Math Problems Vault

Problem 6.2.1

(a)(b)

For a $\Delta$ -complex $K$ , the associated chain complex $(C_i^{\Delta }(K), \partial_i)$ has $C_i^{\Delta }(K)$ the free abelian group with one generator for each $i$ -simplex $[v_0, \ldots , v_i]$ in the $\Delta$ -complex structure (using the order on the vertices that comes with the $\Delta$ -complex structure), and $\partial_0 = 0$ , with $\partial_i : C_i \to C_{i-1}$ for $i > 0$ defined on generators by

\partial [v_0, \ldots , v_i] = \sum _{k=0}^{i} (-1)^k [v_0, \ldots , \widehat{v_k}, \ldots , v_i],

where $\widehat{v_k}$ indicates that the vertex $v_k$ is omitted, extended linearly to all of $C_i^{\Delta }(K)$ .

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(a)

Show that the composition $\partial_1 \partial_2([v_0, v_1, v_2]) = 0$ .

(b)

Show that in general $\partial_i \partial_{i+1} = 0$ .

Remark.

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For part (b): it suffices to check this on a generator $[v_0, \ldots , v_{i+1}]$ . For any $(i-1)$ -simplex formed from this by omitting two vertices, show that this term occurs twice with opposite signs in $\partial_i \partial_{i+1}([v_0, \ldots , v_{i+1}])$ .

Problem 6.2.2

(a)(b)

Suppose $K$ is a path connected $\Delta$ -complex. Recall that the $\Delta$ -homology groups are defined by $H_i^{\Delta }(K) = \ker (\partial_i)/\mathrm{im}(\partial_{i+1})$ , using the chain complex of Exercise 6.2.1.

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(a)

Show that $K^1$ is path connected, and in particular, given any two vertices $v_1, v_2$ , there is a sequence of directed edges $e_1, e_2, \ldots , e_k$ (which give some of the $1$ -simplices, possibly with the opposite directions) which connect $v_1$ with $v_2$ .

(b)

Show that factoring out by $\mathrm{im}(\partial_1)$ serves to identify all vertices in $C_0^{\Delta }(K) = \ker (\partial_0)$ by showing that $[w] - [v] \in \mathrm{im}(\partial_1)$ for any $v_1, v_2$ . Conclude that $H_0^{\Delta }(K) = \mathbb {Z}$ , with a generator coming from each vertex.

Remark.

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For part (a): first show that adding higher-dimensional simplices does not change path connectivity, so that $K^1$ and $K$ have the same path connectivity properties. Then show that a path connecting two vertices may be replaced by a simplicial path.

Problem 6.2.3

Suppose that $K$ is a $\Delta$ -complex with $n$ path components $K_1, \ldots , K_n$ . Show that each chain group $C_i^{\Delta }(K) = C_i^{\Delta }(K_1) \oplus \cdots \oplus C_i^{\Delta }(K_n)$ and the boundary homomorphisms respect this splitting since $\partial_i : C_i^{\Delta }(K_j) \to C_{i-1}^{\Delta }(K_j)$ . Use this to show that $H_i^{\Delta }(K) = H_i^{\Delta }(K_1) \oplus \cdots \oplus H_i^{\Delta }(K_n)$ . Thus the computation of homology just decomposes into the computation of the homology groups of the path components. Conclude that $H_0^{\Delta }(K)$ is the free abelian group with $n$ generators.

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Problem 6.2.4

Use the $\Delta$ -complex structure of the Klein bottle $K$ from Figure 5.16 to compute the $\Delta$ -homology groups of $K$ .

$\Delta -complex structures for T, K.$ \Delta -complex structures for T, K.

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Problem 6.2.5

Use the $\Delta$ -complex structure of the sphere $S$ from Exercise 5.5.3 to compute the $\Delta$ -homology groups of $S$ .

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Problem 6.2.6

Use the $\Delta$ -complex structure of the projective plane $P$ from Exercise 5.5.4 to compute the $\Delta$ -homology groups of $P$ .

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Problem 6.2.7

Show that a compact connected surface $M$ without boundary with a $\Delta$ -complex structure satisfies $H_2^{\Delta }(M) \simeq \mathbb {Z}$ when it is orientable as given in the remark below, and $H_2^{\Delta }(M) \simeq 0$ when it is nonorientable. Here we are using $0$ for the trivial abelian group with one element.

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Remark.

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Orientability of a surface given as a simplicial complex is phrased as follows. First, the condition that the space is a surface (possibly with boundary) is phrased by requiring that, for each $1$ -simplex, there are either exactly two $2$ -simplices with that simplex as a face (for an interior point) or there is exactly one $2$ -simplex with that $1$ -simplex as a face (for a boundary point). A simplicial complex is called orientable if we can choose an orientation for each $2$ -simplex so that, for each interior $1$ -simplex, the two $2$ -simplices with it as a face impose opposite orientations on that $1$ -simplex. In a $\Delta$ -complex where there are identifications on the boundary of a $2$ -simplex, it is also possible to have a single $2$ -simplex where two edges emanating from a vertex are identified (this occurs for the sphere in Exercise 6.2.5). In the $\Delta$ -complex case, each simplex has the orientation given by its ordered vertices, and this determines how boundaries are mapped and the corresponding boundary map for the $\Delta$ -chain complex. Then the condition required for the orientability of the surface is that orientations extend across identifications. This translates to the condition that we can select a sign $\pm 1$ for each ordered $2$ -simplex so that, for any $1$ -simplex, it occurs with opposite signs in the boundary of two different signed $2$ -simplices or occurs with both signs in the boundary of a single $2$ -simplex.

Hint for the exercise itself: show that the only way to get a $2$ -cycle is from a multiple of $\sum \epsilon (i) \sigma_i$ , where $\epsilon (i) = \pm 1$ and we sum over all $2$ -simplices. You may use the fact that connectedness implies that, given any two $2$ -simplices, there is a chain of $2$ -simplices going from one to the other with a common face. Start with an arbitrary sum $\sum n_i \sigma_i$ of $2$ -simplices and show that the only way it could be a cycle is that all $\left|n_i\right|$ are equal.