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The Lefschetz Fixed Point Theorem is wonderful. It generalizes the Fixed Point Theorem of Brouwer, and is an indispensable tool in topological analysis of dynamical systems. The weakest form goes like …

At least in the degree-theoretic world, the index notation appears to be dominant. When we write the Lefschetz-Hopf theorem $$L(f) = \sum_{x \in \text{Fix}(f)} \text{stuff}_x(f),$$ the $\text{stuff} …

In the absence of convex images, one typically relies on algebraic topology as you have guessed. If your set-valued map has a reasonably nice domain and contractible images, then you can easily string …

Your setup defines set-valued dynamics on $X$. More precisely, you have $$X \stackrel{p}{\leftarrow} \overline{G} \stackrel{q}{\rightarrow} X$$ where $p$ and $q$ are the obvious projection maps. The s …

Lovely question! Sadly, the answer is "no" in the sense that the fixed point property is not homotopy-invariant even in the category of finite polyhedra. In fact, it is also not invariant under the op …

Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a maxima …

Background At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is …