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Lefschetz theory for coincidences

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Why do we need to study coincidences?

The Coincidence Problem is concerned with the following question: given two two maps $f,g:N\rightarrow M$ do they have any coincidences, i.e., $$x\in N:f(x)=g(x)?$$

This study is motivated by two separate problems.

First, consider the familiar Fixed Point Problem: If $M$ is a manifold and $f:M\rightarrow M$ is a map, what can be said about the set of points $x\in M$ such that $f(x)=x$? This problem is a coincidence problem with $N=M$ and $g=Id_M$. The main application we have considered is for detecting equilibria of dynamical systems.

We know that the Lefschetz number of $f$ will detect these fixed points of all maps homotopic to $f$.

The methods we have developed, however, may be insufficient if, reflecting noise and other uncertainty, the dynamical system is given by a set-valued map $F$. In that case, the equilibria of this system are solutions of the fixed point problem in the new form: is there $x\in F(x)$?

Set-valued dynamical system.png

But this problem is a coincidence problem if we choose $N=G$ to be the graph of $F$ as a subset of $M \times M$ and $f,g$ to be the two projections $p^M:G \rightarrow M$ and $p^N: G\rightarrow N$.

Second, consider the Surjectivity Problem: If $M$ is a manifold and $f:N\rightarrow M$ is a map, what can be said about the image of $f$? Is it the whole $M$: for each $c\in M$ there is $x\in N$ such that $f(x)=c$?

We know that maps of non-zero degree are surjective.

However, this result applies only to a narrow class of maps, those between manifolds of the same dimension. Our interest in the forward kinematic problem and the reachability of control systems dictates the need for a higher dimensional domain space $N$. For example, the projection of the torus on the circle is such a map:

Collapse of torus.jpg

But we quickly discover that this problem is a coincidence theorem if we choose $g=g_c$ to be the constant map, $g_c(x)=c$ for all $x\in N$.

Our the main tools will be the graded Lefschetz homomorphism $\Lambda_{fg}$ that generalizes both the Lefschetz number $\lambda_{f}$ of a self-map $f$ and the degree of $f$. A part of this homomorphism is still defined as the alternating sum of traces of a certain endomorphism on the homology group of $M$. A Lefschetz type coincidence theorem states that if $\lambda_{fg}\neq0$ then $f,g$ (and any pair homotopic to them) have a coincidence.

The question of when homotopic maps don't have coincidences anymore is also addressed.

Another area of applications is discrete dynamical systems. A dynamical system on a manifold $M$ is determined by a map $f:M\rightarrow M.$ Then the next position $f(x)$ depends only on the current one, $x\in M$. Suppose we have a fiber bundle $F\rightarrow N^{\underrightarrow{~\ \ \ g\ \ \ ~~}}M$ and a map $f:N\rightarrow M.$ Then this is a parametrized dynamical system. Here, where the next position $f(x,s)$ depends not only on the current one, $x\in M,$ but also the current state, $s\in F.$ Then the Coincidence Problem asks whether there are a point and a state such that the point remains fixed, $f(x,s)=x$. A parametrized dynamical system can be a model for

  • a non-autonomous ordinary differential equation: $M$ is the space, $F$ is the time, and $N$ is the space-time; and
  • a control system: $M$ is the state space, $F$ is the input or control.

While studying coincidences one typically stays within one of the following two settings:

  • Case 1: $f:N\rightarrow M$ is a map between two $n$-manifolds.
  • Case 2: $f:N\rightarrow M$ is a map from an arbitrary topological space to an open subset of $\mathbf{R}^{n}$ and all fibers $f^{-1}(y)$ are acyclic, i.e., $H_{k}(f^{-1}(y))=0$ for $k=1,2,...$.

The treatment below is unified.

Detecting coincidences

In the standard, manifold, setting the definition of the Lefschetz number is as follows.

Suppose we are given a pair of maps $f:(N,\partial N)\rightarrow(M,\partial M),$ $g:N\rightarrow M,$ where both $M$ and $N$ are orientable compact connected manifolds with boundaries $\partial M,\partial N,$ and $\dim M=\dim N=n$. The definition relies on the following algebraic fact: if $h:E\rightarrow E$ is a degree $0$ endomorphism of a finitely dimensional graded vector space $E=\{E_{k}\},h_{k}:E_{k}\rightarrow E_{k},$ then its Lefschetz number is $$L(h)=\sum_{k}(-1)^{k}Trace(h_{k}).$$

To apply this formula we let $E=H_{\ast}(M)$, then the Lefschetz number is defined as $$\lambda_{fg}=L(g_{\ast}D_{N}f^{\ast}D_{M}^{-1}),$$ where $$D_{M}:H^{\ast }(M,\partial M)\rightarrow H_{n-\ast}(M),$$ $$D_{N}:H^{\ast}(N,\partial N)\rightarrow H_{n-\ast}(N)$$ are the Poincare duality isomorphisms.

Now suppose $N$ is an arbitrary topological space, $A\subset N$, $M$ is an orientable compact connected manifold, $\dim M=n$, and $f:(N,A)\rightarrow(M,\partial M),g:N\rightarrow M$ are maps. The generalization is based on the fact that since $E=H_{\ast}(M)$ is equipped with the cap product $\frown:E^{\ast }\otimes E\rightarrow E$, one can define the Lefschetz class $L(h)\in E$ of an endomorphism $h$ given by $h_{k}:E_{k}\rightarrow E_{k+m}$ of any degree $m$. There is an explicit formula for it.

Theorem. If $h:H_{\ast}(M)\rightarrow H_{\ast+m}(M)$ is a homomorphism of degree $m$ then $$L(h)=\sum_{k}(-1)^{k(k+m)}\sum_{j}x_{j}^{k}\frown h(a_{j}^{k}), $$ where $\{a_{1}^{k},...,a_{m_{k}}^{k}\}$ is a basis for $H_{k}(M)$ and $\{x_{1}^{k},...,x_{m_{k}}^{k}\}$ the corresponding dual basis for $H^{k}(M).$

For a given $z\in H_{\ast}(N,A),$ suppose $h_{fg}^{z}$ is defined as the composition $$H_{\ast}(M)^{\underrightarrow{\quad D_{M}^{-1}\quad}}H_{n-\ast}(M,\partial M)^{\underrightarrow{\quad f^{\ast}\quad}}H_{n-\ast}(N,A)^{\underrightarrow {\quad\frown z\quad}}H_{k-n+\ast}(N)^{\underrightarrow{\quad g_{\ast}\quad} }H_{k-n+\ast}(M).$$ i.e., $$h_{fg}^{z}=g_{\ast}((f^{\ast}D_{M}^{-1})\frown z)).$$ Its degree is $m=|z|-n.$

Definition. The Lefschetz homomorphism $\Lambda_{fg}:H_{k}(N,A)\rightarrow H_{k-n}(M),$ $k=0,1,...,$ is defined by \[ \Lambda_{fg}(z)=L(h_{fg}^{z}). \]

Proposition. If the degree $m$ of $h$ is zero, then $$L(h)=\sum_{k}(-1)^{k} Trace(h_{k}).$$

In particular, the degree of $h_{fg}^{z}$ is zero for all $z\in H_{n}(N,A).$ If, moreover, $N$ is a orientable compact connected manifold of dimension $n,$ we have $H_{n}(N,\partial N)=\mathbf{Q.}$ It is generated by the fundamental class $O_{N}\in H_{n}(N,\partial N)$ of $N.$ Since $D_{N}(x)=x\frown O_{N},$ we recover the standard Lefschetz number, $\lambda_{fg}=\Lambda_{fg}(O_{N}).$

Theorem (Existence of coincidences). If $\Lambda_{fg}\neq0$ then any pair of maps $f^{\prime },g^{\prime}$ homotopic to $f,g$ has a coincidence.

Especially important are the following corollaries.

Corollary (Existence of fixed points). Let $g:M\times U\rightarrow M$ be a map. Given $v\in H_{\ast}(U),$ suppose the homomorphism $g_{v} :H_{\ast}(M)\rightarrow H_{\ast+k}(M)$ of degree $|v|$ is defined by $$g_{v}(x)=(-1)^{(n-|x|)|v|}g_{\ast}(x\otimes v), x\in H_{\ast}(M).$$ Then, if \[ L(g_{v})\neq0\text{ for some}\ v\in H_{\ast}(U) \] then any map $g^{\prime}:M\times U\rightarrow M$ homotopic to $g$ has a fixed point $x$, $g^{\prime}(x,u)=x$ for some $u.$

Corollary (Sufficient condition of surjectivity). If $$f_{\ast}:H_{n}(N,A)\rightarrow H_{n}(M,\partial M)=\mathbf{Q} \text{ is nonzero}$$ then any map $f^{\prime}:(N,A)\rightarrow(M,\partial M)$ homotopic to $f$ is onto.

Proof. Apply the theorem to the pair $f,c,$ where $c$ is any constant map. $\blacksquare$

The condition of this Corollary is equivalent to non-vanishing of the degree of $f$ in case of manifolds of equal dimensions.

Note: Suppose $N>n.$ Then from dimensional considerations it follows that $\Lambda_{fg}=0$ for maps $f,g:\mathbf{S}^{N}\rightarrow \mathbf{S}^{n}.$

Note: Suppose $z\in H_{i}(M,\partial M)$ and $i>n.$ Then $z=0$, therefore $$\Lambda _{Id,Id}(z)=0.$$

The Lefschetz homomorphism satisfies a naturality property below.

Theorem (Naturality). Let $(Y,B)$ be a topological space and $h:(Y,B)\rightarrow (X,A) $ be a map. Then $$\Lambda _{fh,gh}=\Lambda _{fg}h_{\ast }.$$

Proof. Let $z\in H(Y,B)$. Then we have $$(gh)_{\ast }(fh)_{!}^{z}=g_{\ast }h_{\ast }(h^{\ast }f^{\ast }D^{-1}\frown z)=g_{\ast }(f^{\ast }D^{-1}\frown h_{\ast }(z))=g_{\ast }f_{!}^{h_{\ast }(z)}.\blacksquare$$

This theorem generalizes the well known formula for maps between two $n$-manifolds: $$L(fh,gh)=\deg (h)L(f,g),$$ where $L(f,g)$ is the ordinary Lefschetz number.

Corollary. Suppose $(X,A)=Y\times (M,\partial M),$ $g:X\rightarrow M$ is a map and $p:Y\times (M,\partial M)\rightarrow (M,\partial M)$ is the projection (then $Fix(g)=Coin(p,g)$). Then $$L(g_{u\ast })=\Lambda _{pg}(u\times O_{M}),\ u\in H(Y),$$ where $g_{u}:M\rightarrow M$ is given by $g_{u}(x)=g(u,x).$

Proof. Fix $u\in H(Y)$. Let $p_{u}:Y\times M\rightarrow \{u\}\times M$ be given by $p_{u}(x)=(u,x).$ Let $q:\{u\}\times M\rightarrow M$ be given by $q(u,x)=x$. Then $qp_{u}=p,$ $q_{\ast }=Id$ and $g=g_{u}p_{u}$. Therefore by the Theorem we have $$\Lambda _{pg}(u\times O_{M})=\Lambda _{qg_{u}}(p_{u\ast }(u\times O_{M}))=\Lambda _{qg_{u}}(O_{M})=L(g_{u\ast }q_{!}^{O_{M}})=L(g_{u\ast }).\blacksquare$$

Examples of surjectivity

Let's verify the sufficient condition of surjectivity for several examples: $$(A) f_{\ast}:H_{n}(N,A)\rightarrow H_{n}(M,\partial M)=\mathbf{Q} \text{ is nonzero.}$$

The corollary takes the following form.

Corollary. If $f:(X,X^{\prime })\rightarrow (S,\partial S)$ satisfies condition (A) then $f$, and every map homotopic to $f$, is onto.

We conclude with a few examples of applications of this corollary. These examples are not included in either Case 1 or Case 2.

It is hard to come by an example of coincidences that does not involve manifolds. Yet we can consider a pair $(X,X^{\prime })$ such that $X$ is a manifold (possibly with boundary) but $(X,X^{\prime })$ is not a manifold with boundary, i.e., $X^{\prime }$ is not the boundary of $X$ (nor homotopically equivalent to it). This provides a setting not included in Case 1.

Example. Let $f:({\bf D}^2,\partial e\cup \partial e^{\prime })\rightarrow ({\bf D}^2, {\bf S}^1),$ where $e$ and $e^{\prime }$ are disjoint cells in ${\bf D}^2$, be a map. The it is a matter of simple computation to verify condition (A).

Example. Let $f:({\bf I},\partial {\bf I})\times {\bf S}^1\rightarrow ({\bf D}^2,{\bf S}^1\cup \{0\}),\ {\bf I}=[0,1]$, be the map that takes by identification $\{0\}\times {\bf S}^1$ to $\{0\}.$

It is clear that condition (A) is satisfied for $X^{\prime }=\{1\}\times {\bf S}^1$, therefore, any map homotopic to $f$ is onto. But $f^{-1}(0)={\bf S}^1$ is not acyclic, so this example is not covered by Case 2. On the other hand, even though $f$ maps a manifold to a manifold, it does not map boundary to boundary. Therefore, Case 1 does not include this example.

In a similar fashion we can show that the projection of the torus ${\bf T}^2$ on the circle ${\bf S}^1$ is onto. This is an example of a map between manifolds of different dimensions.

For a negative example of this kind, take the Hopf map $f:{\bf S}^3\rightarrow {\bf S}^2$, then for any $g,$ the Lefschetz number of the pair $(f,g)$ is equal to zero.

Next, non-manifolds.

Example. Let $E$ be a space that is not acyclic and not a manifold, e.g., the figure eight. Consider the projection $f:(X,X^{\prime })=({\bf I},\partial {\bf I})\times E\rightarrow ({\bf I},\partial {\bf I}),{\bf I}=[0,1],$ onto the first coordinate. Then $f$ clearly satisfies (A). Thus $f$ is onto. Observe that $f^{-1}(x)=E$ is not acyclic and $X$ is not a manifold.

Corollary. Suppose $X$ is a topological space, $M$ is an oriented compact closed $(n-1)$ -connected $n$-manifold, $f:X\rightarrow M$ is a map, and

  • (A$^{\prime }$) $f_{\#}:\pi _n(X)\rightarrow \pi _n(M)$ is onto.

Then $f$ is onto.

Proof. As $M$ is $(n-1)$-connected, the Hurewicz homomorphism $h_n:\pi_n(M)\rightarrow H_n(M)$ is onto [p. 488]{Bredon}. Hence $f_{*}$ is onto and $(A)$ is satisfied. $\blacksquare$

Condition (A$^{\prime }$) holds when $f$ is a fibration with $\pi_{n-1}(f^{-1}(y))=0$ (it follows from the homotopy sequence of the fibration [Theorem VII.6.7, p. 453]{Bredon}).

Continuing the discussion in the beginning of this section, what if the acyclicity condition for $f$ fails at $(n-1)$ degree? Then there is no version of the Vietoris Mapping Theorem available to ensure condition (A).

Definition. Let $B(A),A\subset {\bf R}^n,$ denote the bounded component of ${\bf R}^n\backslash A.$ A closed-valued u.s.c. map $\Phi :{\bf D}^n\rightarrow {\bf D}^n$ is called $(n-1)$-spherical, $n>1$, if

  • (i) for every $x\in {\bf D}^n$, $H(\Phi (x))=H({\bf S}^{n-1})$ or $H(point),$
  • (ii) for every $x\in {\bf D}^n$, if $x\in B(\Phi (x))$

then there exists an $\varepsilon $-neighborhood $O_\varepsilon (x)$ of $x$ such that $x^{\prime }\in B(\Phi (x^{\prime }))$ for each $x^{\prime }\in O_\varepsilon(x).$

Corollary. An $(n-1)$-spherical map $\Phi :{\bf D}^n\rightarrow {\bf D}^n,\ n>1,$ has a fixed point.

Proof. We notice, first, that if $\Phi $ has no fixed points and there are no points $x$ such that $x\in B(\Phi (x)),$ then by replacing $\Phi (x)$ with $\Phi ^{\prime }(x)=\Phi (x)\cup B(\Phi (x))$ we obtain an acyclic multifunction without fixed points. Therefore we suppose that such an $x$ exists and for simplicity assume that it is $0$. Now, if $0$ is not a fixed point, then from the upper semicontinuity of $\Phi $ and (ii) above, it follows that there is an $\varepsilon >0$ such that $$|x|<\varepsilon \Rightarrow x\in B(\Phi (x))\text{ and }|\Phi (x)|>2\varepsilon.$$ Let $X$ be the graph of $\Phi $, $K=\{x:|x|\geq 2\varepsilon \},\ f,g$ projections of $X,\ X^{\prime }=f^{-1}({\bf D}^n\backslash K)$. One can see that $f$ is essentially the same as the projection of $({\bf D}^n,{\bf S} ^{n-1})\times {\bf S}^1$ onto $({\bf D}^n,{\bf S}^{n-1})$, and, therefore, induces a surjection $$f_{*}:H_n(({\bf D}^n,{\bf S}^{n-1})\times {\bf S}^1)\rightarrow H_n({\bf D}^n,{\bf S}^{n-1}). $$ Hence condition (A) is satisfied, so $\Phi $ has a fixed point. $\blacksquare$

Example. Let $\Phi :{\bf D}^2\rightarrow {\bf D}^2$ be given by $$\Phi (x)=\{y\in {\bf D}^2:|y-x|=\rho (x)\}\cup \{y\in {\bf S}^1:|y-x|>\rho (x)\}, $$ where $\rho (x)=1-|x|+|x|^2,\ x\in {\bf D}^2.$

This example shows that condition (ii) of the above definition is necessary for existence of a fixed point.

Example. There is a multivalued u.s.c. retraction $\Phi :{\bf D}^n\rightarrow {\bf S}^{n-1}:$ $$\Phi (x)=\{y\in {\bf S}^{n-1}:|y-x|\geq 4|x|^2-3|x|\}. $$

Example. Let ${\bf M}^2$ be the Mobius band, given in cylindrical coordinates by: $z=\theta ,\ -1\leq r\leq 1,\ 0\leq \theta \leq \pi ,$ with the top and bottom edges identified. Let $f:({\bf M}^2,\partial {\bf M}^2)\rightarrow ({\bf D}^2,{\bf S}^1)$ be the projection on the horizontal plane and $g:{\bf M}^2\rightarrow {\bf S}^1$ the projection on the $z$-axis.

Observe that $\Phi $ has no fixed points, while in the last example $g$ is homotopic to a map $g^{\prime }$ such that the pair $(f,g^{\prime })$ has no coincidences. This is reflected in the fact that $\Phi (x)$ fails to be acyclic for $|x|\leq 1/2,$ while in the last example $f_{*}$ misses the fundamental class.

Removing coincidences

In this section our concern is about the ability of a "small" homotopy to remove existing coincidence.

Recall, that two maps $f,g:(N,A)\rightarrow(M,B)$ are called homotopic, $f\sim g,$ if $f$ can be continuously "deformed" into $g$, i.e., there is a map $F:[0,1]\times(N,A)\rightarrow(M,B)$ such that $F(0,\cdot)=f$ and $F(1,\cdot)=g.$ An $\varepsilon$-homotopy is one satisfying $d(F(t,x),F(0,x))<\varepsilon.$ If $f$ and $g$ are homotopic then $f_{\ast }=g_{\ast}.$ If $f$ is homotopic to a constant map then $f_{\ast}$ is trivial, i.e., $f_{\ast}:H_{k}(M)\rightarrow H_{k}(N)$ is zero for $k=1,2,...$.

Suppose $M$ and $N$ are manifolds, $\dim M=n,$ $f:(N,A)\rightarrow(M,\partial M),g:N\rightarrow M$ are maps, $A\ $is a closed subset of $N$.

When $\dim N=\dim M=n>2,$ the vanishing of the Lefschetz number $\lambda_{fg}$ implies that the coincidence set can be removed by homotopies of $f,g$. If $\dim N=n+m,m>0,$ this is no longer true even if $\lambda_{fg}$ is replaced with $\Lambda_{fg}$.

For some $m>1,$ a partial converse of the Lefschetz type theorem above is provided below.

Let $F$ be a closed submanifold of $N.$ We say that $F$ satisfies condition (*) if one of the following three conditions holds:

  • (a1) $M$ is a surface, i.e., $n=2;$ or
  • (a2) $F$ is acyclic, i.e., $H_{k}(F)=0$ for $k=1,2,...$; or
  • (a3) every component of $F$ is a homology $m$-sphere, i.e., $H_{k}(F)=0$ for $k\neq0,m,$ for the following values of $m$ and $n$:
  1. $m=4$ and $n\geq6;$
  2. $m=5$ and $n\geq7;$
  3. $m=12$ and $n=7,8,9,14,15,16,....$.

Theorem (Local removability of coincidences). Suppose condition (*) is satisfied for $F=C_{fg},$ the coincidence set of $f,g$, $f(C_{fg})=g(C_{fg})=\{x\},$ $x\in M\backslash\partial M,$ and $C_{fg}\cap A=\varnothing.$ Then, if $$\Lambda_{fg}(z)=\mathit{\ }\sum_{k}(-1)^{k}Trace(h_{fg}^{z})\in H_{0}(M)=\mathbf{Q} \text{ is zero for all } z\in H_{n}(N,A)$$ then there is a homotopy of $f$ (or $g)$ to a map $f^{\prime}$ (or $g^{\prime})$ such that the new pair has no coincidences. The homotopy can be chosen arbitrarily small and constant on the complement of a neighborhood of $F.$

Especially important are the following corollaries.

Corollary (Local removability of fixed points). Given a parametrized map $g:M\times U\rightarrow M$ with only one fixed point, $g(a,u)=a\in M\backslash\partial M,$ suppose condition (*) is satisfied for the fixed point set $F=\{u\in U:g(a,u)=a\}$ of $g$. Then, if \[ L(g_{1})=\sum_{k}(-1)^{k}Trace(\overline{g}_{\ast k})=0, \] where $\overline{g}(\cdot)=g(\cdot,u_{0}),$ then there is a homotopy of $g$ to a map $g^{\prime}$ such that $g^{\prime}$ has no fixed points$.$ The homotopy can be chosen arbitrarily small and constant on the complement of a neighborhood of $\{a\}\times F.$

Corollary (Local removability of images). Suppose that there is a fiber $F=f^{-1}(x_{0})$ of $f$ satisfying condition (*). Then, if $$f_{\ast}:H_{n}(N,A)\rightarrow H_{n}(M,\partial M)=\mathbf{Q} \text{ is zero}$$ then there is a homotopy $f$ to a map $f^{\prime}$ which is not onto, $x_{0}\notin f(N)$. The homotopy can be chosen arbitrarily small and constant on the complement of a neighborhood of $F.$

Proof. Suppose $c(x,u)=x_{0}$ is the constant map. Next, $f_{\ast}=0$ if and only if $\Lambda_{fc}(z)=0$ for all $z\in H_{n}(N,A)$. Now apply the theorem to the pair $f,c$. Observe also that $C_{fc}=f^{-1}(x_{0}).$ $\blacksquare$

Applications of these results in control theory are considered elsewhere.

Proofs are in

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