Let be a natural number. Give an example of a compact -manifold such that .

# differential topology

# Bounding the projective plane

Is there a compact 3-manifold having as boundary?

# Cutting holes

Let be a compact, connected oriented manifold (without boundary) and let be a closed connected hypersurface. Prove that if is simply connected then is **not** connected.

# Foiled by multiple roots

Let be the set of monic complex polynomials of degree 3 with multiple roots, and regard it as a subspace of by looking at the non-leading coefficients. Prove that is homotopy equivalent to , where is the trefoil knot.

# Your degree is a chance for expansion

Let and be compact oriented smooth manifolds of the same dimension, with connected. Let be a smooth function. Take a regular value of and, for , define as if preserves orientation at and as if it reverses the orientation. We define the degree of as the number .

Prove the following extension theorem: Assume is a -dimensional sphere and is the boundary of a compact oriented -manifold . There exists a smooth extension of if and only if .

# Skip the sphere

Compute the homotopy groups of the compact surfaces.

# Drawable 3-manifolds

For an embedded compact 3-manifold , implies .