Basic topological constructions

 Preliminary exam prep

Let $X,Y$ be topological spaces based at $x_0,y_0$, respectively, and $I=[0,1]$ the unit interval.
\[
\begin{array}{r r c l}
\text{cone} & CX & = & X\times I / X\times \{0\} \\[5pt]
\text{suspension} & \Sigma X & = & X\times I / X\times \{0\}, X\times \{1\}\\[5pt]
\text{reduced suspension} & \widetilde\Sigma X & = & X\times I/X\times\{0\}, X\times \{0\}, \{x_0\}\times I \\[5pt]
\text{wedge} & X\vee Y & = & X\sqcup Y / \{x_0\} \sim \{y_0\} \\[5pt]
\text{smash} & X\wedge Y & = & X\times Y / X\times \{y_0\}, \{x_0\}\times Y \\[5pt]
\text{join} & X * Y & = & X\times Y \times I \left/\begin{array}{l l}
X\times \{y\}\times \{0\} & \forall\ y\in Y \\
\{x\}\times Y \times \{1\} & \forall\ x\in X
\end{array}\right. \\[5pt]
\text{connected sum} & X \# Y & = & (X\setminus D^n_X)\sqcup (Y\setminus D^n_Y) / \partial D^n_X \sim \partial D^n_Y
\end{array}
\]
In the last description, $X$ and $Y$ are assumed to be $n$-manifolds, with $D^n_X$ a closed $n$-dimensional disk in $X$ (similarly for $Y$). The quotient identification may also be made via some non-trivial map. In fact, only the interior of each $n$-disk is removed from $X$ and $Y$, so that the quotient makes sense.

Remark: Some of the above constructions may be expressed in terms of others, for example
\[
X\wedge Y = X\times Y / X\vee Y,
\hspace{1cm}
X*Y = \Sigma(X\wedge Y).
\]
The first is clear by viewing $X = X\times \{y_0\}$ and $Y = \{x_0\}\times Y$ as sitting inside $X\times Y$. The second is clear by letting $X\times \{y\}\times \{0\}$ be identified to $\{x_0\}\times\{y\}\times \{0\}$ for every $y\in Y$, and analogously with $Y$.

Example: Here are some of the constructions above applied to some common spaces.
\begin{align*}
CX & \simeq \text{pt} & \Sigma S^n & = S^{n+1} & S^n \wedge S^m & = S^{n+m}\\
\Sigma X & = S^1 \wedge X & S^n * S^m & = S^{n+m+1}\end{align*}
Remark: We may also calculate the homology of the new spaces in terms of the old ones.
\[
\begin{array}{r c l l}
\widetilde H_k(CX) & = & 0 & \text{via homotopy} \\
\widetilde H_k(\Sigma X) & = & \widetilde H_{k-1}(X) & \text{via Mayer--Vietoris} \\
\widetilde H_k(X\vee Y) & = & \widetilde H_k(X)\oplus \widetilde H_k(Y) & \text{via Mayer--Vietoris}\\
\widetilde H_k(X\wedge S^\ell) & = & \widetilde H_{k-\ell}(X) & \text{via Kunneth} \\
\widetilde H_k(X\# Y) & = & \widetilde H_k(X) \oplus \widetilde H_k(Y) & \text{via Mayer--Vietoris and relative homology}
\end{array}
\]
The last equality holds for $k<n-1$, for $M$ and $N$ both $n$-manifolds, and for $k=n-1$ when at least one of them is orientable.

References: Hatcher (Algebraic Topology, Chapters 0, 2)

(Co)fibrations, suspensions, and loop spaces

 Seminar topic

Recall the exponential object $Z^Y$, which, in the category of topological spaces, is the set of all continuous functions $Y\to Z$. In general, the definition involves a commuting diagram and gives an isomorphism $\Hom(X\times Y,Z)\cong \Hom(X,Z^Y)$. The subspace $F(Y,Z)$ of $Z^Y$ consists of based functions $Y\to Z$.

Definition: Let $F,E,B,X$ be topological spaces. A map $i:F\to E$ is a cofibration if for every map $f:E\to X$ and every homotopy $h:F\times I\to X$, there exists a homotopy $\tilde h:E\times I\to X$ (extending $h$) making either of the equivalent diagrams below commute.

The horizontal maps on the left are the natural inclusion maps $x\mapsto (x,0)$ and the map on the right is the natural evaluation map $\varphi \mapsto \varphi(0)$. Similarly, a map $p:E\to B$ is a fibration if for every map $g:X\to E$ and every homotopy $h:X\times I\to B$, there exists a homotopy $\tilde h:X\times I\to E$ (lifting $h$) making either of the equivalent diagrams below commute.

The horizontal maps on the right are the natural evaluation maps and the map on the right is the natural inclusion map.

Instead of this terminology, often we say the pair $(F,E)$ has the homotopy extension property and the pair $(E,B)$ has the homotopy lifting property. Now, let let $(X,x)$ be a pointed topological space.

Definition: The (reduced) suspension $\Sigma X$ of $X$ is
\[
\Sigma X := X\times I/X\times \{0\} \cup X\times \{1\} \cup \{x\}\times I.
\] 
The unreduced suspension $SX$ of $X$ is
\[
S X := X\times I/X\times \{0\} \cup X\times \{1\}.
\]
The loop space $\Omega X$ of $X$ is
\[
\Omega X := F(S^1,X).
\]
Remark: If $X$ is well-pointed (the inclusion $i:\{x\}\hookrightarrow X$ is a cofibration), then the natural quotient map $SX\to \Sigma X$ is a homotopy equivalence. Moreover, there is an adjunction $F(\Sigma X,Y)\cong F(X,\Omega Y)$. In the fundamental group this gives the adjunction
\[
[\Sigma X,Y]\cong [X,\Omega Y],
\]
where $[A,B]$ is the set of based homotopy classes of maps $A\to B$.

References: May (A concise course in algebraic toplogy, Chapters 6, 7, 8), Aguilar, Gitler, and Prieto (Algebraic topology from a homotopical viewpoint, Chapter 2.10)