MA40040
University of Bath
DEPARTMENT OF MATHEMATICAL SCIENCES
EXAMINATION
MA40040: ALGEBRAIC TOPOLOGY
Friday 20th January 2012, 16.3018.30
No calculators may be brought in and used.
Full marks will be given for correct answers to THREE questions.
Only th
M40: Exercise sheet 3
1. Let f , g Path(X, x, y). Show that f g if and only if f g x .
2. Let x, y Rn . Show that 1 (Rn , x) = cfw_x . Deduce that 1 (Rn , x, y) consists of a
single element.
3. Let x, y X where X is path-connected. Recall that 1 (X, x, y)
M40: Christmas exercise sheet
Some revision questions
1. (1994 exam) Let n N. The map p : S 1 S 1 given by p(eit ) = eint is a covering
map with group of deck translations H. Show that
H Z/nZ.
=
2. (1991 exam) Let f : S 1 S 1 be continuous and not homotop
M40: Exercise sheet 6
1. Show that the retraction dened in the Brouwer Fixed Point Theorem is indeed continuous.
2. Let X be homeomorphic to the closed unit disc in R2 . Show that any continuous map
f : X X has a xed point. (This is an easy application of
M40: Exercise sheet 7
1. Let p : E X be a covering space and consider the action of 1 (X, x0 ) on p1 (cfw_x0 ).
Show:
(a) if E is path-connected, this action is transitive: that is, if e1 , e2 p1 cfw_x0 then
there is [] 1 (X, x0 ) with e1 [] = e2 .
(b) t
M40: Exercise sheet 5
1.
(a) Let A X be a deformation retract of X. Show that the inclusion i : A X is
a homotopy equivalence with the retraction as homotopy inverse.
(b) Let A Rn be convex and a A. Show that cfw_a is a deformation retract of A.
(c) Show
M40: Exercise sheet 4
1. Let 1 , 2 : X Y and 1 , 2 : Y Z be continuous maps. If 1 2 and 1 2 ,
show that 1 1
2 2 . Deduce that homotopy equivalence is an equivalence
relation on topological spaces.
2. Let X be a topological space. Recall that the path comp
M40: Exercise sheet 1 (mostly revision)
1. Let X be a topological space and A, B be closed subsets of X such that A B = X.
Let : X Y be a map into a topological space Y such that |A and |B are
continuous with respect to the induced topologies on A and B r
MA40040
1.
(a)
2.
Let 0 , 1 be paths in a topological space X. What does it mean to say that 0 is
based homotopic to 1 : 0 1 ?
Show that is an equivalence relation.
(b) Let x0 , x1 be points in a topological space X and let : I X be a path from x0
to x1 .
M40: Exercise sheet 2
Fun with topological groups
1. Let X, Y be topological spaces and let A X, B Y have the induced topology.
(a) Show that the product topology on A B is the same as the topology on A B
induced by the product topology on X Y .
(b) If f
MA30231
University of Bath
DEPARTMENT OF MATHEMATICAL SCIENCES
EXAMINATION
M30231: PROJECTIVE GEOMETRY
SAMPLE EXAM, duration 2 hours
No calculators may be brought in and used.
Full marks will be given for correct answers to THREE questions.
Only the best
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M231: Exercise sheet 5
1. Let v1 , . . . , vn be a basis of a vector space V and dene v1 , . . . , vn in V by
vi (vj ) =
1
0
if i = j;
otherwise.
and extending by linearity.
(a) If v =
(b) Prove
n
i=1 i vi , what is vj (v)?
that v1 , . . . , vn is a basis
University of Bath
DEPARTMENT OF MATHEMATICAL SCIENCES
Outline solution to Examination Questions
Unit Code Unit Title
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091E!) (PW) 1 "l 74/1 E333
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M231: Exercise sheet 2
Lines and planes
1. Let A, B, C be three points in a projective plane with homogeneous coordinates
[a0 , a1 , a2 ], [b0 , b1 , b2 ] and [c0 , c1 , c2 ], respectively. Show that A, B, C are collinear1 if
and only if the determinant
a
M231: Exercise sheet 1
1. Let
(a)
(b)
(c)
V be a vector space.
(Very easy) Let U1 , U2 , W V with U1 , U2 W . Show that U1 + U2 W .
Deduce that if X1 , X2 , Y P(V ) with X1 , X2 Y then the join X1 X2 Y .
Show that X1 X2 is the intersection of all projecti
M231: Exercise sheet 4
On projections
1. Let L1 and L2 be distinct projective lines in a projective plane that intersect at A.
Let : L1 L2 be a projective transformation such that A = A.
Show that is a projection from some point of the plane.
Hint: Choose
M231: Exercise sheet 3
1. Here is a perspective picture of train tracks
in a desert. The sleepers on the track are
parallel and 1 metre apart. I have drawn the
rst two sleepers. Explain how to accurately
draw the next sleeper.
2. Show that three points in