Exercises:Saul - Algebraic Topology - 7/Exercise 7.6

Exercises

Exercises 7.6

Question

1. 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]
2. Prove that [ilmath]T^2[/ilmath] and [ilmath]X[/ilmath] are not homotopy equivalent spaces

Solutions

Part I

[ilmath]\Delta[/ilmath]-complex for [ilmath]X[/ilmath]
[ilmath]\xymatrix{ \bullet_{v_0} \ar[dd]_\alpha \ar[dr]^\beta & & \bullet_{v_3} \ar@2{->}[dr]^b \ar@2{->}[dl]_a & & \bullet_{v_5} \ar[dd]^z \\ & \bullet_{v_2} \ar@2{<-}[dr]_a \ar@{}[rr]|(.3){\LARGE{\mathbf{A} } }|(.7){\LARGE{\mathbf{ B} } } & & \bullet_{v_4} \ar@{<-}[ur]^x \\ \bullet_{v_1} \ar[ur]_\gamma & & \bullet_{v_3} \ar@2{->}[ur]_b \ar@2{->}[uu]^c_c & & \bullet_{v_6} \ar[ul]^y }[/ilmath]
[ilmath]0[/ilmath]-simplices:	[ilmath]v_0,v_1,v_2,v_3,v_4,v_5,v_6[/ilmath]	(7)
[ilmath]1[/ilmath]-simplices:	[ilmath]\alpha,\beta,\gamma,a,b,c,x,y,z[/ilmath]	(9)
[ilmath]2[/ilmath]-simplices:	[ilmath]A,B[/ilmath]	(2)

The homology groups of the [ilmath]2[/ilmath]-torus have been computed in previous assignments, the result was:

• [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] 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} \cong\mathbb{Z} [/math]
  • We compute this (if we want to go the hard way) by re-writing [ilmath]\langle v_0,v_1,v_2,v_3,v_4,v_5,v_6\rangle[/ilmath] step by step until it looks like [ilmath]\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], for example:
    • [ilmath]\text{Span}(v_0,v_1)\eq\text{Span}(v_0-v_1,v_1)[/ilmath], we do this by noticing that each "vector" (linear combination) in the span can be made from members in the other span, and visa-versa, so these spans are equal (we show [ilmath]\subseteq[/ilmath] and [math]\supseteq[/math]), and just keep going until the numerator looks like the denominator with some extra terms
• [math]H_n^\Delta(X)\eq\frac{\text{Ker}(\partial_n)}{\text{Im}(\partial_{n+1})}\eq\frac{0}{0}\cong 0[/math] for [ilmath]n\ge 3[/ilmath] and [ilmath]n\in\mathbb{N} [/ilmath] as discussed above

Singular homology groups

By Hatcher Theorem 2.27 found on page 128.6 (with [ilmath]A\eq\emptyset[/ilmath]) we see that the singular homology groups, [ilmath]H_n(S)[/ilmath] for a topological space [ilmath]S[/ilmath] are isomorphic to the simplicial (or delta-complex specifically) homology groups. That is:

• [ilmath]\forall n\in\mathbb{N}_0[H_n^\Delta(S)\cong H_n(S)][/ilmath], so we see:
  • [ilmath]H_0(X)\cong H_0^\Delta(X)\cong\mathbb{Z} [/ilmath] and [ilmath]H_0(T^2)\cong H_0^\Delta(T^2)\cong\mathbb{Z} [/ilmath], then
  • [ilmath]H_1(X)\cong H_1^\Delta(X)\cong\mathbb{Z}\oplus\mathbb{Z}\cong\mathbb{Z}^2 [/ilmath] and [ilmath]H_1(T^2)\cong H_1^\Delta(T^2)\cong\mathbb{Z}\oplus\mathbb{Z}\cong\mathbb{Z}^2 [/ilmath], then
  • [ilmath]H_2(X)\cong H_2^\Delta(X)\cong\mathbb{Z} [/ilmath] and [ilmath]H_2(T^2)\cong H_2^\Delta(T^2)\cong\mathbb{Z} [/ilmath], and lastly
  • for [ilmath]n\in\mathbb{N} [/ilmath] and [ilmath]n\ge 3[/ilmath]:
    • [ilmath]H_n(X)\cong H_n^\Delta(X)\cong 0 [/ilmath] and [ilmath]H_n(T^2)\cong H_n^\Delta(T^2)\cong 0 [/ilmath]

Part II

Observe that both [ilmath]X[/ilmath] and [ilmath]T^2[/ilmath] are path-connected topological spaces. As a result we will write [ilmath]\pi_1(X)[/ilmath] or [ilmath]\pi_1(T^2)[/ilmath] for their fundamental groups (respectively), knowing that for a path-connected space the fundamental groups based anywhere are isomorphic to each other.

Note that:

• [ilmath]\pi_1(T^2)\cong \mathbb{Z}^2[/ilmath] and
• [ilmath]\pi_1(X)\cong \mathbb{Z}\ast\mathbb{Z} [/ilmath]

Recall the following:

Claim:
Hatcher - p37	Proposition 1.18: if [ilmath]\varphi:X\rightarrow Y[/ilmath] is a homotopy equivalence, then the induced homomorphism [ilmath]\varphi_*:\pi_1(X,x_0)\rightarrow\pi_1(Y,\varphi(x_0))[/ilmath] is an isomorphism for all [ilmath]x_0\in X[/ilmath]

Specifically notice that: [ilmath]\text{Homotopy equivalent}\implies\text{isomorphic fundamental groups} [/ilmath], we invoke the contrapositive, which is logically equivalent (if and only if) to this:

• [ilmath]\neg(\text{isomorphic fundamental groups})\implies\neg(\text{Homotopy equivalent})[/ilmath]

or

• fundamental groups are not isomorphic [ilmath]\implies[/ilmath] the spaces are not homotopy equivalent

Clearly [ilmath]\mathbb{Z}^2\ncong\mathbb{Z}\ast\mathbb{Z} [/ilmath], thus [ilmath]X \not\simeq T^2[/ilmath], as required (where [ilmath]X\simeq T^2[/ilmath] would denote that [ilmath]X[/ilmath] (the space) is homotopy equivalent to [ilmath]T^2[/ilmath] and the line through it means "not", like [ilmath]\eq[/ilmath] and [ilmath]\neq[/ilmath])

Appendix

TODO: Lee - Topological manifolds - page 255. Computing the fundamental group of a wedge product