Exercises:Saul - Algebraic Topology - 3/Exercise 3.2

Exercises

Exercise 3.2

Suppose that [ilmath](X,\mathcal{ J })[/ilmath] is a non-empty path-connected topological space, equipped with a [ilmath]\Delta[/ilmath]-complex structure. Show, directly from the definitions (Hatcher, of course...) that [ilmath]H^\Delta_0(X)\cong\mathbb{Z} [/ilmath]

• We may assume without proof that the [ilmath]1[/ilmath]-skeleton is path connected.

Notes

• [math]H^\Delta_n(X):\eq\frac{\text{Ker}(\partial_0)}{\text{Im}(\partial_1)} [/math]

As [ilmath]\partial_0[/ilmath] just sends everything to [ilmath]0[/ilmath] we see [ilmath]\text{Ker}(\partial_0)\eq \Delta_0(X)[/ilmath] - all the vertices. Thus, essentially, [ilmath]\text{Ker}(\partial_0)\cong\mathbb{Z}^{\#\text{vertices} } [/ilmath].

Okay now [ilmath]\partial_1:\Delta_1(X)\rightarrow\Delta_0(X)[/ilmath], what is its image?

• First of all, for [ilmath]f\in\Delta_1(X)[/ilmath] we see [ilmath]\partial_1(X):\eq\partial_1(\sum_{\alpha\in I_1}n_\alpha \sigma_\alpha)\eq\sum_{\alpha\in I_1}n_\alpha\partial_1(\sigma_\alpha)[/ilmath] [ilmath]\eq\sum_{\alpha\in I_1}n_\alpha(\text{Terminal}(\sigma_\alpha)-\text{Initial}(\sigma_\alpha))[/ilmath]
  • where [ilmath]I_1[/ilmath] is the set of [ilmath]1[/ilmath]-simplices involved in [ilmath]X[/ilmath], and [ilmath]n_\alpha\in\mathbb{Z} [/ilmath] with [ilmath]n_\alpha\neq 0[/ilmath] for only finitely many of the [ilmath]\alpha\in I_1[/ilmath]

This shows us that (sort of anyway) the image is spanned by various [ilmath]\text{Terminal}(\sigma_\alpha)-\text{Initial}(\sigma_\alpha) [/ilmath] (which are vertices)

Using [ilmath]X^1[/ilmath] to denote the [ilmath]1[/ilmath]-skeleton (consistent notation be damned) then Caveat:and this is the informal part for any two vertexes of [ilmath]X[/ilmath], say [ilmath]v_0[/ilmath] and [ilmath]v_1[/ilmath], there is a path through the "edges" of [ilmath]X^1[/ilmath], such that [ilmath]\partial_1(\text{that path})\eq v_1-v_0[/ilmath]

• We can make this a bit better! If [ilmath](p_i)_{i\eq 1}^k[/ilmath] is a representation of the path, where [ilmath]p_i[/ilmath] is an edge, such that the initial vertex of [ilmath]p_1[/ilmath] is [ilmath]v_0[/ilmath] and the final vertex of [ilmath]p_k[/ilmath] is [ilmath]v_1[/ilmath] we can represent the path by the formal linear combination: [ilmath]\sum_{i\eq 1}^k p_i[/ilmath].
  • Note the [ilmath]p_i[/ilmath] need not be unique, it could have several loops in it, that wont matter (as the boundaries of cycles are [ilmath]0[/ilmath])

Now what we can sort of do is... well we want that sum to collapse to a few terms. What we can do is consider each [ilmath]\sigma_\alpha[/ilmath] which is an edge - plus the path from its endpoint to some fixed point of our choice. The boundaries then will have the same terminal point, subtracting various initials (and some/one (?) will have the same terminal and initial, giving us our "one less")

It would have to be shown this is in bijection with the edges.