The Maslov Index and the Spectral Flow - revisited

We give an elementary proof of a celebrated theorem of Cappell, Lee and Miller which relates the Maslov index of a pair of paths of Lagrangian subspaces to the spectral flow of an associated path of selfadjoint first-order operators. We particularly pay attention to the continuity of the latter path of operators, where we consider the gap-metric on the set of all closed operators on a Hilbert space. Finally, we obtain from Cappell, Lee and Miller's theorem a spectral flow formula for linear Hamiltonian systems which generalises a recent result of Hu and Portaluri.Comment: 19 page

The Equivariant Spectral Flow and Bifurcation for Functionals with Symmetries -- Part I

We consider bifurcation of critical points from a trivial branch for families of functionals that are invariant under the orthogonal action of a compact Lie group. Based on a recent construction of an equivariant spectral flow by the authors, we obtain a bifurcation theorem that generalises well-established results of Smoller and Wasserman as well as Fitzpatrick, Pejsachowicz and Recht. Finally, we discuss some elementary examples of strongly indefinite systems of PDEs and Hamiltonian systems where the mentioned classical approaches fail but an invariance under an orthogonal action of $\mathbb{Z}_2$ makes our methods applicable and yields the existence of bifurcation

Bifurcation of equilibrium forms of an elastic rod on a two-parameter Winkler foundation

We consider two-parameter bifurcation of equilibrium states of an elastic rod on a deformable foundation. Our main theorem shows that bifurcation occurs if and only if the linearization of our problem has nontrivial solutions. In fact our proof, based on the concept of the Brouwer degree, gives more, namely that from each bifurcation point there branches off a continuum of solutions

An example concerning equivariant deformations

We give an example of $Z_2$-space $X$ with a property that the identity map $\text{\rm{id}}_X:X\to X$ as well as its restriction to the fixed point set of the group action $\text{\rm{id}}^{Z_2}:X^{Z_2}\to X^{Z_2}$ are deformable to fixed point free maps whereas there is no fixed point free map in the equivariant homotopy class of the identity $[\text{\rm{id}}_X]_{ Z_2}$

The Borsuk-Ulam property for cyclic groups

An orthogonal representation $V$ of a group $G$ is said to have the Borsuk-Ulam property if the existence of an equivariant map $f:S(W) \rightarrow S(V)$ from a sphere of representation $W$ into a sphere of representation $V$ implies that $\dim W \leq \dim V$. It is known that a sufficient condition for $V$ to have the Borsuk-Ulam property is the nontriviality of its Euler class ${\text {\bf e}}(V)\in H^{*} (BG;\mathcal R)$. Our purpose is to show that ${\text {\bf e}}(V) \neq 0$ is also necessary if $G$ is a cyclic group of odd and double odd order. For a finite group $G$ with periodic cohomology an estimate for $G$-category of a $G$-space $X$ is also derived