Where can I find the proof of the following fact: If $M$ is a contractible manifold of dimension $n\ge 5$, then the direct product of $M$ and $\mathbb{R}^{n+1}$ is homeomorphic to $R^{2n+1}$ ? 
Robert Ghrist is all about applied topology:  Sensor Network, Signal Processing, and Fluid Dynamics.  (homepage:  http://www.math.upenn.edu/~ghrist/index.html ).  For instance, we want to use the least number of sensors to cover a certain area, such that when we remove one sensor, a part of that area is undetectable.  We can form a complex of these sensors and hence its nerve, and use homology to determine whether there are any gaps in the sensor-collection.  I've met with him in person and he expressed confidence that this is going to be a big thing of the future.

There are also applications of cohomology to Crystallography (see Howard Hiller) and Quasicrystals in physics (see Benji Fisher and David Rabson).  In particular, it uses cohomology in connection with Fourier space to reformulate the language of quasicrystals/physics in terms of cohomology... Extinctions in x-ray diffraction patterns and degeneracy of electronic levels are interpreted as physical manifestations of nonzero homology classes.

Another application is on fermion lattices (http://arxiv.org/abs/0804.0174v2), using homology combinatorially.  We want to see how fermions can align themselves in a lattice, noting that by the Pauli Exclusion principle we cannot put a bunch of fermions next to each other.  Homology is defined on the patterns of fermion-distributions.
