Download Qualifying Exam in Topology

Northeastern University
Department of Mathematics
Qualifying Exam in Topology
September 2010
Do the following six problems. Give proofs or justifications for each statement you make.
Draw pictures when needed. Be as clear and concise as possible. Show all your work.
1. Let Z be the set of integers. An arithmetic progression is a subset of the form Aa,b =
{a + nb | n ∈ Z}, with a, b ∈ Z and b 6= 0.
(a) Prove that the collection of arithmetic progressions, A = {Aa,b | a, b ∈ Z, b 6= 0}, is
a basis for a topology on Z.
(b) Is Z endowed with this topology a Hausdorff space?
(c) Is Z endowed with this topology a compact space?
2. Let X be a locally path-connected space.
(a) Let U be a connected, open subset of X. Show that U is path-connected.
(b) Let f : X → Y be a continuous, surjective map. Is the space Y locally pathconnected?
(c) Let f : X → Y be a quotient map. Is the space Y locally path-connected?
3. (a) Define the two notions: “homotopy between two maps” and “homotopy equivalence
between two topological spaces.”
(b) Give an example of topological spaces X and Y that have the same homotopy type
but are not homeomorphic.
(c) Give an example of (path-connected) topological spaces X and Y that have isomorphic fundamental groups but are not homotopy equivalent.
(d) Give an example of (path-connected) topological spaces X and Y that have isomorphic first homology groups but non-isomorphic fundamental groups.
4. Let
(a)
(b)
(c)
(d)
(e)
S n be the unit sphere in Rn+1 .
Given a continuous map f : S n → S n , define the degree of f .
If f has nonzero degree, show that f is onto.
What are the possible degrees of maps f : S 0 → S 0 ?
Let a : S n → S n , a(x) = −x be the antipodal map. Compute the degree of a.
Show that each sphere S n , n > 0, admits maps of arbitrary degree.
5. Let T 2 be the 2-torus, and let K be the Klein bottle.
(a) Define a two-fold covering map p : T 2 → K.
(b) Pick base-points e ∈ T 2 and b ∈ K with p(e) = b. Determine the induced homomorphism p] : π1 (T 2 , e) → π1 (K, b).
(c) Is p] injective? If not, what is ker(p] )? Is p] surjective? If not, what is coker(p] )?
(d) Determine the induced homomorphism p∗ : H1 (T 2 ; Z) → H1 (K; Z).
(e) Is p∗ injective? If not, what is ker(p∗ )? Is p∗ surjective? If not, what is coker(p∗ )?
6. Let
(a)
(b)
(c)
(d)
X = RP2 × K be the product of the real projective plane with a Klein bottle.
Find a CW-decomposition of X.
Determine the chain complex (C• (X), d) associated to this cell decomposition.
Compute the homology groups H∗ (X, Z).
Compute the homology groups H∗ (X, Z2 ).