Exercises:Saul - Algebraic Topology - 7/Exercise 7.6
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Contents
Exercises
Exercises 7.6
Question
- Compute the singular homology groups of [ilmath]T^2:\eq\mathbb{S}^1\times\mathbb{S}^1[/ilmath] and of [ilmath]X:\eq\mathbb{S}^1\vee\mathbb{S}^1\vee\mathbb{S}^2[/ilmath]
- Prove that [ilmath]T^2[/ilmath] and [ilmath]X[/ilmath] are not homotopy equivalent spaces
Solutions
Part I
- [ilmath]H_0^\Delta(T^2)\cong\mathbb{Z} [/ilmath]
- [ilmath]H_1^\Delta(T^2)\cong\mathbb{Z}\oplus\mathbb{Z}\eq:\mathbb{Z}^2[/ilmath]
- [ilmath]H_2^\Delta(T^2)\cong\mathbb{Z} [/ilmath]
- [ilmath]H_n^\Delta(T^2)\cong 0[/ilmath][Note 1] for [ilmath]n\in\{3,\ldots\}\subset\mathbb{N} [/ilmath]
To calculate the homology groups of [ilmath]X[/ilmath] we must start by computing the boundary of the generators (which are extended to group homomorphisms on free Abelian group by the characteristic property of free Abelian groups):
- [ilmath]2[/ilmath]-simplices:
- [ilmath]\partial_2(A)\eq c[/ilmath], [ilmath]\partial_2(B)\eq c[/ilmath]
- [ilmath]1[/ilmath]-simplices:
- [ilmath]\partial_1(\alpha)\eq v_1-v_0[/ilmath], [ilmath]\partial_1(\beta)\eq v_2-v_0[/ilmath], [ilmath]\partial_1(\gamma)\eq v_2-v_1[/ilmath]
- [ilmath]\partial_1(a)\eq v_2-v_3[/ilmath], [ilmath]\partial_1(b)\eq v_4-v_3[/ilmath], [ilmath]\partial_1(c)\eq 0[/ilmath]
- [ilmath]\partial_1(x)\eq v_4-v_5[/ilmath], [ilmath]\partial_1(y)\eq v_4-v_6[/ilmath], [ilmath]\partial_1(z)\eq v_6-v_5[/ilmath]
- Note that [ilmath]\partial_0:(\text{anything})\mapsto 0[/ilmath], so we do not mention in here, it is also obvious that the entire of the domain is the kernel of [ilmath]\partial_0[/ilmath] from this definition.
Now the images and kernels:
- [ilmath]\text{Im}(\partial_2)\eq \langle c\rangle[/ilmath] - by inspection
- [ilmath]\text{Ker}(\partial_2)\eq \langle A-B \rangle[/ilmath] - by inspection
- [ilmath]\text{Im}(\partial_1)\eq \langle v_0-v_1,\ v_0-v_2,\ v_2-v_3,\ v_3-v_4,\ v_4-v_5,\ v_6-v_4\rangle[/ilmath]
- Calculated by starting with just the first vector, [ilmath]\partial_1(\alpha)[/ilmath], going over each of the remaining vectors and adding it to this linearly-independent set if said vector is not a linear combination of what we have in this set.
- [ilmath]\partial_1(\gamma)\eq\partial_1(\beta)-\partial_1(\beta)[/ilmath], then we see [ilmath]\partial_1(c)[/ilmath] is the identity element, [ilmath]0[/ilmath], so cannot be in a basis set for obvious reasons, and [ilmath]\partial_1(z)\eq\partial_1(x)-\partial_1(y)[/ilmath], hence the chosen basis consists of all but these.
- [ilmath]\text{Ker}(\partial_1)\eq\langle\alpha-\beta+\gamma, c, x-y-z\rangle[/ilmath]
- Computed by "RReffing in [ilmath]\mathbb{Z} [/ilmath]", I may upload a picture of these matrices, but as it is [ilmath]10[/ilmath] columns by [ilmath]7[/ilmath] rows I am not eager to type both the starting matrix and it's reduced form out.
- [ilmath]\text{Ker}(\partial_0)\eq\langle v_0,v_1,v_2,v_3,v_4,v_5,v_6\rangle[/ilmath], as discussed above and from the definition of [ilmath]\partial_0[/ilmath]
Note that for any higher values:
- [ilmath]\text{Ker}(\partial_n)\eq 0[/ilmath] for [ilmath]n\ge 3[/ilmath], [ilmath]n\in\mathbb{N} [/ilmath], as these groups are trivial groups, and a group homomorphism must map the identity to the identity, couple this with the domain of [ilmath]\partial_n[/ilmath] has one element and the result follows.
- [ilmath]\text{Im}(\partial_n)\eq 0[/ilmath] for [ilmath]n\ge 3[/ilmath], [ilmath]n\in\mathbb{N} [/ilmath], as the image of a trivial group must be the identity element of the co-domain group.
Homology groups of [ilmath]X[/ilmath]
- [math]H_2^\Delta(X):\eq\frac{\text{Ker}(\partial_2)}{\text{Im}(\partial_3)}\eq\frac{\langle A-B \rangle}{0}\cong \langle A-B \rangle\cong\mathbb{Z} [/math]
- [math]H_1^\Delta(X):\eq\frac{\text{Ker}(\partial_1)}{\text{Im}(\partial_2)}\eq\frac{\langle\alpha-\beta+\gamma, c, x-y-z\rangle}{\langle c\rangle}\cong\langle\alpha-\beta+\gamma,x-y-z\rangle\cong\mathbb{Z}^2 [/math]
- [math]H_0^\Delta(X):\eq\frac{\text{Ker}(\partial_0)}{\text{Im}(\partial_1)}\eq\frac{\langle v_0,v_1,v_2,v_3,v_4,v_5,v_6\rangle}{\langle v_0-v_1,\ v_0-v_2,\ v_2-v_3,\ v_3-v_4,\ v_4-v_5,\ v_6-v_4\rangle} [/math]
Part II
Notes
- ↑ Here [ilmath]0[/ilmath] denotes the trivial group
References