Download Problems for the exam

Prof. Rahul Pandharipande
ETH Zürich
Algebraic Topology I
HS 2014
Problems for the exam
1. Let p ∈ CP2 and q ∈ RP3 . Is there a compact surface which is homotopy equivalent to CP2 \ {p} ? Is there a compact surface which is
homotopy equivalent to RP3 \ {q} ?
2. Does the Borsuk-Ulam Theorem hold for the torus? In other words,
is it true that for every continuous map
f : S 1 × S 1 → R2 ,
there exists (v, w) ∈ S 1 × S 1 such that
f (v, w) = f (−v, −w) ?
3. Let Σ1,3 be the surface of genus 1 with three points removed. The
fundamental group of Σ1,3 is a free group. Compute its rank.
4. Let X be the topological space obtained from the unit sphere S 2 ⊂ R3
by identifying the north and south poles:
X = S2/ ∼
where (0, 0, 1) ∼ (0, 0, −1) .
Calculate all the homology groups H∗ (X, Z).
5. What is the fundamental group of S 1 × S 1 ? Viewing S 1 ⊂ C as the
unit complex numbers, define a covering map
φ : R1 × S 1 → S 1 × S 1
by φ(x, ξ) = (e2πix , ξ 2 ). What is the subgroup of the fundamental
group of S 1 × S 1 corresponding to the covering map φ?
6. Let L1 , L2 , L3 be pairwise disjoint straight lines in R3 . Calculate all
S
the homology groups H∗ (R3 \ 3i=1 Li , Z).
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7. Calculate all the homology groups H∗ (R4 \{P1 , P2 }, Z) where P1 is the
plane
P1 = {(x, y, z, w)|x = y = 0}
and P2 is the plane
P2 = {(x, y, z, w)|z = w = 0}.
8. Is it possible to realize CP2 as a finite CW-complex with an even
number of cells in every dimension?
9. Viewing S 1 ⊂ C2 as the unit complex numbers, define a continuous
map
φ : S1 × S1 → S1 × S1
by φ(ξ1 , ξ2 ) = (ξ1 , ξ1 ξ2 ). Is φ homotopic to the identity map?
10. Let ι : S 1 ,→ S 2 be the equator. Let p ∈ S 2 be the north pole. Calcalate the relative homology groups H∗ (S 2 , S 1 ) and H∗ (S 2 \ {p}, S 1 ).
11. Let p : X → Y be a covering map of connected topological spaces. Is
p∗ : H2 (X, Z) → H2 (Y, Z)
always injective? Is it always surjective? Prove or provide counterexamples.
12. Let S 2 ⊂ R3 be the unit sphere. Let
φ : S2 → S2
be a continuous map satisfying φ(v) = φ(−v) for all v ∈ S 2 . Can φ
have non-zero degree?
13. Let X be a finite CW-complex of Euler characteristic 3, let p : Y → X
be a 5-sheeted covering map. What is the Euler characteristic of Y ?
14. Let φ : S 3 → S 3 be a continuous map which is homotopic to the identity. Must φ have a fixed point? Prove or provide a counterexample.
∼
15. Is it true that if Hk (X, Z) = Hk (Y, Z) for all k then X and Y are
homotopy equivalent? Prove or provide a counterexample.
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