3.1 Differentiable Structures

To make sense of differentiation on a topological manifold, we need to provide additional structure. A pair $$(U,\varphi )$$ with U open in $$\textsf{M}^m$$ and $$\varphi$$ a homeomorphism from U to some open subset of $$\mathbb {R}^{m}$$ is called a chart. Then, for any $$p\in U$$, $$\varphi (p)=(x^1(p),\dots ,x^m(p))\in \mathbb {R}^m$$ are said to be local coordinates of p on U, with the inverse map sometimes called a parametrization. A pair of charts $$(U_1,\varphi _1)$$ and $$(U_2,\varphi _2)$$ is $$C^r$$-compatible when either $$U_1\cap U_2 = \emptyset$$ or $$\varphi _2\circ \varphi _1^{-1}:\varphi _1(U_1\cap U_2)\rightarrow \varphi _2(U_1\cap U_2)$$ is $$C^r$$-smooth. A collection of charts $$\{(U_i,\varphi _i)\}_{i\in \mathcal {I}}$$ such that $$\textsf{M}=\cup _{i\in \mathcal {I}}U_i$$ and all charts are $$C^r$$-compatible is called a $$C^r$$ -smooth atlas. Now we can define a $$C^r$$-smooth maximal atlas, denoted $$\bar{\mathscr {A}}$$, as the atlas that contains all charts $$C^r$$-compatible with the elements of $$\bar{\mathscr {A}}$$. Then, we say that $$\textsf{M}$$ is a $$C^r$$-smooth manifold, or simply a $$C^r$$ manifold, when $$\textsf{M}$$ admits a $$C^r$$-smooth maximal atlas $$\bar{\mathscr {A}}=\{(U_i,\varphi _i)\}_{i\in \mathcal {I}}$$ for some $$r\in \mathbb {N}\cup \{\infty \}\cup \{\omega \}$$. As such, one can call a topological manifold a $$C^0$$ manifold. See [27, Example 1.4, Example 1.31] for the construction of a smooth structure on $$\mathbb {S}^{n-1}$$. It is imperative, however, to point out that one rarely constructs atlases explicitly, their mere existence usually suffices. We speak of smooth manifolds when $$r\ge 1$$. There is no need to further classify these spaces as for $$r\ge 1$$, every $$C^r$$ manifold is $$C^r$$ diffeomorphic to a $$C^{\infty }$$ manifold [18, Theorem 2.2.10].

Given a smooth manifold $$\textsf{M}^m$$, then $$T_p\textsf{M}^m$$ denotes the tangent space of $$\textsf{M}^m$$ at the point $$p\in \textsf{M}^m$$, that is, $$T_p\textsf{M}^m=\{\dot{\gamma }(t)|_{t=0}:t\mapsto \gamma (t)\in \textsf{M}$$ is a curve differentiable at 0 with $$\gamma (0)=p\}$$. Now by considering equivalence classes of curves, with respect to $$\dot{\gamma }(t)_{t=0}$$ in coordinates, one can show that $$T_p\textsf{M}^m$$ has a m-dimensional vector space structure [27, Chap. 3]. The disjoint union $$T\textsf{M}^m=\sqcup _{p\in \textsf{M}^m}T_p\textsf{M}^m$$ is the tangent bundle of $$\textsf{M}^m$$ and is a smooth 2m-dimensional manifold itself [27, Proposition 3.18].

3.2 Submanifolds and Transversality

Given two smooth manifolds $$\textsf{M}^m$$ and $$\textsf{N}^n$$ with $$m\ge n$$, let $$G:\textsf{M}^m\rightarrow \textsf{N}^n$$ be a smooth map, then, $$q=G(p)\in \textsf{N}^n$$ is a regular value if the differential of G at p, $$DG_p:T_p \textsf{M}^m\rightarrow T_{G(p)}\textsf{N}^n$$, is surjective for all p such that $$G(p)=q$$. The points $$p\in \textsf{M}^m$$ where this surjectivity condition fails are called critical points of G on $$\textsf{M}^m$$. Now it follows from Sard’s theorem that regular values are genericFootnote 1 [27, Theorem 6.10]. Similarly, one can define regular points and critical values. The critical points of a smooth function $$g:\textsf{M}\rightarrow \mathbb {R}$$ are all points $$p\in \textsf{M}$$ such that $$Dg_p=0$$.

Again, let $$\textsf{M}$$ and $$\textsf{N}$$ be smooth manifolds and let $$G:\textsf{M}\rightarrow \textsf{N}$$ be a smooth map. The map G is called a smooth submersion when $$DG_p$$ is surjective for all $$p\in \textsf{M}$$. Similarly, G is a smooth immersion when $$DG_p$$ is injective for all $$p\in \textsf{M}$$. The map G is called a smooth embedding when G is an immersion and $$\textsf{M}$$ is homeomorphic to its image $$G(\textsf{M})$$. Let the subset $$\textsf{S}\subseteq \textsf{M}$$ be a manifold under the subspace topology, then, $$\textsf{S}$$ is said to be an embedded submanifold when the inclusion $$\iota _{\textsf{S}}:\textsf{S}\hookrightarrow \textsf{M}$$ is a smooth embedding. When irrelevant or unknown, the adjective “embedded” is omitted, the same is true for the declaration of a particular map $$\iota _{\textsf{S}}$$, the mere existence of some embedding usually suffices, i.e., we simply write $$\textsf{S}\hookrightarrow \textsf{M}$$.

Similar to the kernel of a linear map, the preimage of a regular value, under a smooth map $$G:\textsf{M}^m\rightarrow \textsf{N}^n$$, is a submanifold of dimension $$m-n$$, e.g., think of $$\mathbb {S}^{n-1}$$. The generalization beyond points (regular values) turns out to be remarkably useful.

Let $$G:\textsf{M}^m\rightarrow \textsf{N}^n$$ be a smooth map between smooth, boundaryless, manifolds and let $$\textsf{S}^s\subset \textsf{N}^n$$ be some smooth, boundaryless, submanifold. Then, G is said to be transverse to $$\textsf{S}^s$$, denoted $$G\pitchfork \textsf{S}^s$$, when either $$G(\textsf{M}^m)\cap \textsf{S}^s=\emptyset$$ or

\begin{aligned} \textrm{im}(DG_p) + T_{G(p)}\textsf{S}^s=T_{G(p)}\textsf{N}^n \end{aligned}
(3.1)

for all $$p\in G^{-1}(\textsf{S}^s)$$, see Fig. 3.1(i). Evidently but importantly, (3.1) trivially holds for G being a smooth submersion. When the transversality conditions holds, then by the implicit function theorem, the preimage of $$\textsf{S}^s$$, that is $$G^{-1}(\textsf{S}^s)$$, is also a submanifold, of dimension $$m-n+s$$ [16, p. 28]. Two submanifolds $$\textsf{S}_1\subseteq \textsf{N}$$ and $$\textsf{S}_2\subseteq \textsf{N}$$ are called transverse when the inclusion map of one of them is transverse to the remaining submanifold. This boils down to the condition that $$T_q\textsf{S}_1 + T_q \textsf{S}_2 = T_q\textsf{N}$$ for all $$q\in \textsf{S}_1\cap \textsf{S}_2$$, which has a clear geometric interpretation. A particularly useful implication is that when $$\textsf{S}_1\pitchfork \textsf{S}_2$$, then $$\textsf{S}_1\cap \textsf{S}_2$$ is a submanifold itself, with $$\textrm{codim}(\textsf{S}_1\cap \textsf{S}_2)=\textrm{codim}(\textsf{S}_1)+\textrm{codim}(\textsf{S}_2)$$ [16, p. 30].

Generalizing transversality to maps over domains with a boundary, i.e., $$\partial \textsf{M}\ne \emptyset$$, requires the restriction $$G\vert _{\partial \textsf{M}}:\partial \textsf{M}\rightarrow \textsf{N}$$ to be also transverse to $$\textsf{S}$$ for $$G^{-1}(S)$$ to be a manifold with boundary that satisfies $$\partial \{G^{-1}(S)\}=G^{-1}(S)\cap \partial \textsf{M}$$, see [16, p. 60], consider for example an ellipsoid in a disk as in Fig. 3.1(ii).

Then, the power of transversality is captured by the following two results.

Theorem 3.1

(Thom’s (parametric) transversality theorem [16, p. 68]) Let $$H:\textsf{T}\times \textsf{M}\rightarrow \textsf{N}$$ be a smooth map over the manifolds $$\textsf{T}$$, $$\textsf{M}$$ and $$\textsf{N}$$, with only $$\textsf{M}$$ possibly having a boundary. Define the family of maps $$\{ G_t: {t\in \textsf{T}} \}$$ by $$G_t(p)=H(t,p)$$ and let $$\textsf{S}\subseteq \textsf{N}$$ be a smooth submanifold. If $$H\pitchfork \textsf{S}$$ and $$H\vert _{\partial \textsf{M}}\pitchfork \textsf{S}$$, then, for almost every $$t\in \textsf{T}$$ also $$G_t\pitchfork \textsf{S}$$ and $$G_t\vert _{\partial \textsf{M}}\pitchfork \textsf{S}$$.

Using the language of jets, one can show that transversality generalizes regular values in the sense that transverse maps are also genericFootnote 2 [13, Theorem II.4.9, Corollary II.4.12]. Again, this corresponds to geometric intuition, drawing two lines at random in $$\mathbb {R}^2$$, they will be almost surely transverse. Technically, one can prove this by showing that for appropriate $$\textsf{T}$$, the map H is easily constructed to be a perturbation of G, yet, submersive, and hence transversal to any $$\textsf{S}$$.

Theorem 3.2

(Transversality homotopy extension theorem [16, pp. 72–73]) Let $$\textsf{S}$$ be a smooth submanifold of $$\textsf{N}$$, both without boundary, and consider a closed subset $$A\subseteq \textsf{M}$$ of the smooth manifold $$\textsf{M}$$. Let $$G:\textsf{M}\rightarrow \textsf{N}$$ be a smooth map such that $$G\pitchfork \textsf{S}$$ on A and $$G\vert _{\partial M}\pitchfork \textsf{S}$$ on $$A\cap \partial \textsf{M}$$. Then, there is a smooth map $$G':\textsf{M}\rightarrow \textsf{N}$$, homotopic to G such that $$G'\pitchfork \textsf{S}$$, $$G'\vert _{\partial \textsf{M}}\pitchfork \textsf{S}$$ and $$G=G'$$ on a neighbourhood of A.

By taking $$A=\partial \textsf{M}$$, Theorem 3.2 implies in particular that if $$g:\partial \textsf{M}\rightarrow \textsf{N}$$ is transverse to $$\textsf{S}\subseteq \textsf{N}$$ and g extends to $$\textsf{M}$$, then there is an extension $$G:\textsf{M}\rightarrow \textsf{N}$$ such that $$G\pitchfork \textsf{S}$$.

3.3 Bundles

Given two topological spaces $$\textsf{E}$$ and $$\textsf{B}$$, the total and base space, respectively, and a continuous surjective map $$\pi :\textsf{E}\rightarrow \textsf{B}$$, then, the triple $$(\pi ,\textsf{E},\textsf{B})$$ is called a vector bundle when for each $$b\in \textsf{B}$$ the fiber $$\pi ^{-1}(b)$$ has the structure of a real vector space, say $$\mathbb {R}^k$$. Moreover, for any $$b\in \textsf{B}$$, $$\textsf{E}$$ must be locally trivial over some open neighbourhood U of b, that is, there is a homeomorphism $$\varphi :\pi ^{-1}(U)\rightarrow U\times \mathbb {R}^k$$. Additionally, $$\varphi$$ should preserve the base and fiber structure, i.e., for $$\pi _U(U,\mathbb {R}^k)=U$$, $$\pi _{U}\circ \varphi = \pi$$ and for each $$b'\in U$$, $$\varphi (\pi ^{-1}(b'))$$ is linearly isomorphic to $$\mathbb {R}^k$$. Given a vector bundle $$\pi :\textsf{E}\rightarrow \textsf{B}$$, a section is a continuous map $$\sigma :\textsf{B}\rightarrow \textsf{E}$$ such that $$\pi \circ \sigma =\textrm{id}_{\textsf{B}}$$. Sections, denoted $$\Gamma (\textsf{E})$$, will aid in describing feedback laws later on. A section of interest is the zero section $$Z_{\pi }\in \Gamma (\textsf{E})$$, defined by mapping $$b\in \textsf{B}$$ to the zero element of the fiber $$\pi ^{-1}(b)$$, i.e., $$Z_{\pi }(\textsf{B}){\simeq }_t \textsf{B}$$, see also Fig. 3.2(i).

Example 3.1

(Vector bundle retraction) Consider a vector bundle $$\pi :\textsf{E}\rightarrow \textsf{B}$$, then $$Z_{\pi }(\textsf{B})$$ is a deformation retract of $$\textsf{E}$$. Conceptually, one retracts $$\textsf{E}$$ along the fibers to $$\textsf{B}$$. To show this, we first need to establish how to transition between two homeomorphisms $$\varphi :\pi ^{-1}(U)\rightarrow U\times \mathbb {R}^k$$ and $$\psi :\pi ^{-1}(V)\rightarrow V\times \mathbb {R}^k$$ with $$U\cap V \ne \emptyset$$. It follows from the structure preservation that $$\pi _{U\cap V}\circ \varphi \circ \psi ^{-1}=\pi _{U\cap V}$$ and as such for any $$b\in U\cap V$$ we have $$\varphi \circ \psi ^{-1}(b,x)=(b,g(b,x))$$ for some $$g:(U\cap V)\times \mathbb {R}^k\rightarrow \mathbb {R}^k$$. By the properties of $$\varphi ,\psi$$, the map $$x\mapsto g(b,x)$$ must be a linear bijection, that is, $$g(b,x)=A(b)x$$ for $$A(b)\in \textsf{GL}(k,\mathbb {R})$$. Evidently, this means that the transition $$\varphi \circ \psi ^{-1}$$ is linear in $$x\in \mathbb {R}^k$$. Now, as we can let $$\textsf{E}$$ be locally trivial over some neighbourhood U of $$b\in \textsf{B}$$, construct, in local coordinates the homotopy $$H(t,(b,x))=(t,(b,(1-t)x))$$. As we just saw, the local transition maps are also linear in x, as such, this construction is well-defined over the entire vector bundle and indeed yields a deformation retract from E onto $$Z_{\pi }(\textsf{B})$$.

Example 3.1 also shows why vector bundles admit zero sections; the transition $$\varphi \circ \psi ^{-1}$$ maps 0 to 0. For more information, see [18, Chap. 4].

Then, to characterize neighbourhoods of embedded submanifolds, it is useful to introduce the following. The vector bundle $$\pi _{\textsf{S}}:\textsf{S}\rightarrow \textsf{B}$$ is a subbundle of the vector bundle $$\pi :\textsf{E}\rightarrow \textsf{B}$$ when $$\textsf{S}\subseteq \textsf{E}$$, $$\pi _{\textsf{S}}=\pi \vert _{\textsf{S}}$$ and for all $$b\in \textsf{B}$$ one has that $$\pi _{\textsf{S}}^{-1}(b)=S\cap \pi ^{-1}(b)$$ is a linear subspace of the fiber $$\pi ^{-1}(b)$$. A subbundle of particular interest is the normal bundle of an embedded (or immersed) submanifold $$\textsf{M}^n\subseteq \mathbb {R}^d$$, denoted $$T\textsf{M}^{\perp }$$ or $$N\textsf{M}$$. This bundle is the orthogonal complement, under the Euclidean inner-product inherited from $$\mathbb {R}^d$$, of the tangent bundle in the embedding space. In particular, let $$\textsf{S}\subseteq \textsf{M}$$ be an embedded submanifold of a smooth manifold $$\textsf{M}$$. For simplicity assume $$\textsf{M}$$ is itself embedded into Euclidean space. Then, the normal bundle $$T\textsf{S}^{\perp }$$, with respect to $$T\textsf{S}$$, is given by $$T\textsf{S}^{\perp }=\sqcup _{s\in \textsf{S}}T_s\textsf{S}^{\perp }$$, for $$T_s\textsf{S}^{\perp }$$ the orthogonal complement, with respect to the Euclidean metric, to $$T_s\textsf{S}$$. Algebraically, $$T\textsf{S}^{\perp }\subseteq T\textsf{M}$$ is given by the quotient $$T\textsf{M}\vert _{\textsf{S}}/T\textsf{S}$$ [18, Chap. 4.2].

Now, following [27], consider some embedded submanifold $$\textsf{M}\subseteq \mathbb {R}^d$$ and define the map $$w:N\textsf{M}\rightarrow \mathbb {R}^d$$ by $$w(p,n)=p+n$$. Additionally, define the set $$V=\{(p,n)\in N\textsf{M}:\Vert n\Vert <\delta (p)\}$$ for some continuous function $$\delta :\textsf{M}\rightarrow \mathbb {R}_{>0}$$ such that V is open. Then, a neighbourhood U of $$\textsf{M}$$ in $$\mathbb {R}^d$$ that is diffeomorphic to w(V) is said to be a tubular neighbourhood of $$\textsf{M}$$. The Tubular neighbourhood theorem states that every embedded submanifold of $$\mathbb {R}^d$$ has a tubular neighbourhood [27, Theorem 6.24]. By exploiting the diffeomorphism $$w:V\rightarrow U$$, one can show the following, see also the $$\epsilon$$-neighbourhood theorem [16, p. 69] and Fig. 3.2(ii).

Proposition 3.1

(Tubular neighbourhood retraction [27, Proposition 6.25]) If U is a tubular neighbourhood of some smooth embedded submanifold $$\textsf{M}\subseteq \mathbb {R}^d$$, there is a smooth map $$r:U\rightarrow \textsf{M}$$ that is both a retraction and a submersion.

Proposition 3.1 will be useful in a later stage, it also allows for showing the following well-known result by Whitney, here, G is understood to be $$\delta$$-close to F.

Theorem 3.3

(Whitney’s Approximation Theorem [27, Theorem 6.26]) Let $$\textsf{M}$$ and $$\textsf{N}$$ be smooth manifolds, with only $$\textsf{M}$$ possibly having boundary. For any continuous map $$F:\textsf{M}\rightarrow \textsf{N}$$ there is a smooth map $$G:\textsf{M}\rightarrow \textsf{N}$$ homotopic to F.

Indeed, G can even be chosen such that $$G\pitchfork \textsf{S}$$ for any $$\textsf{S}\hookrightarrow \textsf{N}$$ [27, Theorem 6.36].

In what follows we mostly study continuous maps on smooth manifolds. However, most of the results at our disposal require some degree of smoothness to be proven, not merely continuity. At the same time, most of these results are invariant under homotopy. As such, Theorem 3.3 allows from bridging the gap between continuity and smoothness, which would be otherwise non-trivial. See also [18, Lemma 5.1.5].

We end this section with a comment on a generalization of vector bundles. Instead of demanding that the fibers are vector spaces, one can relax this to the demand that $$\pi ^{-1}(b)$$ is homeomorphic to some topological space $$\textsf{F}$$, while $$\textsf{E}$$ must still be locally trivial. In this case, the 4-tuple $$(\pi ,\textsf{E},\textsf{B},\textsf{F})$$ represents what is called a fiber bundle, e.g., $$(\pi ,\mathbb {S}^3,\mathbb {S}^2,\mathbb {S}^1)$$ is arguably the most influential example and is called the Hopf fibration. Importantly, for fiber bundles the existence of a continuous section is not immediate. A section does exist when $$\textsf{F}$$ is contractible [39, Part III]. We return to this topic in Chap. 6. Additionally, one can specify the structure group, e.g., instead of $$\textsf{GL}(k,\mathbb {R})$$ in Example 3.1, one could consider $$\textsf{O}(k,\mathbb {R})$$ and so forth.

3.4 Intersection and Index Theory

The practical classification of manifolds and maps over these manifolds relies on topological invariants (frequently, homotopy invariants). Using the previous results on transversality we provide a brief overview of the construction of the key topological invariant for this work: the Euler characteristic. In this chapter this in done through the lens of differential topology and in the next chapter we highlight arguments from algebraic topology. We point out that the material in this section is instrumental to appreciate later chapters and sections.

Let $$\textsf{M}$$ be a smooth compact manifold and let the smooth map $$G:\textsf{M}\rightarrow \textsf{N}$$ be transverse to the closed submanifold $$\textsf{S}\subseteq \textsf{N}$$. Suppose that $$\textrm{dim}(\textsf{M})+\textrm{dim}(\textsf{S})=\textrm{dim}(\textsf{N})$$ such that $$\textrm{dim}(G^{-1}(\textsf{S}))=0$$, i.e., $$G^{-1}(\textsf{S})$$ is a finite set of points. Let $$\#(\cdot )$$ denote the number or points, then, define the mod 2 intersection number of the pair $$(G,\textsf{S})$$ as

\begin{aligned} I_2(G,\textsf{S}) = \#(G^{-1}(\textsf{S})) \text { mod }2\in \mathbb {Z}/2\mathbb {Z}. \end{aligned}
(3.2)

For a general map, recall from Thom’s transversality theorem (Theorem 3.1) that transversality is generic such that for any $$G':\textsf{M}\rightarrow \textsf{N}$$, we let $$I_2(G',\textsf{S})=I_2(G,\textsf{S})$$ for any G homotopic to $$G'$$ that also satisfies the transversality condition $$G\pitchfork \textsf{S}$$. The following result shows why this is well-defined.

Theorem 3.4

(Mod 2 intersection homotopy invariance [16, p. 78]) Let $$\textsf{M}^m$$ be compact and let $$\textsf{S}^s\subseteq \textsf{N}^n$$ be a closed submanifold such that $$m+s=n$$, all manifolds being boundaryless and smooth. Then, for any pair of smooth maps $$G_1,G_2:\textsf{M}^m\rightarrow \textsf{N}^n$$ being homotopic, one has $$I_2(G_1,\textsf{S}^s)=I_2(G_2,\textsf{S}^s)$$.

As this result exemplifies upcoming material, a proof from [16] is collected.

Proof

By definition we have $$I_2(G',\textsf{S}^s)=I_2(G,\textsf{S}^s)$$ such that $$G{\simeq }_h G'$$ and $$G\pitchfork \textsf{S}^s$$. Hence, by the transitive property of homotopies, without loss of generality, let $$G_1\pitchfork \textsf{S}^s$$ and $$G_2\pitchfork \textsf{S}^s$$. Then, let $$H:[0,1]\times \textsf{M}^m\rightarrow \textsf{N}^n$$ be the homotopy between $$G_1$$ and $$G_2$$. By construction, $$H\vert _{\partial \{[0,1]\times \textsf{M}^m\}}$$ is transverse to $$\textsf{S}^s$$. By the homotopy transversality extension theorem (Theorem 3.2) we can assume that $$H\pitchfork \textsf{S}^s$$, i.e., as the map can be extended. This implies that $$H^{-1}(\textsf{S}^s)$$ is a one-dimensional manifold with boundary, defined via $$\partial \{H^{-1}(\textsf{S}^s)\}=(\{0\}\times G_1^{-1}(\textsf{S}^s))\cup (\{1\}\times G_2^{-1}(\textsf{S}^s))$$. Then the result follows by observing that one-dimensional manifolds have boundaries with an even number of points, motivating the definition of $$I_2(\cdot ,\cdot )$$, see [29, Appendix].

For example, if we construct the constant map $$C:\textsf{M}^m\rightarrow \textsf{N}^n$$, for $$n>0$$, defined by $$C:p\mapsto q'$$ for all $$p\in \textsf{M}^m$$ and some $$q'\in \textsf{N}^n\setminus \textsf{S}^s$$, then the transversality condition (3.1) holds trivially and indeed $$I_2(C,\textsf{S}^s)=0$$. We point out that exactly results like Theorem 3.4 are the reason why we discussed (transversality in the context of) manifolds with boundaries, i.e., see that the manifolds under consideration in that theorem are all boundaryless themselves while in the proof we exploit $$[0,1]\times \textsf{M}^m$$.

Again, one can use the inclusion map to define the intersection number between manifolds. Let both $$\textsf{S}_1\subseteq \textsf{N}$$ and $$\textsf{S}_2\subseteq \textsf{N}$$ be compact and boundaryless, and of complementary dimension, that is $$\textrm{dim}(\textsf{S}_1)+\textrm{dim}(\textsf{S}_2)=\textrm{dim}(\textsf{N})$$. Then, $$I_2(\textsf{S}_1,\textsf{S}_2)=I_2(\iota _{\textsf{S}_1},\textsf{S}_2)$$, for $$\iota _{\textsf{S}_1}:\textsf{S}_1\hookrightarrow \textsf{N}$$. If $$\textsf{S}_1\pitchfork \textsf{S}_2$$, then, by construction $$I_2(\textsf{S}_1,\textsf{S}_2)=\# (\iota ^{-1}_{\textsf{S}_1}(\textsf{S}_2)) \text { mod }2$$, which is simply the number of intersection points, modulo 2. Due to the homotopy invariance, when $$I_2(\textsf{S}_1,\textsf{S}_2)=1$$, any manifold homotopic to $$\textsf{S}_1$$ intersects $$\textsf{S}_2$$, e.g., consider two circles on the torus. In that sense, $$I_2(\cdot ,\cdot )$$ is robust.

Lemma 3.1

(Compact manifolds are generally not contractible) Compact, boundaryless manifolds $$\textsf{N}^n$$ with $$n\ge 1$$ are not contractible.

Proof

Suppose $$\textsf{N}^n$$ is contractible, then $$\textrm{id}_{\textsf{N}^n}$$ is homotopic to some constant map $$C:q\mapsto q'$$ over $$\textsf{N}^n$$ for some $$q'\in \textsf{N}^n$$. Then, given any compact $$\textsf{M}^m$$ and closed submanifold $$\textsf{S}^s\subseteq \textsf{N}^n$$ such that $$m+s=n$$, pick any smooth map $$G:\textsf{M}^m\rightarrow \textsf{N}^n$$ transverse to $$\textsf{S}^s$$. It follows by homotopy equivalence and composition that $$I_2(G,\textsf{S}^s)=I_2(\textrm{id}_{\textsf{N}^n}\circ G,\textsf{S}^s)=I_2(C\circ G,\textsf{S}^s)=0$$ (if needed, perturb $$\textsf{S}^s$$ not to contain $$q'$$). In particular, consider the setting of G being the identity map on $$\textsf{N}^n$$, forcing $$\textsf{S}^s$$ to be a point, i.e., $$s\in \textsf{N}^n\setminus \{q'\}$$. One is led to a contradiction as $$\textsf{N}^n\cap \textsf{S}^s=\{s\}$$, yet, $$I_2(\textrm{id}_{\textsf{N}^n},\textsf{S}^s)=0$$. For a reference, see [16, Exercises 5–6, p. 83].

Manifolds of dimension 0 can be contractible and indeed, the proof fails for $$n=0$$ as in that case $$I_2(C\circ G,\textsf{S}^s)$$ is not necessarily 0. When $$\partial \textsf{N}^n\ne \emptyset$$ contractability might hold. In that case, the proof fails as the homotopy invariance of $$I_2$$ does not necessarily carries over, e.g., consider the unit interval moving through to the unit circle.Footnote 3 One can also employ Poincaré duality (see Sect. 4) to show Lemma 3.1.

We introduce one more concept. Let $$G:\textsf{M}^m\rightarrow \textsf{N}^n$$ be a smooth map from a compact to a connected manifold with $$m=n$$, both boundaryless.Footnote 4 Then, for any $$q\in \textsf{N}^n$$, the number $$I_2(G,\{q\})$$ is the mod 2 degree of G, denoted $$\textrm{deg}_2(G)$$. Note that this number is a homotopy invariant by Theorem 3.4 and it is the same for any $$q\in \textsf{N}^n$$ [16, p. 81]. This means that given a regular value q, $$\textrm{deg}_2(G)=\# (G^{-1}(q))\text { mod }2$$.

An obstruction related to Lemma 3.1 holds for deformation retracts.

Lemma 3.2

(Compact manifolds admit no proper deformation retract) Let $$\textsf{M}$$ be a boundaryless, compact, connected, manifold. Then, there is no proper subset A of $$\textsf{M}$$ such that $$\textsf{M}$$ deformation retracts onto A.

Proof

For the sake of contradiction, assume there would be such a deformation retract, let $$r:\textsf{M}\rightarrow A$$ be the retraction and let $$\iota _{\textsf{A}}:A\hookrightarrow \textsf{M}$$ be the inclusion map. Now clearly, $$\textrm{deg}_2( \iota _{\textsf{A}}\circ r)=0$$ as one can consider the preimage of any point in $$\textsf{M}\setminus A$$. However, by assumption $$\iota _{\textsf{A}}\circ r {\simeq }_h\textrm{id}_{\textsf{M}}$$, such that $$0=\textrm{deg}_2( \iota _{\textsf{A}}\circ r)=\textrm{deg}_2(\textrm{id}_{\textsf{M}})\ne 0$$.

For example, by Lemma 3.2, the group $$\textsf{SO}(3,\mathbb {R})$$ cannot deformation retract onto $$\textsf{SO}(2,\mathbb {R})\hookrightarrow \textsf{SO}(3,\mathbb {R})$$. Spot again the boundaryless assumption.

Due to the binary evaluation, however, the insights gained from mod 2 intersection theory are limited. Endowing a space with an orientation allows for a different manipulation of $$\# G^{-1}(\textsf{S})$$ with far reaching ramifications.

A smooth manifold $$\textsf{M}^m$$ is said to be oriented when an admissible smooth orientation is selected (see below). All orientations will be with respect to the standard orientation on $$\mathbb {R}^m$$. Given a vector space $$V^m$$, if a basis B for $$V^m$$ is isomorphic to $$\mathbb {R}^m$$ by means of an orientation-preserving map, that is, a linear map with strictly positive determinant, then $$V^m$$ is said to be positively oriented under B. Otherwise, $$V^m$$ is negatively oriented. For manifolds with boundary, the orientation on the boundary is the one induced by the outward normal. For 1-dimensional manifolds with boundary, the domain of the local coordinates needs to be altered for this to work, that is, allow for mapping to $$\mathbb {R}_{\le 0}$$. Following [41, Example 21.8], given two charts $$(U_0,\varphi _0)$$ and $$(U_1,\varphi _1)$$ on [0, 1] defined by $$U_0=[0,1)$$, $$\varphi _0(x)=x$$ and $$U_1=(0,1]$$, $$\varphi _1(x)=x-1$$, observe that $$\varphi _1\circ \varphi _0^{-1}=x-1$$ and $$\varphi _0\circ \varphi _1^{-1}=x+1$$, as such, [0, 1] is orientable, that is, the transition functions preserve orientation. This does, however, mean that the outward induced orientation assigns $$-1$$ to the point 0 and $$+1$$ to the point 1 as shown in Fig. 3.3(i). This example will also aid in illustrating why the (oriented) degree is defined for boundaryless manifolds below.

It turns out that transversality naturally leads to an orientation on the manifold of interest. Let $$G:\textsf{M}\rightarrow \textsf{N}$$ be smooth, with $$\textsf{S}_{\textsf{N}}\subseteq \textsf{N}$$, $$\textsf{N}$$ and $$\textsf{S}_{\textsf{N}}$$ boundaryless, $$G\pitchfork \textsf{S}_{\textsf{N}}$$, $$G\vert _{\partial \textsf{M}}\pitchfork \textsf{S}_{\textsf{N}}$$, and $$\textsf{M},\textsf{N},\textsf{S}_{\textsf{N}}$$ all oriented. Let $$\textsf{S}_{\textsf{M}}=G^{-1}(\textsf{S}_{\textsf{N}})$$ and define $$T^{\perp }_p(\textsf{S}_{\textsf{M}};\textsf{M})$$ to be complementary to $$T_p\textsf{S}_{\textsf{M}}$$, that is, (3.3a) below must be satisfied, e.g., $$T_p^{\perp }(\textsf{S}_{\textsf{M}};\textsf{M})$$ is the orthogonal complement when a metric is defined (which is irrelevant for the proceeding discussion). By combining transversality (3.1) with $$T_p\textsf{S}_{\textsf{M}}$$ being the preimage of $$T_{G(p)}\textsf{S}_{\textsf{N}}$$ under $$DG_p$$, we obtain for all $$p\in G^{-1}(\textsf{S}_{\textsf{N}})$$ that

\begin{aligned} T_p^{\perp }(\textsf{S}_{\textsf{M}};\textsf{M})\oplus T_p \textsf{S}_{\textsf{M}}&= T_p\textsf{M} \end{aligned}
(3.3a)
\begin{aligned} DG_pT_p^{\perp }(\textsf{S}_{\textsf{M}};\textsf{M})\oplus T_{G(p)} \textsf{S}_{\textsf{N}}&= T_{G(p)}\textsf{N}. \end{aligned}
(3.3b)

Then, as by transversality, the kernel of $$DG_p$$ must be contained in $$T_p\textsf{S}_{\textsf{M}}$$ (in the preimage of $$T_{G(p)}\textsf{S}_{\textsf{N}}$$), (3.3b) induces an orientation on $$T_p^{\perp }(\textsf{S}_{\textsf{M}};\textsf{M})$$ and subsequently, via (3.3a) an orientation on $$T_p\textsf{S}_{\textsf{M}}$$, called the preimage-orientation.

Now the oriented intersection number is defined similar as $$I_2$$, yet we add up orientation numbers, with respect to the preimage-orientation, of all $$p\in G^{-1}(\textsf{S})=\textsf{S}_{\textsf{M}}$$, denoted $$I(G,\textsf{S})$$. In particular, let $$G:\textsf{M}\rightarrow \textsf{N}$$ be smooth and transverse to $$\textsf{S}\subseteq \textsf{N}$$. Under compactness of $$\textsf{M}$$, closedness of $$\textsf{S}$$ and a complementary dimension condition, $$G^{-1}(\textsf{S})$$ is a finite set of points. Now also assume that $$\textsf{M},\textsf{N}$$ and $$\textsf{S}$$ are oriented. As $$G\pitchfork \textsf{S}$$ one has for any $$p\in G^{-1}(\textsf{S})$$ that $$DG_p T_p \textsf{M}\oplus T_{G(p)}\textsf{S}=T_{G(p)}\textsf{N}$$. It follows from (3.3) that the orientation number at p is defined and equals $$+1$$ when the orientation on both sides of the equation agrees, whereas the orientation number is $$-1$$ when they do not. Note that 0-dimensional manifolds also have an orientation attached to them and that (3.3a) only implies that $$T_p^{\perp }(p;\textsf{M})$$ and $$T_p\textsf{M}$$ are isomorphic.

Most importantly, oriented intersection numbers are also homotopy invariant [16, p. 108]. Indeed, this result is similar to Theorem 3.4 and to that end we clarified above how to define orientation on a boundary. More specifically, in the mod 2 intersection setting, the proof of Theorem 3.4 relied on the sum of boundary points of one-dimensional manifolds, modulo 2, always being 0. Recalling Fig. 3.3, in the oriented case, the sum of the orientation numbers of exactly those points is also always 0. Given this homotopy invariance, we define the oriented intersection number of general smooth maps $$G'$$ via $$I(G',\textsf{S})=I(G,\textsf{S})$$ for a map G that does satisfy the aforementioned conditions and is homotopic to $$G'$$. Exploiting this observation regarding one-dimensional manifolds one can show the following.

Lemma 3.3

(Intersection number extension lemma [16, p. 108]) Let $$g:\partial \textsf{M}\rightarrow \textsf{N}$$ be a smooth map transverse to a closed, smooth, boundaryless submanifold $$\textsf{S}\subseteq \textsf{N}$$ of complementary dimension. If g extends to the entire smooth, compact manifold $$\textsf{M}$$, then $$I(g,\textsf{S})=0$$.

Proof

(sketch) Let $$G:\textsf{M}\rightarrow \textsf{N}$$ be the extension, by construction $$G\vert _{\partial \textsf{M}}\pitchfork \textsf{S}$$ and by Theorem 3.2 one can take $$G\pitchfork \textsf{S}$$ such that $$G^{-1}(\textsf{S})\subseteq \textsf{M}$$ is a one-dimensional manifold. The result follows by recalling that by transversality $$\partial \{ G^{-1}(\textsf{S})\}=g^{-1}(\textsf{S})$$.

Similar as before, for any smooth map between boundaryless manifolds $$G:\textsf{M}^m\rightarrow \textsf{N}^n$$ with $$m=n$$, $$\textsf{M}^m$$ compact and $$\textsf{N}^n$$ being connected,Footnote 5 we define the degree of G as $$\textrm{deg}(G)=I(G,\{q\})$$ for any $$q\in \textsf{N}^n$$ (recall Theorem 3.1). In this case the orientation number can be computed using the same reasoning as before, i.e., for $$p\in G^{-1}(q)$$ we check if $$DG_p$$ will preserve or reverse the orientation on $$T_p\textsf{M}^m$$. If $$\textsf{M}^m$$ is endowed with the canonical positive orientation, then, for any regular value q

\begin{aligned} \textrm{deg}(G) = \textstyle \sum \limits _{p\in G^{-1}(q)} \textrm{sgn}\,\textrm{det}(D G_p). \end{aligned}
(3.4)

The form of (3.4) goes back to Kronecker and the intuition of (3.4) is that the number $$\textrm{deg}(G)$$ represents how many times (net) the domain “wraps around” the codomain under the map G. See also Sect. 4 for the homological viewpoint.

Example 3.2

(Homotopies and the n-sphere) Consider the n-sphere $$\mathbb {S}^n\subset \mathbb {R}^{n+1}$$, a smooth manifold $$\textsf{X}$$ and a smooth map $$G:\textsf{X}\rightarrow \mathbb {S}^n$$. Now, let $$G':\textsf{X}\rightarrow \mathbb {S}^n$$ be such that $$\Vert G(p)-G'(p)\Vert _2< 2$$ for all $$p\in \textsf{X}$$. This condition implies that G is homotopic to $$G'$$ as $$H:[0,1]\times \textsf{X}\rightarrow \mathbb {S}^n$$ defined by

\begin{aligned} H(t,p) = \frac{(1-t)G(p)-tG'(p)}{\Vert (1-t)G(p)-tG'(p)\Vert _2} \end{aligned}

is the corresponding homotopy. This construction concurrently shows the robustness of $$\textrm{deg}(\cdot )$$. Note, the $$\Vert \cdot \Vert _2$$ condition is not necessary as on $$\mathbb {S}^1\subset \mathbb {R}^2$$ one can rotate the identity map to its negation. In fact, Hopf’s degree theorem states that if $$\textsf{M}$$ is compact, connected, boundaryless and oriented, then any continuous map $$g:\textsf{M}\rightarrow \mathbb {S}^n$$ is homotopic to $$g':\textsf{M}\rightarrow \mathbb {S}^n$$ if and only if $$\textrm{deg}(g)=\textrm{deg}(g')$$ [29, Sect. 7].

As before, given two submanifolds $$\textsf{S}_1$$ and $$\textsf{S}_2$$, we can also define $$I(\textsf{S}_1,\textsf{S}_2)$$ via their respective inclusion maps. Note, however, that by no means $$I(\textsf{S}_1,\textsf{S}_2)$$ is necessarily equal to $$I(\textsf{S}_2,\textsf{S}_1)$$. In particular, one can show [16, pp. 113–115] that if $$\textsf{S}^{s}$$ and $$\textsf{R}^r$$ are compact submanifolds of $$\textsf{N}^n$$ of complementary dimension, then,

\begin{aligned} I(\textsf{S}^s,\textsf{R}^r)= (-1)^{s\cdot r}I(\textsf{R}^r,\textsf{S}^s). \end{aligned}
(3.5)

We can now define the central invariant of this work. Let $$\Delta _{V}$$ denote the diagonal of $$V\times V$$, that is, $$\Delta _V=\{ (v,v) : v\in V\}$$, and let $$\textsf{M}$$ be a smooth, boundaryless, compact and orientable manifold, then, its Euler characteristic is defined as

\begin{aligned} \chi (\textsf{M}) = I(\Delta _{\textsf{M}},\Delta _{\textsf{M}}). \end{aligned}
(3.6)

Note, (3.5) implies that $$\chi (\textsf{M}^m)=0$$ when m is odd. One should interpret the self-intersection number (3.6) with the aforementioned transversality conditions and homotopy invariance taken into account. For example, for $$\textsf{M}=\mathbb {S}^1$$, think of $$\Delta _{\mathbb {S}^1}$$ as a particular circle on $$\mathbb {S}^1\times \mathbb {S}^1=\mathbb {T}^2$$. Then, $$\chi (\mathbb {S}^1)$$ captures to what extent a homotopy of $$\Delta _{\mathbb {S}^1}$$ remains entangled with $$\Delta _{\mathbb {S}^1}$$, see Fig. 3.3(ii). See also Sect. 4.2 for $$\chi (\textsf{M})$$ through the lens of algebraic topology. We will follow a constructive approach in showing how $$\chi (\textsf{M})$$ relates to qualitative properties of vector fields on $$\textsf{M}$$. We do not start with vector fields, but with maps. This simplifies the analysis and in contrast to combinatorial/algebraic approaches, this makes it possible to relate some upcoming material to discrete-time systems via time-one maps cf. Section 8.1.

Let $$G:\textsf{M}\rightarrow \textsf{M}$$ be a smooth map over a smooth, boundaryless, compact, orientable manifold $$\textsf{M}$$. The existence of fixed points of G is for instance captured by the Lefschetz number $$L(G)=I(\Delta _{\textsf{M}},\textrm{graph}(G))$$ being different from 0. Indeed, $$L(\textrm{id}_{\textsf{M}})=\chi (\textsf{M})$$. A map G is called a Lefschetz map when $$\textrm{graph}(G)\pitchfork \Delta _{\textsf{M}}$$, yielding robust fixed-point properties. Now, one can derive that G being Lefschetz over $$\textsf{M}^m$$ is equivalent to $$DG_p-I_m$$ being invertible (the reader is invited to visualize this). Fixed points $$p\in \textsf{M}^m$$ of G such that $$DG_p-I_m$$ is invertible are called Lefschetz fixed points and for Lefschetz maps one can compute L(G) via local Lefschetz numbers, that is $$L(G)=\sum \nolimits _{p=G(p)}L_p(G)$$, where the sign of the corresponding isomorphism defines the local Lefschetz numbers, i.e., the orientation numbers of the Lefschetz fixed point. By comparing orientations, one can show the following.

Proposition 3.2

(Lefschetz number of Lefschetz fixed point [16, p. 121]) Let $$G:\textsf{M}^m\rightarrow \textsf{M}^m$$ be a smooth Lefschetz map over a smooth, boundaryless compact, orientable manifold $$\textsf{M}^n$$, then

\begin{aligned} L(G) = \textstyle \sum \limits _{p=G(p)}\textrm{sgn}\,\textrm{det}(DG_p-I_m). \end{aligned}
(3.7)

Equation (3.7) is appealing, but only valid for Lefschetz fixed points. In the dynamical systems context, by only considering Lefschetz fixed points we are ignoring a set of structurally unstable fixed points. To make sure that L(G) is well-defined, and computable, we need to refine the notion of $$L_p(G)$$ for generic maps.

We start in the Euclidean setting. Let $$G:\mathbb {R}^m\rightarrow \mathbb {R}^m$$ be smooth with a fixed point $$p\in \mathbb {R}^m$$ and define for some closed neighbourhood $$\textrm{cl}\,{U}\subseteq \mathbb {R}^m$$ around p, containing no other fixed points, the map $$g:\partial {U}\rightarrow \mathbb {S}^{m-1}$$ by

\begin{aligned} g:q \mapsto g(q)= \frac{G(q)-q}{\Vert G(q)-q\Vert _2}. \end{aligned}
(3.8)

Then, let the generalized local Lefschetz number $$\widetilde{L}_p(G)$$ be equal to the degree of this map, that is $$\widetilde{L}_p(G)=\textrm{deg}(g)$$. For this construction to be of any use, the degree must be invariant under a change of neighbourhood U. Pick any other closed neighbourhood $$\textrm{cl}\,{U}'$$, strictly contained in $$\textrm{cl}\,{U}$$, then, as g extends to $$\textrm{cl}\,{U}\setminus {U}'$$, the degree vanishes on the boundary of this set by means of Lemma 3.3. However, by construction, this implies that the degree under both neighbourhoods must be equal (as the induced orientations must be the opposite), see also [16, p. 127]. If $$\textrm{cl}\,{U}$$ and $$\textrm{cl}\,{U}'$$ merely have a non-empty intersection, then, one first finds a larger closed neighbourhood $$\textrm{cl}\,{U}''$$ that contains both sets and the previous argument extends. Moreover, it can be shown that when p is a Lefschetz fixed point $$\widetilde{L}_p(G)=L_p(G)$$ [16, p. 128]. The following result is instrumental in linking local Lefschetz numbers.

Proposition 3.3

(On local Lefschetz numbers [16, pp. 126–129]) Let the smooth map $$G:\mathbb {R}^m\rightarrow \mathbb {R}^m$$ have some isolated fixed point $$p^{\star }$$ and let $$\mathbb {B}^m$$ be an open ball containing $$p^{\star }$$ such that $$\textrm{cl}\,\mathbb {B}^m$$ does not contain any other fixed points of G. Next, pick a map $$G'$$ that equals G outside of some compact subset of $$\mathbb {B}^m$$ and has all its fixed points in $$\mathbb {B}^m$$ being of the Lefschetz type. Then, the pair $$(G,G')$$ satisfies $$\widetilde{L}_{p^{\star }}(G)=\sum \nolimits _{p=G'(p)}{L}_p(G')$$ for $$p\in \mathbb {B}^m$$.

Proof

(sketch) First, $$\widetilde{L}_{p^{\star }}(G)$$ equals the degree of the map $$g:\partial \mathbb {B}^m\rightarrow \mathbb {S}^{m-1}$$ defined via (3.8). By construction, on $$\mathbb {R}^m\setminus \mathbb {B}^m$$, G can be replaced with $$G'$$. Let $$p_1,\dots ,p_k$$ be the set of Lefschetz fixed points of $$G'$$ in $$\mathbb {B}^m$$ and let $$(\mathbb {B}^m_{r_i})_i\subseteq \mathbb {B}^m$$ be a disjoint set of sufficiently small balls around those points such that $$\partial \mathbb {B}^m \cap (\cup _i \mathbb {B}^m_{r_i})=\emptyset$$. Let $$\mathbb {B}'=\mathbb {B}^m\setminus \cup _i \mathbb {B}^m_{r_i}$$, considering (3.8) for $$G'$$ over $$\partial \mathbb {B}'\rightarrow \mathbb {S}^{n-1}$$, by construction this map extends to $$\mathbb {B}'$$ such that by Lemma 3.3 the degree of this map must equal 0. Then the claims follows by observing that $$\partial \mathbb {B}'$$ consists out of $$\partial \mathbb {B}^m$$ and $$\partial \{\cup _i \mathbb {B}_{r_i}^m\}$$, with for the latter set(s) the orientation being flipped with respect to $$\mathbb {B}^m$$.

Now, given a general smooth map $$G:\textsf{M}^m\rightarrow \textsf{M}^m$$ with isolated fixed point $$p^{\star }$$, let $$\psi :U\rightarrow \mathbb {R}^m$$ be a diffeomorphism around $$p^{\star }$$ mapping $$p^{\star }$$ to 0 and consider $$\psi \circ G\circ \psi ^{-1}:\mathbb {R}^m\rightarrow \mathbb {R}^m$$. First, assume that $$p^{\star }$$ is a Lefschetz fixed point, hence, $$DG_{p^{\star }}-I_m$$ is an isomorphism. See that in coordinates one has $$D(\psi \circ G\circ \psi ^{-1})_0-I_m=D\psi _{p^{\star }}\circ (DG_{p^{\star }}-I_m)\circ D\psi _0^{-1}$$ such that the local Lefschetz number is preserved. For generic fixed points, employ Proposition 3.3. Homotopy invariance, generality of Lefschetz maps and Propositions 3.23.3 lead to the following generalization.

Theorem 3.5

(General Lefschetz numbers [16, p. 130], [17, Sect. 2.C]) Let $$G:\textsf{M}^m\rightarrow \textsf{M}^m$$ be a smooth map over a smooth, boundaryless, compact, oriented manifold $$\textsf{M}^m$$ with finitely many fixed points. Then,

\begin{aligned} L(G) = \textstyle \sum _{p=G(p)}\widetilde{L}_p(G). \end{aligned}
(3.9)

Remark 1

(Axiomatic degree theory [9, Appendix B], [33, Chap. IV]) Degree theory on closed manifolds is powerful, yet sometimes restrictive. It turns out that the concept can be generalized axiomatically to closed subsets of $$\mathbb {R}^n$$. Inspired by the properties of $$\textrm{deg}(\cdot )$$ one can derive a map d(GDq) with these desirable features like (3.4), where now $$G:D\rightarrow \mathbb {R}^n$$ is a $$C^{r\ge 0}$$ map over some bounded set $$D\subset \mathbb {R}^n$$ and q is a regular value such that $$G^{-1}(q)\notin \partial D$$. We will not appeal to this construction, but regarding further reading this is important to be aware of.

3.5 Poincaré–Hopf and the Bobylev–Krasnosel’skiĭ theorem

Lefschetz fixed point theory allows for analyzing flows and discrete-time dynamical systems, however, we are ultimately interested in continuous-time dynamical systems and particularly vector fields. The reason being, first principles possibly provide one with a differential equation approximatingFootnote 6 some phenomenon, having access to an explicit solution (flow), however, is rare. Hence, we switch from maps to vector fields on $$\textsf{M}$$, where the set of $$C^r$$-smooth vector fields on $$\textsf{M}$$ is denoted by $$\mathfrak {X}^r(\textsf{M})$$, see Sect. 5.1 for more details on continuous-time dynamical systems.

We will start again on $$\mathbb {R}^n$$ and define the vector field analogue of the local Lefschetz number, as introduced by Kronecker/Poincaré and formalized by Hopf.

Definition 1

(Index of a zero) Consider some open set $$\Omega \subseteq \mathbb {R}^n$$ and let $$p^{\star }\in \Omega$$ be an isolated zero of the smooth vector field $$X\in \mathfrak {X}^{\infty }(\Omega )$$. Let U be a neighbourhood of $$p^{\star }$$ such that $$p^{\star }$$ is the only zero of X over $$\textrm{cl}\,U$$ and define the map $$v:\partial U\rightarrow \mathbb {S}^{n-1}$$ by $$v:p\mapsto X(p)/\Vert X(p)\Vert _2$$. Then, the index of $$p^{\star }$$ is defined as

\begin{aligned} \textrm{ind}_{p^{\star }}(X) = \textrm{deg}(v). \end{aligned}
(3.10)

Indeed, if one is not aware of $$X\in \mathfrak {X}^{\infty }(\Omega )$$ having a zero on $$U\subseteq \Omega$$, index computations provide a partial answer. To lift the construction to manifolds, one can show that (3.10) is invariant under diffeomorphisms [29, p. 33]. This will be shown after establishing the link between vector field indices and local Lefschetz numbers.

Proposition 3.4

(Vector field indices and Lefschetz numbers [16, pp. 135–136]) Let X be a smooth vector field over some open neighbourhood $$\Omega \subseteq \mathbb {R}^n$$ of the origin, only vanishing at 0. Let $$\{\varphi _X^t:t\ge 0\}$$ be a family of mutually homotopic maps, smoothly mapping $$\Omega$$ to itself, with $$\varphi _X^0=I_n$$ and for $$t\ne 0$$, $$\varphi ^t_X$$ having no fixed points besides 0. If X(p) is tangent to $$\varphi _X^t(p)$$ at $$t=0$$ for all $$p\in \Omega$$, then

\begin{aligned} \textrm{ind}_0(X) = \widetilde{L}_0(\varphi _X^t). \end{aligned}
(3.11)

Proof

(sketch) First, by direct integration, one can show that if g(t) is a smooth function, then there is a smooth function r(t) such that $$g(t)=g(0)+t (\textrm{d}/\textrm{d}s)g(s)|_{s=0}+t^2r(t)$$. Then, fix $$p\in \Omega$$ and apply this coordinate-wise to $$\varphi _X^t(p)$$ as seen as function in t, that is $$\varphi ^t_X(p)=\varphi ^0_X(p)+t({\textrm{d}}/{\textrm{d}s})\varphi _X^s(p)\vert _{s=0}+t^2 R(t,p)$$, for R(tp) the vector-valued remainder term. Rearranging yields $$\varphi _X^t(p)-p = t X(p) + t^2 R(t,p)$$. By construction, for $$p\ne 0$$ and $$t\ne 0$$ we have $$\varphi ^t_X(p)-p\ne 0$$ and as such

\begin{aligned} \frac{\varphi ^t_X(p)-p}{\Vert \varphi ^t_X(p)-p\Vert _2} = \frac{X(p)+tR(t,p)}{\Vert X(p)+tR(t,p)\Vert _2} \end{aligned}
(3.12)

is well-defined. The left part of (3.12) can be identified with $$\widetilde{L}_0(\varphi ^t_X)$$, whereas the right part defines a homotopy in t. For $$t=0$$ we recover $$\textrm{ind}_0(X)$$ and as the degree is homotopy invariant the result follows (only use this last argument on the right).

Proposition 3.4 is of interest in its own right, but in particular, to show the following invariance. Let X be a smooth vector field over $$\textsf{M}$$ with some isolated equilibrium point $$p^{\star }$$. Let U be a neighbourhood of $$p^{\star }$$ and let $$\psi$$ be a diffeomorphism from U onto a neighbourhood V of 0. As such, the pushforward $$D\psi \circ X\circ \psi ^{-1}=\psi _{*}X$$ defines a vector field in local coordinates. In particular, we have

\begin{aligned} \textrm{ind}_0(\psi _{*}X) = \widetilde{L}_0(\varphi ^t_{\psi _{*}X})= \widetilde{L}_{p^{\star }}(\varphi ^t_X)=\textrm{ind}_{p^{\star }}(X), \end{aligned}
(3.13)

where we exploit that $$\textsf{M}$$ can be embedded in some Euclidean space. Most importantly, this shows that the index is well-defined on manifolds. Also see that (3.10) is purely local and does not rely on $$\textsf{M}$$ being orientable. See Section 5.1 for the formal introduction of the pushforward of a vector field X under a smooth map $$\psi$$.

Given the aforementioned discussion, let a vector field X, with only isolated zeroes, give rise to a flow $$\varphi _X^t$$. Indeed, the fixed points of this flow are the zeros of X. Moreover, for sufficiently small t, $$\varphi _X^t$$ behaves similar to the identity map. In this case the Lefschetz number collapses to the Euler characteristic $$\chi (\textsf{M})$$; this is the Poincaré–Hopf theorem, named after its initiator and key contributor.

Theorem 3.6

(Poincaré–Hopf theorem [29, p. 35]) Let $$\textsf{M}$$ be a smooth, compact, oriented, boundaryless manifold. Then, for any smooth vector field $$X\in \mathfrak {X}^{\infty }(\textsf{M})$$ with only isolated zeroes $$\{p^{\star }_i\}_{i\in \mathcal {I}}\subset \textsf{M}$$ one has

\begin{aligned} \chi (\textsf{M}) = \textstyle \sum \limits _{i\in \mathcal {I}} \textrm{ind}_{p^{\star }_i}(X). \end{aligned}
(3.14)

Proof

Hopf preferred a combinatorial/algebraic approach [19, Chap. 1], we, however, follow [16, p. 137], embed $$\textsf{M}$$ in some Euclidean space $$\mathbb {R}^{d}$$ (which we can do with $$d\le 2\textrm{dim}(\textsf{M})+1$$ by our topological assumptions on $$\textsf{M}$$ [27, Chap. 6]) and construct a tubular neighbourhood U of $$\textsf{M}$$. Then, by Proposition 3.1 we know we can find a U such that the normal projection $$\pi :U\rightarrow \textsf{M}$$ is a $$C^{\infty }$$ retraction. As $$\textsf{M}$$ is compact, $$p+t X(p)$$ will be contained in U for sufficiently small $$t>0$$ and any $$p\in \textsf{M}$$. Then, construct the map $$\varphi ^t(p)=\pi (p+tX(p))$$. For any $$p\in \textsf{M}$$ we have

\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\varphi ^t(p)\vert _{t=0} = X(p), \end{aligned}

such that Theorem 3.5 and Proposition 3.4 apply if we can show that the fixed points of $$\varphi ^t$$ equal the equilibrium points of X. Any zero of X leads to a fixed point of $$\varphi ^t$$. Since $$\pi$$ is a normal projection, any other fixed point must have tX(p) being perpendicular to $$T_p\textsf{M}$$, which implies that X(p) must be 0. Then as $$\varphi ^0$$ is $$\textrm{id}_{\textsf{M}}$$ and homotopic to $$\varphi ^t$$ (for sufficiently small $$t>0$$), the result follows as $$L(\textrm{id}_{\textsf{M}})=I(\Delta _{\textsf{M}},\Delta _{\textsf{M}})=\chi (\textsf{M})$$.

It is known that the Poincaré–Hopf theorem can be used to assess if vector fields have unique equilibria over compact sets, early remarks of this nature can be found in [36, 43]. In the context of dynamical control systems, similar tools have been used in [2, 3, 7, 8, 12, 21, 32, 37, 42] and more recently in [20, 24, 25, 44]. This approach can also be seen in the context of economics [10, 28], quantum mechanics [1] and optimization [38]. We will also exploit the theorem extensively.

Corollary 3.1

(Poincaré–Hopf theorem for continuous vector fields) Let $$\textsf{M}$$ be as in Theorem 3.6. Then, for any continuous vector field $$X\in \mathfrak {X}^{r\ge 0}(\textsf{M})$$ with only isolated zeroes $$\{p^{\star }_i\}_{i\in \mathcal {I}}\subset \textsf{M}$$, (3.14) holds.

Proof

(sketch) As illustrated on [34, p. 23], Theorem 3.3 asserts the existence of a smooth map $$Y:\textsf{M}\rightarrow T\textsf{M}$$ being $$\delta$$-close and homotopic to X. We cannot simply assume that this map will also be a vector field. However, let $$\varphi =\pi \circ Y$$ for the natural projection $$\pi :T\textsf{M}\rightarrow \textsf{M}$$, which we can assume to be a diffeomorphism. Now let $$X^{\infty }=Y\circ \varphi ^{-1}:M\rightarrow T\textsf{M}$$ and see that $$\pi \circ X^{\infty }=\textrm{id}_{\textsf{M}}$$. Then the result follows from the homotopy invariance of I, i.e., $$I(X,Z_{\pi }(\textsf{M}))=I(X^{\infty },Z_{\pi }(\textsf{M}))=\chi (\textsf{M})$$.

For linear systems $$\dot{x}(t){=}Ax(t)$$, with $$\textrm{det}(A)>0$$, one has $$\textrm{ind}_0(Ax)=\textrm{sgn}\,\textrm{det}(A)$$ [22, Theorem 6.1]. This can be extended to nonlinear systems by appealing to the Hartman-Grobman theorem [22, Theorem 6.3]. One show this by showing that orientation preserving diffeomorphisms are homotopic to the identity map cf. [29], see also Example 3.3 below. The take-away is that hyperbolic (structurally stable) equilibrium points have index $$\pm 1$$.

Next we provide a typical non-trivial example with a degenerate differential at 0. Therefore, one cannot appeal to the hyperbolic formula from above.

Example 3.3

(Degenerate equilibria) Consider a vector field $$X\in \mathfrak {X}^{\infty }(\mathbb {R}^2)$$ given by $$X(p)=(p_1^2,-p_2)$$. Clearly, $$p^{\star }=0$$ is the only equilibrium point. To compute $$\textrm{ind}_0(X)$$, see from Fig. 3.4(iii) that $$y=(1,1)$$ is regular value of the map $$v(X(p))=X(p)/\Vert X(p)\Vert _2$$ for both $$p=2^{-1/2}(-1,-1)$$ and $$p=2^{-1/2}(1,-1)$$. Hence, from (3.4), we get $$\textrm{ind}_0(X)=1-1=0$$.

On $$\mathbb {R}^2$$, the vector field index corresponds to the so-called winding number,Footnote 7 also called the Cauchy index or Poincaré index [11]. In particular, one computes for $$X(p)=(p_1^2,-p_2)=(X_1,X_2)$$, as in Example 3.3, the index of 0 as,

\begin{aligned} \textrm{ind}_0(X)= \frac{1}{2\pi }\int \limits _{\mathbb {S}^1}\frac{X_1\textrm{d}X_2-X_2\textrm{d}X_1}{X_1^2+X_2^2} = \frac{1}{2\pi }\int \limits _{0}^{2\pi }\frac{-\cos (\theta )^3-2\sin (\theta )^2\cos (\theta )}{\cos (\theta )^4+\sin (\theta )^2}\textrm{d}\theta = 0. \end{aligned}

So, in dimension 2, the index corresponds to how often the vector $$X_{\gamma (t)}$$ rotates counter-clockwise when moving along a path $$\gamma (t)$$ counter-clockwise (in line with the standard orientation on $$\mathbb {R}^2$$) around the isolated equilibrium point [16, p. 192], see Fig. 3.4 for some more examples. See [11, Sect. 1.2] for more on the relation between the degree as defined via intersection theory or differential forms.

Next we provide as an example the Bobylev–Krasnosel’skiĭ theorem—that will play a central role in the remainder of this work.

Example 3.4

(Index of isolated locally asymptotically stable equilibrium points: the Bobylev–Krasnosel’skiĭ theorem) Consider $$X\in \mathfrak {X}^r(\textsf{M}^n)$$ with $$p^{\star }\in \textsf{M}^n$$ being some isolated locally asymptotically stable equilibrium point. We cannot assume $$p^{\star }$$ to be hyperbolic and follow [23, Chap. II], [22, Theorem 52.1] [40]. Without loss of generality consider X in local coordinates and assume 0 to be an isolated asymptotically stable equilibrium point. Let $$\textrm{cl}\,r\mathbb {B}^n$$ be a sufficiently small closed ball around 0 (occasionally written using $$\mathbb {D}^n=\textrm{cl}\,\mathbb {B}^n$$) such that X has no other zeroes on $$\textrm{cl}\,r\mathbb {B}^n$$. To aid the computation of the index, we recall that the degree is invariant under homotopy. A similar notion holds for the index. We say that two vector fields, as seen as maps, are vector field homotopic when the entire homotopy itself does not vanish.Footnote 8 This means the vector fields themselves must be nondegenerate over their domain. Akin to Example 3.2, one can show that if this is true, the corresponding indices agree [23, Theorem 5.5]. For example, consider in Fig. 3.4 scenario (i) and (ii), the corresponding maps, from $$\mathbb {S}^1$$ to itself, are not homotopic and indeed, the indices do not agree. A particularly useful ramification is the following, given two nondegenerate vector fields $$X_1$$ and $$X_2$$ over W. If $$X_1$$ and $$X_2$$ are never oppositely directed, that is,

\begin{aligned} \frac{X_1(w)}{\Vert X_1(w)\Vert _2}\ne - \frac{X_2(w)}{\Vert X_2(w)\Vert _2}\qquad \forall w\in W, \end{aligned}

then as in Example 3.2, a convex combination of $$X_1$$ and $$X_2$$ entails a vector field homotopy, see also [23, Theorem 5.6]. Now consider the vector field $$-X$$ and its relation to the flow $$\varphi _X^t$$

\begin{aligned} \lim _{t\downarrow 0}\frac{p-\varphi _X^t(p)}{t}=-X(p). \end{aligned}
(3.15)

By continuity and the fact that t is nonnegative, (3.15) implies that $$p-\varphi _X^t(p)$$ and $$-X(p)$$ will not be of opposite sign for sufficiently small $$t>0$$. However, by asymptotic stability $$p-\varphi _X^t(p)\ne 0$$ for $$t>0$$. Hence, we have constructed a vector field homotopy. However, asymptotic stability also implies that for $$t\rightarrow +\infty$$ the map $$p\mapsto p-\varphi ^t(p)$$ eventually tends to the identity map. This proves that $$\textrm{ind}_0(-X)=\textrm{ind}_0(\textrm{id}_{\textrm{cl}\,r \mathbb {B}^{n}})=1$$ cf. (3.12). For $$\textrm{ind}_0(X)$$, observe that for a map g over some n-dimensional domain $$\textrm{deg}(g)=(-1)^n\textrm{deg}(-g)$$ such that $$\textrm{ind}_{0}(X)=(-1)^n$$. Then, by the invariance under diffeomorphisms (3.13) we get that $$\textrm{ind}_{p^{\star }}(X)=(-1)^n$$.

The index result from Example 3.4 appeared for the first time in [4] and was largely extended and popularized by Krasnosel’skiĭ and Zabreĭko [22]. However, it is likely that the results where known, e.g., to Poincaré [35, Chap. XVIII] and Anosov [4], presumably since it appeared to be an “obvious fact” [4, p. 1043]. With respect to that body of literature is important to remark that the rotation of a vector field was the invariant of choice. For all practical purposes in this work, that concept is the same as the vector field index. For the subtle difference see [46].

Results analogous to Example 3.4 for Lyapunov stable- or attractive isolated equilibrium points are less transparent, but motivated by the work of Zabczyk [45] we provide a short remark. One severe complication is that asymptotic stability is locally of interest, just as Lyapunov stability, solely zooming in on attractivity, however, is mostly interesting on the global level due to the interplay with the (global) topology. For example, consider globally attractive isolated equilibrium points on $$\mathbb {S}^1$$ and $$\mathbb {S}^2$$. By the Poincaré–Hopf theorem, those equilibrium points must have vector field index 0 and 2, respectively. If those points would be merely locally attractive, the indices could be $$-1$$ and 1, respectively. As was already pointed out in [22], it is known that equilibrium points that are merely Lyapunov stable can in general have any index [5]. For attractivity the statement is also subtle and depends on the domain over which the system is attractive. From Example 3.4 we see that locally, the arguments of local asymptotic stability carry over. However, see that in both cases we exploit the properties of a continuous flow. In [30] this is relaxed, the vector field is continuous, but solutions are not necessarily unique nor do they necessarily depend continuously on initial conditions. See also the proof of Theorem 6.2 for a way around exploiting the direct existence of a flow.

Example 3.5

(Case study Sect. 1.3: Lie groups) Consider any compact Lie group $$\textsf{G}^n$$ with $$n\ge 1$$. As one can construct a non-vanishing smooth vector field on $$\textsf{G}^n$$ by pushing-forward any fixed non-zero $$v\in T_e\textsf{G}^n$$ under some left translation $$L_g$$ [27, Theorem 8.37], it follows from Theorem 3.6 that $$\chi (\textsf{G}^n)=0$$, compactness is important here. We return to this frequently.

To make use of the Poincaré–Hopf theorem one needs to assert that an appropriate vector field exists. This can be shown using Thom’s transversality theorem.

Proposition 3.5

(Existence of vector fields with isolated equilibrium points) On every smooth compact manifold $$\textsf{M}$$ there exists a vector field with only finitely many isolated zeroes.

Recall Example 3.3, the following result due to Hopf shows that, up to homotopy, equilibrium points with vector field index 0 can be ignored. Compactness is key here.

Proposition 3.6

(Nowhere-vanishing vector fields [16, p. 146]) A compact, connected, oriented, smooth manifold $$\textsf{M}$$ has $$\chi (\textsf{M})=0$$ if and only if there exists a continuous nowhere-vanishing vector field X on $$\textsf{M}$$.

See Example 5.1 for an application of Proposition 3.6 and see Chapter 6 and Section 8.2 for further generalizations of Theorem 3.6.

For references on differential topology, see for example [6, 13, 16, 18, 26, 27, 29]. For more on degree theory in particular, see [11, 16, 18, 33, 46] and [9, 31] in the context of control theory. See [14] for an exposition on the generality of index theory and [15] for a general, beyond continuity, axiomatic treatment of index theory.