problem sheets
1. Homotopy and the properties of the fundamental group
Homotopy
1. Show that any non-surjective map f:X→Snis homotopic to a constant map.
2. Let f, g :X→Snbe such that for any x∈X,f(x) and g(x) are not antipodal points on the
sphere. Show that f≃g.
3. Show that when nis odd, the antipodal map Sn→Sn, given by negation of unit vectors
x7→ −x, is homotopic to the identity map of Sn.
4. A space which is homotopy-equivalent to a point is called contractible. Show that a space is
contractible if and only if its identity map is homotopic to a constant map.
5. The M¨obius strip Mis defined as I×Iquotiented by the relation (x, 0) ∼(1 −x, 1),∀x∈I.
Prove that S1×Iis homotopy-equivalent to the M¨obius strip M.
6. Show that R3−S1(the complement of the unit circle in the (x, y)-plane) is homotopy-
equivalent to the one-point union (obtained by identifying one point from each) S1∨S2.
7. Classify the capital letters of the alphabet up to homeomorphism and up to homotopy-
equivalence! (Assume that S1, S 1∨S1and a point are not homotopy-equivalent to one another.)
8. (Tricky but important!) Let f, g :S1→Xbe two maps from the circle to a topological space
X. Define a space P=X∪fB2by “attaching a disc along f”: form the disjoint union X∐B2and
then identify each point x∈S1=∂B2with its image f(x)∈X. Define Q=X∪gB2similarly.
Prove that if f≃g, then P≃Q; thus, “the homotopy type of X∪fB2depends only on the
homotopy class of the attaching map”.
Properties of the fundamental group
9. Let Xbe a path-connected, simply-connected (having trivial fundamental group) space, and
let x, y be points of X. Show that all paths from xto yare homotopic rel {0,1}.
10. Let Xand Ybe topological spaces, let Aa subspace of Xand let f:A→Ybe a map.
A map F:X→Yis said to be an extension of fif its restriction to Ais given by f. Show that
the fundamental group of a path-connected space Xis trivial if and only if every continuous map
f:S1→Xhas an extension to a continuous map F:B2→X.
11. Show that π1(X×Y, (x0, y0)) ∼
=π1(X, x0)×π1(Y, y0).
12. Let Nand Sbe the poles of the sphere Sn. Supposing that n≥2, prove that any path in
Snmay be written as a composite of finitely many paths, each of which is contained in Sn− {N}
or Sn− {S}, and consequently that π1(Sn) = 1 for n≥2.
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