problem sheets
6. Homology I
Simplicial homology
1. For each of the following spaces, describe a simplicial (or ∆-) complex representing it, compute
its homology groups, and describe their generators: The figure “8”, S2,RP2, the sphere with an
equatorial disc glued to it, the torus with a disc attached inside it.
2. Let Kbe a simplicial complex with a fixed base vertex a0, and let E(K, a0) be the edge-group
defined as follows. Consider the set of edge-loops i.e. strings a0a1. . . an, where each aiis a vertex,
an=a0, and each (ai, ai+1) is a (0- or 1-) simplex. Define the obvious composition, and work
modulo the two relations of edge-homotopy, which say that
a0a1. . . ai−1aiai+1 . . . an∼a0a1. . . ai−1ai+1 . . . an
provided (ai−1, ai, ai+1) spans a (0-, 1- or 2-) simplex, and that ai+1 may be cancelled from the
string if it equals ai. Show that there is a surjection E(K, a0)→H1(K) with kernel the commutator
subgroup, and prove E(K, a0)∼
=π1(|K|, a0) by using van Kampen’s theorem (it will probably help
to take a maximal tree).
3. Let Kbe a simplicial complex with no simplexes of dimension higher than n. Suppose that
every (n−1)-simplex is a face of exactly two n-simplexes, and that any two n-simplexes may be
connected by a finite sequence of n-simplexes, each adjacent by an (n−1)-face to the last. Show
that (simplicial) Hn(K) is either trivial or isomorphic to Z; in the latter case a generator being the
sum of all the n-simplexes (suitably oriented).
Singular homology
4. Let Xbe a path-connected space with basepoint x0.
(a) Show that there is a natural map h:π1(X, x0)→H1(X) which is a homomorphism.
(b) Show that it factors through the abelianisation of π1(X, x0); that is, the group obtained by
quotienting by the subgroup generated by all commutators aba−1b−1.
(c) Show that his surjective. (Hint: given a 1-cycle z, look at the set of endpoints of its singular
1-simplexes, and choose a path joining each one to x0.)
(d) Show that his injective. (Hint: if some loop αis the boundary of a 2-chain u, join the corners
of the 2-simplexes in uto the basepoint in a similar way, so as to write αas a large composite of
loops. Then try to show that by reordering these (working modulo the commutator subgroup) it
can be made null-homotopic.)
5. Let a singular homology class c∈Hn(X) be represented by a singular n-cycle z. Use the
idea of question 3 to construct a simplicial complex K, a homology class µ∈Hn(K), and a map
f:|K| → Xsuch that f∗(µ) = c. Show that |K|, while not necessarily an n-manifold, is an
n-circuit, i.e. that its set of singular points (ones without a neighbourhood homeomorphic to Rn)
is a subcomplex of dimension less than or equal to n−2. This gives rise to a useful and very visual
representation of homology classes by manifolds with singularities of codimension ≥2.
Some algebra
6. Let 0 →Ai
→Bp
→C→0 be a short exact sequence of abelian groups. Show that the following
conditions are equivalent:
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