}^{m})$, denoted $\mathrm{dist}\left(\cdot,\mathcal{G}_{d}(\mathbb{R}^{m})\right)$. Let $\mathrm{med}\left(\mathcal{G}_{d}(\mathbb{R}^{m})\right)$ denote the medial axis of $\mathcal{G}_{d}(\mathbb{R}^{m})$. It consists in the points $A\in M(\mathbb{R}^{m})$ which admit at least two projections on $\mathcal{G}_{d}(\mathbb{R}^{m})$:

	$\displaystyle\mathrm{med}\left(\mathcal{G}_{d}(\mathbb{R}^{m})\right)=\{A\in M(\mathbb{R}^{m}),\exists P,P^{\prime}$	$\displaystyle\in\mathcal{G}_{d}(\mathbb{R}^{m}),P\neq P^{\prime},$
		$\displaystyle\left\|A-P\right\|_{\mathrm{F}}=\left\|A-P\right\|_{\mathrm{F}}=\mathrm{dist}\left(A,\mathcal{G}_{d}(\mathbb{R}^{m})\right)\}.$