\begin{align*} {\mathbb{D}}^n = \mathbb{B}^n &\coloneqq\left\{{ \mathbf{x} \in {\mathbf{R}}^{n} {~\mathrel{\Big\vert}~}{\left\lVert {\mathbf{x}} \right\rVert} \leq 1}\right\} {\mathbb{S}}^n &\coloneqq\left\{{ \mathbf{x} \in {\mathbf{R}}^{n+1} {~\mathrel{\Big\vert}~}{\left\lVert {\mathbf{x}} \right\rVert} = 1}\right\} = {{\partial}}{\mathbb{D}}^n \\ .\end{align*}

Note: I’ll immediately drop the blackboard notation, this is just to emphasize that they’re “canonical” objects.

The sphere can be constructed in several equivalent ways:

• \(S^n \cong D^n / {{\partial}}D^n\): collapsing the boundary of a disc is homeomorphic to a sphere.
• \(S^n \cong D^n \coprod_{{{\partial}}D^n} D^n\): gluing two discs along their boundary.

Note the subtle differences in dimension: \(S^n\) is a manifold of dimension \(n\) embedded in a space of dimension \(n+1\).

Low Dimensional Discs/Balls vs Spheres

Constructed in one of several equivalent ways:

• \(S^n/\sim\) where \(\mathbf{x} \sim -\mathbf{x}\), i.e. antipodal points are identified.
• The space of lines in \({\mathbf{R}}^{n+1}\).

One can also define \({\mathbf{RP}}^ \infty \coloneqq\directlim_{n} {\mathbf{RP}}^n\). Fits into a fiber bundle of the form

Defined in a similar ways,

• Taking the unit sphere in \({\mathbf{C}}^n\) and identifying \(\mathbf{z} \sim -\mathbf{z}\).
• The space of lines in \({\mathbf{C}}^{n+1}\)

Can similarly define \({\mathbf{CP}}^ \infty \coloneqq\directlim_n {\mathbf{CP}}^n\). Fits into a fiber bundle of the form

The \(n{\hbox{-}}\)torus, defined as \begin{align*} T^n \coloneqq\prod_{j=1}^n S^1 = S^1 \times S^1 \times \cdots .\end{align*}

The real Grassmannian, \({\operatorname{Gr}}(n, k)_{/{\mathbf{R}}}\), i.e. the set of \(k\) dimensional subspaces of \({\mathbf{R}}^n\). One can similar define \({\operatorname{Gr}}(n, k)_{{\mathbf{C}}}\) for complex subspaces. Note that \({\mathbf{RP}}^n = {\operatorname{Gr}}(n, 1)_{{\mathbf{R}}}\) and \({\mathbf{CP}}^n = {\operatorname{Gr}}(n, 1)_{/{\mathbf{C}}}\).

The Stiefel manifold \(V_{n}(k)_{{\mathbf{R}}}\), the space of orthonormal \(k{\hbox{-}}\)frames in \({\mathbf{R}}^n\)?

Lie Groups:

• The general linear group, \(\operatorname{GL}_{n}({\mathbf{R}})\)
  • The special linear group \(SL_{n}({\mathbf{R}})\)
• The orthogonal group, \(O_{n}({\mathbf{R}})\)
  • The special orthogonal group, \(SO_{n}({\mathbf{R}})\)
• The real unitary group, \(U_{n}({\mathbf{C}})\)
  • The special unitary group, \(SU_{n}({\mathbf{R}})\)
• The symplectic group \(Sp(2n)\)

Some other spaces that show up, but don’t usually have great algebraic topological properties:

• Affine \(n\)-space over a field \({\mathbf{A}}^n(k) = k^n \rtimes GL_{n}(k)\)
• The projective space \({\mathbf{P}}^n(k)\)
• The projective linear group over a ring \(R\), \(PGL_{n}(R)\)
• The projective special linear group over a ring \(R\), \(PSL_{n}(R)\)
• The modular groups \(PSL_{n}({\mathbf{Z}})\)
  • Specifically \(PSL_{2}({\mathbf{Z}})\)

\(K(G, n)\) is an Eilenberg-MacLane space, the homotopy-unique space satisfying \begin{align*} \pi_{k}(K(G, n)) = \begin{cases} G & k=n, \\ 0 & \text{else} \end{cases} \end{align*}

Some known examples:

• \(K({\mathbf{Z}}, 1) = S^1\)
• \(K({\mathbf{Z}}, 2) = {\mathbf{CP}}^\infty\)
• \(K({\mathbf{Z}}/2{\mathbf{Z}}, 1) = {\mathbf{RP}}^\infty\)

\(M(G, n)\) is a Moore space, the homotopy-unique space satisfying \begin{align*} H_{k}(M(G, n); G) = \begin{cases} G & k=n, \\ 0 & k\neq n. \end{cases} \end{align*}

Some known examples:

• \(M({\mathbf{Z}}, n) = S^n\)
• \(M({\mathbf{Z}}/2{\mathbf{Z}}, 1) = {\mathbf{RP}}^2\)
• \(M({\mathbf{Z}}/p{\mathbf{Z}}, n)\) is made by attaching \(e^{n+1}\) to \(S^n\) via a degree \(p\) map.

Write \(D(k, X)\) for the space \(X\) with \(k\in {\mathbb{N}}\) distinct points deleted, i.e. the punctured space \(X - \left\{{x_{1}, x_{2}, \ldots x_{k}}\right\}\) where each \(x_{i} \in X\).

The “generalized uniform bouquet”? \(\mathcal{B}^n(m) = \bigvee_{i=1}^n S^m\). There’s no standard name for this, but it’s an interesting enough object to consider!

Possible modifications to a space \(X\):

• Remove a line segment
• Remove an entire line/axis
• Remove a hole
• Quotient by a group action (e.g. antipodal map, or rotation)
• Remove a knot
• Take complement in ambient space

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