/hatcher-reading-group-notes

Information put together while reading Hatchers book Algebraic Topology

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\subsubsection*{Annulus} Idea: Some plaintext.\

Definition: Some formal foo $y=x^2$ bar.\

\subsubsection*{Antipodal} \subsubsection*{Attached along a map} $A$ ... topological subspace of $Y$

$f\colon A\to X$

$X\cup_f Y = (X\amalg Y) / \sim_f$

TODO: Be more precise about the definition\

\subsubsection*{Boundary}

\subsubsection*{Cantor set} \subsubsection*{Cartesian space ${\mathbb R}^n$} \subsubsection*{Cell complex} \subsubsection*{Cell $e^n$} \subsubsection*{Cell structure} \subsubsection*{Component} \subsubsection*{Compact} \subsubsection*{Cone} \subsubsection*{Contractible space} \subsubsection*{CW-pair} \subsubsection*{CW complex}

\subsubsection*{Deformation retraction} A homotopy $(I\times X)\to X$ between the identity $\mathrm{id}_X:X\to X$ and a retract $r:X\to A$.

I.e. a deforming of the identity by shrinking its image.

\url{https://en.wikipedia.org/wiki/Retraction_(topology)}

\subsubsection*{Dimension of a CW complex} \subsubsection*{Disc ${\mathbb D}^n$} Notes:\ $I:=D^1$ is the interval used for the definition of a homotopy.

\subsubsection*{Disjoint union}

\subsubsection*{Equivalence relation}

\subsubsection*{Genus} \subsubsection*{Graph} x

\subsubsection*{Homeomorphic} \subsubsection*{Homotopy} Continuous $H\colon (I\times X)\to Y$.\

This can also be expressed as\ $\bullet$ families into continuous functions $h\colon I\to (X\to Y)$ (such that $H\colon (I\times X)\to Y$ is also continuous.)\ resp.\ $\bullet$ neighboring paths of point $p\colon X\to (I\to Y)$ (such that $H\colon (I\times X)\to Y$ is also continuous.)

Here $I=[0,1]$ is the interval and we speak of a homotopy between the functions $h(0)$ and $h(1)$. Then $h(0)$ and $h(1)$ are homotopic.

\url{https://en.wikipedia.org/wiki/Homotopy}

\subsubsection*{Homotopy equivalence} A pair $f\colon X\to Y$ and $g\colon Y\to X$ such that \ $g\circ f$ is homotopic to the identity on $X$ \ and also $f\circ g$ is homotopic to the identity on $Y$.

I.e. both composition maps are allowed to be continuous deformations of the identity.

In contrast, one gets homeomorphisms (continuous bijections) when one requires the compositions to be exactly identities.

\subsubsection*{Homotopy extension property} \subsubsection*{Homotopy Type} \subsubsection*{House with two rooms}

\subsubsection*{Inclusion map} ...

$A\subset X$

$\iota\colon A\to X$

$\iota(x):=x$

Is injective. In a diagram, the arrow is often written with a hook (as in $\hookrightarrow$).

\subsubsection*{Infinite sphere ${\mathbb S}^\infty$}

\subsubsection*{Join}

\subsubsection*{Klein bottle}

\subsubsection*{Möbius band} \subsubsection*{Mapping cylinder}

\subsubsection*{Neighborhood} \subsubsection*{Null-homotopic} A function is null-homotopy if it's at an end of a null-homotopy.

\subsubsection*{Null-homotopy} A homotopy with one end a constant function (mapping everything into a point)

\subsubsection*{Path-component} \subsubsection*{Product of cell complexes} \subsubsection*{Projection n-space ${\mathbb C}P^n$} \subsubsection*{Projection n-space ${\mathbb R}P^n$}

\subsubsection*{Quotient map} \subsubsection*{Quotient space} TODO

\subsubsection*{Reduced suspension} \subsubsection*{Rel} \subsubsection*{Relation of homotopy among maps $X\to Y$} \subsubsection*{Retraction} A left-inverse to an inclusion.

$r\colon X\to A$

$r\circ\iota=\mathrm{id}_A$

Essentially dual to a section.

\url{https://en.wikipedia.org/wiki/Retraction_(topology)}

\subsubsection*{Simplex} \subsubsection*{Skeleton} \subsubsection*{Subcomplex} \subsubsection*{Topological subspace} $\langle X, T_X\rangle$

The topological subspace is the topological spaces $\langle A, T_A\rangle$ where $A\subset X$ and $T_A\equiv {U\cap A\mid U\in T_X}$

\subsubsection*{Topological torus ${\mathbb T}^2$} Up to homeomorphism,

${\mathbb S}^1 \times {\mathbb S}^1$ with the box topology.\

Also ${\mathbb R}^2/\sim$ with $\sim$ given via\ $\langle x,y\rangle\sim\langle x+1,y\rangle\sim\langle x,y+1\rangle$\

Equivalently for ${\mathbb T}^n$.

\subsubsection*{Smash product} \subsubsection*{Topological sphere ${\mathbb S}^n$} Up to homeomorphism,

${\mathbb S}^n = \left{ x \in {\mathbb R}^{n+1} \mid \vert\vert x \vert\vert = 1 \right}$ with the subset topology induced from ${\mathbb R}^{n+1}$.\

TODO: describe induced topology\

With $S^{n}$ the (radius 1/2, i.e. smaller) sphere with south pole at $0\in {\mathbb R}^{n+1}$ and north pole at $\langle 0,\dots, 0, 1\rangle$,\ $P_N\colon S^{n}\to({\mathbb R}^n \cup {{\mathrm{pt}}})$\ $P_N(x_1=0,\dots,x_{n-1}=0,x_n=1):={\mathrm{pt}}$\ $P_N(a, x_n):=\frac{1}{1-x_n}\cdot a$

\subsubsection*{Suspension}

\subsubsection*{Wedge sum}