Constructing More Spaces

September 3, 2024
By Aathreya Kadambi
Expanding on Lectures by Professor Alexander Givental and Fomenko and Fuchs

Given two topologies, we can find a “product topology” whose open sets have a base as usual.

Given the interval $I = [0,1]$, we can define the cylinder $ZX = X \times I$. The cone $CX = X \times I / (x,0)\sim (x',0)$ basically it’s the cylinder quotienting by something. The suspension of $X$ is

$\Sigma X := CX/(X \times I)$

We can now talk about the category of topological spaces, where if we have some morphism $X \xrightarrow{f} Y$,

$\text{Cyl}(f) := \frac{X\times I \sqcup Y}{(x,q) \sim f(x)}$ $\text{Con}(f) := \text{Cyl}(f) / X \times 0$

Now we will talk about joins. A join is $X * Y$. We can see it as a quotient:

$X * Y = \frac{X \times I \times Y}{(x,0,y) \sim (x,0,y'), (x,1,y)\sim (x',1,y)}$ and now a question is whether joins are associative: $(X * Y) * Z =^? X * (Y * Z)$ and the answer is yes, as long as $X, Y, Z$ are Housdorff and some stuf according to the book. However, there are some more subtlties to think about.

Now as a result of this issue about associativity, we consider the following. See some products of cones:

$CX_1 \times CX_2 \times CX_3 \times ... = \{ (t_1,x_1), (t_2,x_2),..., (t_k, x_k),... \}$ If we consider the subset here defined by the condition that $\sum t_k = 1$, then this is how the infinite join can be defined in a symmetric way. This construction is a priori symmetric to stuff so this is another way to do something I don’t completely get yet.

We can now consider base point spaces which are basically what we have already considered, but with a base point. Then it is natural to think of the cone as a function from spaces without base points to base points because teh cone has a natural base point being the vertex of the cone.

He then said some stuff about $(X,A)$ and $(X,A) \rightarrow X/A$, and then asked something about $(X,\varnothing)$ and how $X/\varnothing = X \sqcup \text{pt} = X^+$ the space with an extra base point added.

Now before we considered suspensions, we can consider the suspension of this $X^+$. Then we get some moon looking thing that I can’t draw here, but yeah. Apparently we’ll see soon why it is useful.

Next we consider some $\prod_\alpha (X_\alpha, x_\alpha^\circ) = \{(x_\alpha) \mid x_\alpha \in X_\alpha, \text{almost all} x_\alpha = x_\alpha^\circ\}$ ($\alpha$ is in some indexing et $J$). This is the infinite product.

There is also something called the wedge product.

$\bigvee_\alpha (X_\alpha, x_n^n) := \bigsqcup_{\alpha} X_\alpha / \bigsqcup_\alpha \{x_\alpha^\circ\}$

This is also known as a boquet. Another important construction called the smash product is

$X \# Y := \frac{X \times Y}{X \cup Y}$

He proceeded to draw a weird picture with a cone looking thing, a circle, a plane, and a sphere, and wrote that

$S^{m+n+1} = S^m * S^n$ $S^m \# S^n \simeq S^{m+n}$ This is an exercise for us later.

Given some spaces $X$ and $Y$, $C(X,Y)$ is the set of continuous maps from $X$ to $Y$. For this we get a compact open topology with $K$ compacy subset of $X$ and $O$ open subset of $Y$. We get:

$\{f : X \rightarrow Y : f(K) \subseteq O\}$ this is the base of the topology.

If $Y$ is something (matrix?, I can’t read the hanadwriting), then we get uniform convergence on compact subsets. A compact notation for $C(X,Y)$ is $Y^X$ in topology. It’s just a generalization of the notation $Y^N$ where we might be mapping from some $N$ points to $Y$, which gives $N$ (not necessarily distinct) points of $Y$.

A good result is:

$(Z^Y)^X = Z^{Y \times X}$

Now let us consider the following space, called the path space,

$EX = C(I,X)$ For based spaces, we can consider specifically paths starting at the basepoint. $E(X, x_0) = \{\gamma \in C(I,X) : \gamma(0) = x_0\}$ In the case where $x_0 = x_1$, we have a special notation which si called the loop space of $X$ (denoted by I think $\Omega(X, x_0)$). These spaces are also base point spaces because we can juts keep the base point.

So from this idea, we can actually rethink of the the weird picture (the suspension thing) form before is actually a collection of loops. We are thinking of loops parametrized by $X$ and that go “around” $Y$ sort of. There is basically a map from $X$ to the loop space of $Y$ or something. This is an important identity.

$$C(\Sigma X, Y) = C(X, \Omega Y)

We are now in a good place to talk about the next topic from the title of the textbook, which is “homotopies”.