A slope, i . e . unoriented isotopy class of simple closed curves on these boundaries, thus has a well-defined length by taking the minimal Euclidean length over all curves in the isotopy class.
12.
Whereas the real projective plane describes the set of all unoriented lines through the origin in "'R "'3, the "'oriented projective plane "'describes lines with a given orientation.
13.
Given a point " p " of the sphere, we get an unoriented geodesic by intersecting the sphere with the plane passing through the origin that is perpendicular to the chord joining the origin to " p ".
14.
To see that " A " and " B " are unoriented equivalent, simply note that they both may be constructed from the same pair of disjoint knot projections as above, the only difference being the orientations of the knots.
15.
The set of cobordism classes of closed unoriented " n "-dimensional manifolds is usually denoted by \ mathfrak { N } _ n ( rather than the more systematic \ Omega _ n ^ { \ text { O } } ); it is an abelian group with the disjoint union as operation.
16.
The cobordism class [ M ] \ in \ mathfrak { N } _ n of a closed unoriented " n "-dimensional manifold " M " is determined by the Stiefel Whitney characteristic numbers of " M ", which depend on the stable isomorphism class of the tangent bundle.
17.
From the point of view of spectra, unoriented cobordism is a product of Eilenberg MacLane spectra " MO " = " H " ( ? " ( " MO " ) ) while oriented cobordism is a product of Eilenberg MacLane spectra rationally, and at 2, but not at odd primes : the oriented cobordism spectrum " MSO " is rather more complicated than " MO ".
18.
If " M " is an oriented manifold, Aut ( " M " ) would be the orientation-preserving automorphisms of " M " and so the mapping class group of " M " ( as an oriented manifold ) would be index two in the mapping class group of " M " ( as an unoriented manifold ) provided " M " admits an orientation-reversing automorphism.
