Prove that is surjective for every integer but is not necessarily surjective.

# 1 is to q as n! is to…

If one believes that is a vector space over the field with one element , then a permutation of is just a complete flag. Concretely, a permutation corresponds to the chain of subspaces .

As we all know, there are permutations of , or equivalently complete flags of . Give a formula for the number of complete flags of , which will be a -analogue for the factorial.

# 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.

# Rank isn’t like dimension

Let be the free group in the generators and . Consider the group morphism defined by . Prove that is a free group of infinite rank. (Hint: Think topologically!)

# Faithful representations and tensor products

Let be a finite group (or a compact Lie group). Prove that if is a faithful finite dimensional complex representation of then any irreducible representation embeds in some tensor product of .

# 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.

# Not an average calculus problem

Prove that for any complex numbers one can find a nonempty subset such that