Tools of homotopy

 Preliminary exam prep

Let $X,Y$ be topological spaces and $A$ a subspace of $X$. Recall that a path in $X$ is a continuous map $\gamma:I\to X$, and it is closed (or a loop), if $\gamma(0)=\gamma(1)$. When $X$ is pointed at $x_0$, we often require $\gamma(0)=x_0$, and call such paths (and similarly loops) based.

Definitions

Definition:

• $X$ is connected if it is not the union of two disjoint nonempty open sets.
• $X$ is path connected if any two points in $X$ have a path connecting them, or equivalently, if $\pi_0(X)=0$.
• $X$ is simply connected if every loop is contractible, or equivalently, if $\pi_1(X)=0$.
• $X$ is semi-locally simply connected if every point has a neighborhood whose inclusion into $X$ is $\pi_1$-trivial.

Path connectedness and simply connectedness have local variants. That is, for $P$ either of those properties, a space is locally $P$ if for every point $x$ and every neighborhood $U\owns x$, there is a subset $V\subset U$ on which $P$ is satisfied.

Remark: In general, $X$ is $n$-connected whenever $\pi_r(X)=0$ for all $r\leqslant n$. Note that 0-connected is path connected and 1-connected is simply connected and connected. Also observe that the suspension of path connected space is simply connected.

Definition:

• A retraction (or retract) from $X$ to $A$ is a map $r:X\to A$ such that $r|_A = \id_A$.
• A deformation retraction (or deformation retract) from $X$ to $A$ is a family of maps $f_t:X\to X$ continuous in $t,X$ such that $f_0 = \id_X$, $f_1(X) = A$, and $f_t|_A = \id_A$ for all $t$.
• A homotopy from $X$ to $Y$ is a family of maps $f_t:X\to Y$ continuous in $t,X$.
• A homotopy equivalence from $X$ to $Y$ is a map $f:X\to Y$ and a map $g:Y\to X$ such that $g\circ f \simeq \id_X$ and $f\circ g \simeq \id_Y$.

Definition: A pair $(X,A)$, where $A\subset X$ is a closed subspace, is a good pair, or has the homotopy extension property (HEP), if any of the following equivalent properties hold:

• there exists a neighborhood $U\subset X$ of $A$ such that $U$ deformation retracts onto $A$,
• $X\times \{0\}\cup A\times I$ is a retract of $X\times I$, or
• the inclusion $i:A\hookrightarrow X$ is a cofibration.

In some texts such a pair $(X,A)$ is called a neighborhood deformation retract pair, and HEP is reserved for any map $A\to X$, not necessarily the inclusion, that is a cofibration. For more on cofibrations, see a previous blog post (2016-07-31, "(Co)fibrations, suspensions, and loop spaces").

Definition: There is a functor $\pi_1:\text{Top}_*\to \text{Grp}$ called the fundamental group, that takes a pointed topological space $X$ to the space of all pointed loops on $X$, modulo path homotopy.

This may be generalized to $\pi_n$, which takes $X$ to the space of all pointed embeddings of $S^n$.

Definition: Let $G,H$ be groups. The free product of $G$ and $H$ is the group
\[
G*H = \{a_1\cdots a_n\ :\ n\in \Z_{\>0}, a_i\in G\text{ or }H, a_i\in G(H)\implies a_{i+1}\in H(G)\},
\]
with group operation concatenation, and identity element the empty string $\emptyset$. We also assume $e_Ge_H=e_He_G=e_G=e_H=\emptyset$, for $e_G$ ($e_H$) the identity element of $G$ ($H$).

The above construction may be generalized to a collection of groups $G_1*\cdots*G_m$, where the index may be uncountable. If every $G_\alpha=\Z$ (equivalently, has one generator), then $*_{\alpha\in A} G_\alpha$ is called the free group on $|A|$ generators.

Theorems

Theorem: (Borsuk-Ulam) Every continuous map $S^n\to \R^n$ takes a pair of antipodal points to the same value.

Theorem: (Ham Sandwich theorem) Let $U_1,\dots,U_n$ be bounded open sets in $\R^n$. There exists a hyperplane in $\R^n$ that divides each of the open sets $U_i$ into two sets of equal volume.

Volume is taken to be Lebsegue measure. The Ham sandwich theorem is an application of Borsuk-Ulam (see Terry Tao's blog post for more).

Theorem: If $X$ and $Y$ are path-connected, then $\pi_1(X\times Y)\cong \pi_1(X)\times \pi_1(Y)$.

Now suppose that $X = \bigcup_\alpha A_\alpha$ is based at $x_0$ with $x_0\in A_\alpha$ for all $\alpha$. There are natural inclusions $i_\alpha:A_\alpha\to X$ as well as $j_\alpha:A_\alpha\cap A_\beta \to A_\alpha$ and $j_\beta:A_\alpha\cap A_\beta \to A_\beta$.

Both $i_\alpha$ and $j_\alpha$ induce maps on the fundamental group, each (and all) of the $i_{\alpha*}:\pi_1(A_\alpha)\to \pi_1(X)$ extending to a map $\Phi:*_\alpha \pi_1(A_\alpha)\to \pi_1(X)$.

Theorem: (van Kampen)

• If $A_\alpha\cap A_\beta$ is path-connected, then $\Phi$ is a surjection. 
• If $A_\alpha\cap A_\beta\cap A_\gamma$ is path connected, then $\ker(\Phi) = \langle j_{\alpha*}(g)(j_{\beta*}(g))^{-1}\ |\ g\in \pi_1(A_\alpha\cap A_\beta,x_0)\rangle$.

As a consequence, if triple intersections are path connected, then $\pi_1(X) \cong *_\alpha A_\alpha /\ker(\Phi)$. Moreover, if all double intersections are contractible, then $\ker(\Phi)=0$ and $\pi_1(X)\cong *_\alpha A_\alpha$.

Proposition: If $\pi_1(X)=0$ and $\widetilde H_n(X)=0$ for all $n$, then $X$ is contractible.

References: Hatcher (Algebraic topology, Chapter 1), Tao (blog post "The Kakeya conjecture and the Ham Sandwich theorem")

The Eilenberg-Steenrod axioms

The category $\text{Top}$ of topological spaces may be generalized to the category $\text{Top}_*$ of pointed topological spaces. This in turn may be generalized to the category $\text{Top}_{rel}$ of pairs $(X,A)$, where $X\in\Obj(\text{Top})$ and $A$ is a subspace of $X$. The morphisms of $\text{Top}_{rel}$ on $(X,A)$ are the morphisms of $\text{Top}$ on $X$ paired with their restrictions to $A$. We write $(X)$ for $(X,\emptyset)$.

Definition 1: Let $X,Y\in\Obj(\text{Top}_*)$. Then $f\in\Hom_{\text{Top}_*}(X,Y)$ is an $n$-equivalence if the induced map on homotopy groups $f_*:\pi_k(X,x)\to \pi_k(Y,f(x))$ is an isomorphism for $k<n$ and an epimorphism for $k=n$. Further, $f$ is a weak equivalence if it is an $n$-equivalence for all $n\geqslant 1$. Similarly, $f\in \Hom_{\text{Top}_{rel}}((X,A),(Y,B))$ is a weak equivalence if $f\in \Hom_{\text{Top}_*}(X,Y)$ and $f|_A\in \Hom_{\text{Top}_*}(A,B)$ are weak equivalences.

Definition 2: Let $C,D$ be two categories. A functor $\mathcal F:C\to D$ is an assignment $\mathcal F(X)\in \Obj(D)$ for every $X\in \Obj(C)$, and $\mathcal F(f)\in \Hom_D(\mathcal F(X),\mathcal F(Y))$ for every $f\in\Hom_C(X,Y)$. This assignment satisfies the following relations:
          $\mathcal F(g\circ f) = \mathcal F(g)\circ \mathcal F(f)$ for every $f\in \Hom_C(X,Y)$ and $g\in \Hom_C(Y,Z)$
          $\mathcal F(\id_X) = \id_{\mathcal F(X)}$ for every $X\in\Obj(C)$

Definition 3: Let $C$ be any category and $\mathcal F:\text{Top}\to C$ a functor. Then $\mathcal F$ is homotopy invariant if $f\simeq g$ in $\text{Top}$ implies $\mathcal F(f)=\mathcal F(g)$ in $C$, where $\simeq$ is the homotopy of maps.

Definition 4: A (relative) homology theory of topological spaces is a collection of homotopy-invariant functors $H_n:\text{Top}_{rel}\to \text{Ab}$ and a collection of natural transformations $d_n:H_n(X,A) \to H_{n-1}(A)$.

The Eilenberg-Steenrod axioms are properties a relative homology theory may satisfy. The number of axioms depends on how general a view of homology theories one would like. Eilenberg and Steenrod (7), May (4), Aguilar, Gitler, and Prieto (4), Wikipedia (5), and other sources (6,8) have all different numbers of axioms. The order of the axioms below is alphabetical.

For any $(X,A)\in\Obj(\text{Top}_{rel})$ and all $n$:

Axiom 1: Additivity. If $(X,A)=\bigoplus_i(X_i,A_i)$, then $H_n(X,A) \cong \bigoplus_iH_n(X_i,A_i),$ where the isomorphism is induced by the inclusions $(X_i,A_i)\hookrightarrow (X,A)$.

Axiom 2: Exactness. There is a long exact sequence
\[ \cdots \to H_{n+1}(X,A)\tov{d_{n+1}}H_n(A)\tov{\ \ }H_n(X)\tov{\ \ }H_n(X,A)\tov{d_n}H_{n-1}(A)\tov{\ \ }\cdots \]
where $H_n(A)\to H_n(X)$ and $H_n(X)\to H_n(X,A)$ are induced by the inclusions $(A)\hookrightarrow (X)$ and $(X)\hookrightarrow (X,A)$, respectively.

Axiom 3: Excision. If there exists a subset $U$ of $X$ with $\text{cl}(U)\subset \text{int}(A)$, then there is an isomorphism $H_n(X\setminus U,A\setminus U)\cong H_n(X,A)$ induced by the inclusion $(X\setminus U,A\setminus U)\hookrightarrow (X,A)$.

Axiom 4: Dimension. $H_n(*)=0$ for all $n\neq 0$.

Axiom 5: Weak equivalence. If $f\in\Hom_{\text{Top}_{rel}}((X,A),(Y,B))$ is a weak equivalence, then the induced map on homology $f_*:H_n(X,A)\to H_n(Y,B)$ is an isomorphism.

Singular homology is a homology theory that satisfies all the axioms above. $K$-theory is a homology theory that does not satisfy the dimension axiom.

References: May (A Concise course in Algebraic Topology, Chapter 13.1), Aguilar, Gitler, and Prieto (Algebraic Topology from a Homotopical Viewpoint, Chapter 5.3)