Krasnoselskii genus

We follow the definition given by Coffman.^[2]

Let

• ${\displaystyle E}$ be a Banach space,
• ${\displaystyle {\mathcal {A}}=\{A\subset E:A{\text{ closed}},;A=-A\}}$ be the collection of symmetric closed subsets of ${\displaystyle E}$,
• ${\displaystyle C(A,\mathbb {R} ^{n})}$ the space of continuous functions ${\displaystyle A\to \mathbb {R} ^{n}}$.

For ${\displaystyle A\in {\mathcal {A}}}$ define the set

${\displaystyle K_{A}=\{n\in \mathbb {N}$ :\exists f\in C(A,\mathbb {R} ^{n}\setminus {0}),;f(-x)=-f(x)\}}

Then the Krasnoselskii genus of ${\displaystyle A}$ is defined as^[3]

${\displaystyle \gamma (A)={\begin{cases}\inf K_{A}&{\text{if }}K_{A}\neq \emptyset ,\\\infty &{\text{if }}K_{A}=\emptyset ,\\0&{\text{if }}A=\emptyset .\end{cases}}}$

In other words, if ${\displaystyle \gamma (A)=n}$ then there exists a continuous odd function ${\displaystyle \varphi :A\to \mathbb {R} ^{n}}$ such that ${\displaystyle 0\notin \varphi (A)}$. Moreover ${\displaystyle n}$ is the minimal possible dimension, i.e. there exists no such function ${\displaystyle \theta :A\to \mathbb {R} ^{d}}$ with ${\displaystyle d<n}$.

• Let ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ be a bounded symmetric neighborhood of ${\displaystyle 0}$ in ${\displaystyle \mathbb {R} ^{n}}$. Then the genus of its boundary is ${\displaystyle \gamma (\partial \Omega )=n}$.^[4]
• For ${\displaystyle A,B\in {\mathcal {A}}}$, the following holds:^[5]

1. If there exists an odd function ${\displaystyle f\in C(A,B)}$, then ${\displaystyle \gamma (A)\leq \gamma (B)}$.
2. If ${\displaystyle A\subset B}$, then ${\displaystyle \gamma (A)\leq \gamma (B)}$.
3. If there exists an odd homeomorphism between ${\displaystyle A}$ and ${\displaystyle B}$, then ${\displaystyle \gamma (A)=\gamma (B)}$.

Combining these statements, it follows immediately that if there exists an odd homeomorphism between ${\displaystyle A}$ and ${\displaystyle \partial \Omega }$ then ${\displaystyle \gamma (A)=n}$.