Algebraic Topology

Algebraic Topology
I took these notes in fall 2016 for Math 215A, taught by Professor Gunnar Carlsson.
Fundamental group
A homotopy between $f: Y \to X$ and $g : Y \to X$ is $h_t : Y \to X$ such that $h_0 = f$ and $h_1 = g$ .
- A homotopy rel $A$ is constant on $A \subseteq X$ .
- A homotopy of points is a path $\varphi_t \in X$ , with $Y = \{ 0 \}$ .
- A homotopy of paths is $h_t : I \to X$ , with $Y = I$ .
- A homotopy of loops is $h_t : S^1 \to X$ , with $Y = S^1$ .
- Homotopy equivalence is an isomorphism in the homotopy category $\mathrm{hTop}$ : $X$ and $Y$ are homotopy equivalent if $f \circ g$ and $g \circ f$ are homotopic to the identity.
The fundamental group $\pi_1(X, x)$ of $X$ is the set of loops ( $\varphi : S^1 \to X$ ) at a base point $x$ modulo equivalence under homotopy, with the group operation of concatenation.
- If $X$ is path-connected, then $\pi_1(X, x_0) = \pi_1(X, x_1)$ .
- If $X$ is simply-connected, then $\pi_1(X) = 0$ , by definition.
- If $X$ is convex, then $\pi_1(X) = 0$ .
- $\pi_1(S^1) = \mathbb{Z}$ .
- $\pi_1(X \times Y) = \pi_1(X) \times \pi_1(Y)$ for $X$ and $Y$ path-connected
- van Kampen theorem: $\pi_1(U \cup V) = \pi_1(U) * \pi_1(V)$ assuming $U$ , $V$ , $U \cup V$ , and $U \cap V$ are all open and path-connected and the base point is in $U \cap V$ . The free product is taken with amalgamation with respect to $\pi_1(U \cap V)$ .
- $\pi_1(X \vee Y) = \pi_1(X) * \pi_1(Y)$ if $X$ and $Y$ are locally contractible.
- CW complexes
Induced homomorphism: A map $\varphi : (X,x) \to (Y,y)$ induces a homomorphism $\varphi_* : \pi_1(X, x) \to \pi_1(Y, y)$ , since $\varphi$ maps loops to loops.
- Concretely: $\varphi_*([f]) = [\varphi \circ f]$
- $\mathrm{id}_* = \mathrm{id}$
- $\pi_1 : \,\mathrm{Top}_* \to \mathrm{Grp}$ is a functor.
- If $X$ retracts into $A$ , then $\iota_*$ is injective for the inclusion $\iota : A \hookrightarrow X$ .
  - A retraction is a map $r : X \to A$ such that $r(a) = a$ .
- If $X$ has a deformation retract into $A$ , then $\iota_*$ is an isomorphism.
  - A deformation retract is a homotopy from $h_0 = \mathrm{id}$ to $h_1$ such that $h_1(X) = A$ and $h_t$ is a retraction.
  - We can also think of it as a homotopy rel $A$ from $\mathrm{id}$ to a retraction into $A$ .
For a covering space with projection $p : \tilde{X} \to X$ , every point in $X$ has a neighborhood $U$ for which $p^{-1}(U)$ breaks up $\tilde{X}$ into disjoint open sets, each homeomorphic to $U$ by $p$ . (Stack of pancakes!)
- An isomorphism between two covering spaces $p_1 : \tilde{X}_1 \to X$ and $p_2 : \tilde{X}_2 \to X$ is a homeomorphism $f : \tilde{X}_1 \to \tilde{X}_2$ such that $p_1 = p_2 \circ f$ .
- A basepoint-preserving isomorphism is analogous for pointed spaces.
- Homotopy lifting property: Given a homotopy $h_t : Y \to X$ , it’s possible to lift this uniquely to a homotopy $\tilde{h}_t : Y \to \tilde{X}$ of the covering space once we specify a compatible base point in the covering space $\tilde{h}_0 : Y \to \tilde{X}$ .
  - The lifted homotopy satisfies $p \circ \tilde{h}_t = h_t$ .
- Lifting criterion: Given a map $f : (Y, y) \to (X, x)$ , it’s possible to lift this to a map $\tilde{f} : (Y, y) \to (\tilde{X}, \tilde{x})$ if and only if $\mathrm{im}\,f_* \subseteq \mathrm{im}\,p_*$ , assuming that $Y$ is path-connected and locally path-connected.
  - If $\tilde{f}_1(x) = \tilde{f}_2(x)$ , then $\tilde{f}_1 = \tilde{f}_2$ .
- $p_*$ is injective.
- The number of sheets of $p$ is the index of $\mathrm{im}\, p_*$ in $\pi_1(X, x)$ .
The subgroups of $\pi_1(X, x)$ correspond to path-connected covering spaces $p : \tilde{X} \to X$ (assuming $X$ is path-connected, locally path-connected, and semilocally simply-connected):
- The path-connected covering space $p: (\tilde{X}, \tilde{x}) \to (X, x)$ (up to basepoint-preserving isomorphism) corresponds bijectively with the subgroup $\pi_1(\tilde{X}, \tilde{x}) \cong \mathrm{im}\, p_* \le \pi_1(X, x)$ .
- If $(\tilde{X}, \tilde{x}_1) \to (X, x)$ corresponds to $H$ , then $(\tilde{X}, \tilde{x}_2) \to (X, x)$ corresponds to $g^{-1}Hg$ , where $g$ is the class of loops whose lift is a path from $\tilde{x}_1$ to $\tilde{x}_2$ .
- The universal covering space of $(X, x)$ is the simply-connected covering space, which can be constructed as $\tilde{X} = \{ [\gamma] \,|\, \gamma : I \to X, \gamma(0) = x \}$ with $p([\gamma]) = \gamma(1)$ .
- The covering space corresponding to the subgroup $H$ can be constructed as the quotient of the universal covering space with the relation $[\gamma_1] \sim [\gamma_2]$ iff $\gamma_1(1) = \gamma_2(1)$ and $[\gamma_1 \gamma_2^{-1}] \in H$ .
- \todo Normal subgroups iff normal covering spaces
- \todo The covering spaces (not necessarily connected) can be represented by permutations.
Homological algebra
A chain complex $C_\bullet$ is a sequence of abelian groups (or objects in any abelian category) and homomorphisms $\cdots \to^\partial C_{n+1} \to^\partial C_n \to^\partial C_{n-1} \to^\partial \cdots$
where $\partial^2 = 0$ .
- $C_n$ represents the $n$ -simplices of a space, called $n$ -chains.
- $\partial_n : C_n \to C_{n-1}$ is the boundary map, which takes a simplex to its boundary.
- $Z_n = \mathrm{ker}\, \partial_n$ are $n$ -dimensional cycles.
- $B_n = \mathrm{im}\, \partial_{n+1}$ are $n$ -dimensional boundaries, indicating $n$ -dimensional cycles that become homotopic after adding $(n+1)$ -simplices.
- $B_n \subseteq Z_n \subseteq C_n$ , and $0 \to B_n \to Z_n \to H_n \to 0$ is a short exact sequence.
A chain map $f : C_\bullet \to D_\bullet$ is a morphism of chain complexes, i.e. a sequence of homomorphisms $f_n : C_n \to D_n$ such that $f\partial = \partial f$ .
- A chain map $f$ induces homomorphisms $f_* : H_n(C_\bullet) \to H_n(D_\bullet)$ , taking $f_*([x]) = [f(x)]$ .
- $\partial$ is a chain map.
- For any sequence of homomorphisms $s_n : C_n \to D_n$ , the map $s\partial + \partial s$ is a chain map, but it induces the zero homomorphism. Thus, if the difference between two chain maps $f - g = s\partial + \partial s$ for some map $s_n$ , called a chain homotopy, then $f_* = g_*$ , and $f$ and $g$ are chain-homotopic.
The homology of a chain complex are defined as $H_n(C_\bullet) = \mathrm{ker}\,\partial_n / \mathrm{im}\, \partial_{n+1}$ .
- $H_n : \,\mathrm{Comp} \to \mathrm{Ab}$ is a functor.
- $H_n$ represents “holes” enclosed by $n$ -dimensional sheets.
- $H_0$ represents connected components.
An exact sequence is a chain complex with trivial homology, i.e. $\ker\partial_n \cong \mathrm{im}\, \partial_{n+1}$ .
- Note that this condition already implies that $\partial^2 = 0$ .
- $0 \to A \hookrightarrow^\alpha B$ is exact iff $\alpha$ is injective.
- $Y \twoheadrightarrow^\beta Z \to 0$ is exact iff $\beta$ is surjective.
- $0 \to A \to^\alpha Z \to 0$ is exact iff $\alpha$ is an isomorphism.
A short exact sequence is an exact sequence of the form $0 \to A \hookrightarrow^{\iota_A} B \twoheadrightarrow^{\pi_C} C \to 0$ .
- First isomorphism theorem: $C \cong B/A$ (viewing $A \subseteq B$ ), since $A \cong \mathrm{im}\, \iota_A \cong \ker \pi_C$ .
- Splitting lemma: $B \cong A \oplus C$ with $\iota_A$ and $\pi_C$ canonical, iff $\iota_A$ has a left inverse $\pi_A$ , iff $\pi_C$ has a right inverse $\iota_C$ .
- If $C$ is free, then the sequence splits.
- Snake lemma, five-lemma, nine-lemma.
Zig-zag lemma: if $0 \to A_{\bullet} \to^\alpha B_{\bullet} \to^\beta C_{\bullet} \to 0$ is a short exact sequence of chain complexes, then the following sequence of homology groups is exact: $\cdots \to H_{n}(A_{\bullet}) \to^{\alpha_*} H_n(B_{\bullet}) \to^{\beta_*} H_n(C_{\bullet}) \to^\delta H_{n-1}(A_{\bullet}) \to \cdots$ with the connecting homomorphism $\delta$ .
Homology of spaces
In simplicial homology, we set $C_n^\Delta$ to be the free abelian group generated by the $n$ -simplices of a $\Delta$ -complex $X$ and boundary map generated by $\partial_n [ v_0, \ldots, v_n] = \sum_{i = 0}^n (-1)^i [v_0, \ldots, \hat{v}_i, \ldots, v_n].$
In singular homology, we set $C_n$ to be the free abelian group generated by all maps $\sigma : \Delta^n \to X$ and boundary map generated by $\partial_n (\sigma) = \sum_{i=1}^n (-1)^i \sigma|_{[v_0, \ldots, \hat{v}_i, \ldots, v_n]}.$
- These define functors $C_\bullet : \,\mathrm{Top} \to \mathrm{Comp}$ and induce $H_n : \,\mathrm{Top} \to \mathrm{Ab}$ .
- A map $f: X \to Y$ induces a chain map $f_\sharp : C_\bullet(X) \to C_\bullet(Y)$ by mapping $\sigma$ to the composition $\Delta^n \to^\sigma X \to^f Y$ (which in turn induces homomorphisms $H_n(X) \to H_n(Y)$ ).
- If $f$ and $g$ are homotopic, then $f_\sharp$ and $g_\sharp$ are chain-homotopic (and $f_* = g_*$ ).
- If $X$ and $Y$ are homotopy equivalent, then $H_n(X) \cong H_n(Y)$ .
  - This motivates the category $\mathrm{hComp}$ .
Additionally, reduced homology sets $C_{-1} = \mathbb{Z}$ , with $\epsilon : C_0 \to \mathbb{Z}$ defined as $\sigma \mapsto 1$ extended linearly. The homology groups are denoted $\tilde{H}_n$ and satisfy $\tilde{H}_n = H_n$ except $\tilde{H}_0 \oplus \mathbb{Z} = H_0$ .
$\tilde{H}_n(*) = 0$ .
$\tilde{H}_n(S^n) = \mathbb{Z}$ and $\tilde{H}_k(S^n) = 0$ for $k \ne n$ .
The relative chain complex $C_\bullet(X, A)$ for $A \subseteq X$ is defined as $C_n(X, A)= C_n(X) / C_n(A)$ , with boundary maps $\partial : C_n(X,A) \to C_{n-1}(X,A)$ given by $\partial([x]) = [\partial x]$ (well-defined by a diagram chase).
The relative homology groups $H_n(X,A)$ are the homology groups of $C_\bullet(X,A)$ .
- Chains in $A$ become trivial.
- $H_n(X, \varnothing) = H_n(X)$ , and $H_n(X, *) = \tilde{H}_n(X)$ .
- A map $f : (X,A) \to (Y,B)$ , i.e. $f : X \to Y$ with $f(A) \subseteq B$ , induces a chain map $f_\sharp : C_\bullet(X,A) \to C_\bullet(Y,B)$ .
- Good pair: $H_n(X, A) \cong \tilde{H}_n(X/A)$ if $A \ne \varnothing$ is closed and a deformation retract of some neighborhood of $X$ , with the isomorphism induced by $\pi : (X,A) \to (X/A, A/A)$ .
- Excision theorem: If $\mathrm{cl}(Z) \subseteq \mathrm{int}(A) \subseteq X$ , then $H_n(X,A) \cong H_n(X - Z, A-Z)$ with the isomorphism induced by $\iota : (X - Z, A-Z) \to (X,A)$ .
  - Equivalently, if $\mathrm{int}(A) \cup \mathrm{int}(B) = X$ , then $H_n(X, A) \cong H_n(B,A \cap B)$ .
- Long exact sequence of a pair: Applying the zig-zag lemma to the short exact sequence $0 \to C_\bullet(A) \to^\iota C_\bullet(X) \to^\pi C_\bullet(X,A) \to 0,$ the following sequence is exact: $\cdots \to H_n(A) \to^{\iota_*} H_n(X) \to^{\pi_*} H_n(X,A) \to^{\partial_*} H_{n-1}(A) \to \cdots,$ with connecting homomorphism $\partial_*([\alpha]) = [\partial \alpha]$ for $[\alpha] \in H_n(X,A)$ (and thus $\alpha \in C_n(X,A)$ is a relative cycle).
- Long exact sequence of a triple: If $Z \subseteq A \subseteq X$ , we can easily generalize using the short exact sequence $0 \to C_\bullet(A,Z) \to C_\bullet(X,Z) \to C_\bullet(X,A) \to 0.$
- Mayer-Vietoris sequence: Suppose $\mathrm{int}(A) \cup \mathrm{int}(B) = X$ . Applying the zig-zag lemma to the short exact sequence $0 \to C_\bullet(A \cap B) \to^{(1,-1)} C_\bullet(A) \oplus C_\bullet(B) \to^{+} C_\bullet(A) + C_\bullet(B) \to 0,$ the following sequence is exact: $\cdots \to H_n(A \cap B) \to H_n(A) \oplus H_n(B) \to H_n(X) \to^{\partial_*} H_{n-1}(A \cap B) \to \cdots,$ with connecting homomorphism $\partial([a + b]) = [\partial a] = [-\partial b]$ for $a \in C_n(A)$ and $b \in C_n(B)$ .
\todo Example of calculating simplicial homology
Cellular homology
Brouwer fixed point theorem
$R^m = R^n$
Relate fundamental group and homology
Cohomology
The cochain complex is obtained by applying the contravariant functor $\mathrm{Hom}(-, G)$ to a chain complex $C$ : $\cdots \leftarrow^d C^{n+1} \leftarrow^d C^n \leftarrow^d C^{n-1} \leftarrow^d \cdots,$ with coboundary map $d = \partial^*$ , so $d^2 = 0$ .
The cohomology groups are $H^n(C; G) = \ker d/\mathrm{im}\, d$ .
- $\mathrm{Hom}(\mathbb{Z}, G) = G$ .
- $\mathrm{Hom}(\mathbb{Z}_n, \mathbb{Z}) = 0$ .
- $\mathrm{Hom}(\mathbb{Z}_a, \mathbb{Z}_b) = \mathbb{Z}_{\mathrm{gcd}(a,b)}$ .
- $\mathrm{Hom}(\mathbb{Z}_n, G) = \ker(G \to^n G)$ .
- $\mathrm{Hom}(A \oplus B, G) = \mathrm{Hom}(A, G) \oplus \mathrm{Hom}(B, G)$ .
Universal coefficient theorem: The homology and cohomology groups are related by the split exact sequence $0 \to \mathrm{Ext}(H_{n-1}(C), G) \to H^n(C; G) \to \mathrm{Hom}(H_n(C), G) \to 0.$
- A free resolution $F$ of an abelian group $H$ is an exact sequence of the form $\cdots \to F_1 \to F_0 \to H \to 0,$ where $F_i$ is free. Homomorphisms $\alpha : H \to H'$ can be extended to a chain map $\alpha : F \to F'$ , unique up to chain-homotopy. For any two free resolutions $F$ and $F'$ of the same abelian group, $H^n(F;G) \cong H^n(F'; G)$ .
- Every abelian group has a free resolution of the form $0 \to F_1 \hookrightarrow F_0 \twoheadrightarrow H \to 0,$ with $F_0$ freely generated by generators of $H$ , and $F_1$ the kernel of the surjection.
- Ext functor: $\mathrm{Ext}(H, G)$ is the first cohomology group of a free resolution of $H$ excluding $H$ , i.e. the first homology group of $0 \leftarrow \mathrm{Hom}(F_1, G) \leftarrow \mathrm{Hom}(F_0,G) \leftarrow 0.$
- $\mathrm{Ext}(H \oplus H', G) \cong \mathrm{Ext}(H,G) \oplus \mathrm{Ext}(H',G)$ .
- $\mathrm{Ext}(H, G) = 0$ if $H$ is free.
- $\mathrm{Ext}(\mathbb{Z}_n, G) = G/nG$ .
- If $H_i(X; \mathbb{Z}) \cong \mathbb{Z}^{\beta_i} \oplus T_i$ where $T_i$ is the torsion part, then $H^i(X; \mathbb{Z}) \cong \mathbb{Z}^{\beta_i} \oplus T_{i-1}$ .