### Conley index orientations

#### Abstract

The homotopy Conley index along heteroclinic

solutions of certain parabolic evolution equations is zero

under appropriate assumptions.

This result implies that the so-called connecting homomorphism

associated with a heteroclinic solution is an isomorphism. Hence, using $\mathbb{Z}$-coefficients

it can be viewed as either $1$ or $-1$ - depending on the choice of generators

for the homology Conley index. We develop a method to choose such generators,

and compute the connecting homomorphism

relative to these generators.

solutions of certain parabolic evolution equations is zero

under appropriate assumptions.

This result implies that the so-called connecting homomorphism

associated with a heteroclinic solution is an isomorphism. Hence, using $\mathbb{Z}$-coefficients

it can be viewed as either $1$ or $-1$ - depending on the choice of generators

for the homology Conley index. We develop a method to choose such generators,

and compute the connecting homomorphism

relative to these generators.

#### Keywords

Conley index theory; Morse decompositions; reaction diffusion equations

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