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)