In the 3-manifold case, a lens space can be visualized as the result of gluing two solid tori together by a homeomorphism of their boundaries.
12.
A lens space is not determined by its fundamental group ( there are non-homeomorphic lens spaces with isomorphic fundamental groups ); but any other spherical manifold is.
13.
A lens space is not determined by its fundamental group ( there are non-homeomorphic lens spaces with isomorphic fundamental groups ); but any other spherical manifold is.
14.
In modern terms, lens spaces are determined by " simple " homotopy type, and there are no normal invariants ( like characteristic classes ) or surgery obstruction.
15.
Use a Massey product to show that the homotopy type of the configuration space of two points in a lens space depends non-trivially on the simple homotopy type of the lens space.
16.
Use a Massey product to show that the homotopy type of the configuration space of two points in a lens space depends non-trivially on the simple homotopy type of the lens space.
17.
Another invariant is the homotopy type of the configuration spaces showed that homotopy equivalent but not homeomorphic lens spaces may have configuration spaces with different homotopy types, which can be detected by different Massey products.
18.
This is in contrast to projective polyhedra the sphere does cover projective space ( and also lens spaces ), and thus a tessellation of projective space or lens space yields a distinct notion of polyhedron.
19.
This is in contrast to projective polyhedra the sphere does cover projective space ( and also lens spaces ), and thus a tessellation of projective space or lens space yields a distinct notion of polyhedron.
20.
Other lens spaces have even the same homotopy type ( and thus isomorphic fundamental groups and homology ), but not the same homeomorphism type; they can thus be seen as the birth of geometric topology of manifolds as distinct from algebraic topology.
