Examples: Algebraic Topology

Standard Spaces and Modifications

\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.

  • \({\mathcal{M}}\simeq S^1\) where \({\mathcal{M}}\) is the Mobius band.
  • \({\mathbf{CP}}^n = {\mathbf{C}}^n \coprod {\mathbf{CP}}^{n-1} = \coprod_{i=0}^n {\mathbf{C}}^i\)
  • \({\mathbf{CP}}^n = S^{2n+1} / S^n\)
  • \(S^n / S^k \simeq S^n \vee \Sigma S^k\).

In low dimensions, there are some “accidental” homeomorphisms:

  • \({\mathbf{RP}}^1 \cong S^1\)
  • \({\mathbf{CP}}^1 \cong S^2\)
  • \({\operatorname{SO}}(3) \cong {\mathbf{RP}}^2\)?

Modifying Known Spaces

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

Low Dimensional Homology Examples

\begin{align*} \begin{array}{cccccccccc} S^1 &= &[&{\mathbf{Z}}, &{\mathbf{Z}}, &0, &0, &0, &0\rightarrow & ]\\ {\mathcal{M}}&= &[&{\mathbf{Z}}, &{\mathbf{Z}}, &0, &0, &0, &0\rightarrow & ]\\ {\mathbf{RP}}^1 &= &[&{\mathbf{Z}}, &{\mathbf{Z}}, &0, &0, &0, &0\rightarrow & ]\\ {\mathbf{RP}}^2 &= &[&{\mathbf{Z}}, &{\mathbf{Z}}_{2}, &0, &0, &0, &0\rightarrow & ]\\ {\mathbf{RP}}^3 &= &[&{\mathbf{Z}}, &{\mathbf{Z}}_{2}, &0, &{\mathbf{Z}}, &0, &0\rightarrow & ]\\ {\mathbf{RP}}^4 &= &[&{\mathbf{Z}}, &{\mathbf{Z}}_{2}, &0, &{\mathbf{Z}}_{2}, &0, &0\rightarrow & ]\\ S^2 &= &[&{\mathbf{Z}}, &0, &{\mathbf{Z}}, &0, &0, &0\rightarrow & ]\\ {\mathbb{T}}^2 &= &[&{\mathbf{Z}}, &{\mathbf{Z}}^2, &{\mathbf{Z}}, &0, &0, &0\rightarrow & ]\\ {\mathbb{K}}&= &[&{\mathbf{Z}}, &{\mathbf{Z}}\oplus {\mathbf{Z}}_{2}, &0, &0, &0, &0\rightarrow & ]\\ {\mathbf{CP}}^1 &= &[&{\mathbf{Z}}, &0, &{\mathbf{Z}}, &0, &0, &0\rightarrow & ]\\ {\mathbf{CP}}^2 &= &[&{\mathbf{Z}}, &0, &{\mathbf{Z}}, &0, &{\mathbf{Z}}, &0\rightarrow & ]\\ \end{array} .\end{align*}

Table of Homotopy and Homology Structures

The following is a giant list of known homology/homotopy.

\(X\)\(\pi_*(X)\)\(H_*(X)\)CW Structure\(H^*(X)\)
\({\mathbf{R}}^1\)\(0\)\(0\)\({\mathbf{Z}}\cdot 1 + {\mathbf{Z}}\cdot x\)0
\({\mathbf{R}}^n\)\(0\)\(0\)\(({\mathbf{Z}}\cdot 1 + {\mathbf{Z}}\cdot x)^n\)0
\(D(k, {\mathbf{R}}^n)\)\(\pi_*\bigvee^k S^1\)\(\bigoplus_{k} H_* M({\mathbf{Z}}, 1)\)\(1 + kx\)?
\(B^n\)\(\pi_*({\mathbf{R}}^n)\)\(H_*({\mathbf{R}}^n)\)\(1 + x^n + x^{n+1}\)0
\(S^n\)\([0 \ldots , {\mathbf{Z}}, ? \ldots]\)\(H_*M({\mathbf{Z}}, n)\)\(1 + x^n\) or \(\sum_{i=0}^n 2x^i\)\({\mathbf{Z}}[{}_{n}x]/(x^2)\)
\(D(k, S^n)\)\(\pi_*\bigvee^{k-1}S^1\)\(\bigoplus_{k-1}H_*M({\mathbf{Z}}, 1)\)\(1 + (k-1)x^1\)?
\(T^2\)\(\pi_*S^1 \times \pi_* S^1\)\((H_* M({\mathbf{Z}}, 1))^2 \times H_* M({\mathbf{Z}}, 2)\)\(1 + 2x + x^2\)\(\Lambda({}_{1}x_{1}, {}_{1}x_{2})\)
\(T^n\)\(\prod^n \pi_* S^1\)\(\prod_{i=1}^n (H_* M({\mathbf{Z}}, i))^{n\choose i}\)\((1 + x)^n\)\(\Lambda({}_{1}x_{1}, {}_{1}x_{2}, \ldots {}_{1}x_{n})\)
\(D(k, T^n)\)\([0, 0, 0, 0, \ldots]\)?\([0, 0, 0, 0, \ldots]\)?\(1 + x\)?
\(S^1 \vee S^1\)\(\pi_*S^1 \ast \pi_* S^1\)\((H_*M({\mathbf{Z}}, 1))^2\)\(1 + 2x\)?
\(\bigvee^n S^1\)\(\ast^n \pi_* S^1\)\(\prod H_* M({\mathbf{Z}}, 1)\)\(1 + x\)?
\({\mathbf{RP}}^1\)\(\pi_* S^1\)\(H_* M({\mathbf{Z}}, 1)\)\(1 + x\)\({}_{0}{\mathbf{Z}}\times {}_{1}{\mathbf{Z}}\)
\({\mathbf{RP}}^2\)\(\pi_*K({\mathbf{Z}}/2{\mathbf{Z}}, 1)+ \pi_* S^2\)\(H_*M({\mathbf{Z}}/2{\mathbf{Z}}, 1)\)\(1 + x + x^2\)\({}_{0}{\mathbf{Z}}\times {}_{2}{\mathbf{Z}}/2{\mathbf{Z}}\)
\({\mathbf{RP}}^3\)\(\pi_*K({\mathbf{Z}}/2{\mathbf{Z}}, 1)+ \pi_* S^3\)\(H_*M({\mathbf{Z}}/2{\mathbf{Z}}, 1) + H_*M({\mathbf{Z}}, 3)\)\(1 + x + x^2 + x^3\)\({}_{0}{\mathbf{Z}}\times {}_{2}{\mathbf{Z}}/2{\mathbf{Z}}\times {}_{3}{\mathbf{Z}}\)
\({\mathbf{RP}}^4\)\(\pi_*K({\mathbf{Z}}/2{\mathbf{Z}}, 1)+ \pi_* S^4\)\(H_*M({\mathbf{Z}}/2{\mathbf{Z}}, 1) + H_*M({\mathbf{Z}}/2{\mathbf{Z}}, 3)\)\(1 + x + x^2 + x^3 + x^4\)\({}_{0}{\mathbf{Z}}\times ({}_{2}{\mathbf{Z}}/2{\mathbf{Z}})^2\)
\({\mathbf{RP}}^n, n \geq 4\) even\(\pi_*K({\mathbf{Z}}/2{\mathbf{Z}}, 1)+ \pi_*S^n\)\(\prod_{\text{odd}~i < n} H_*M({\mathbf{Z}}/2{\mathbf{Z}}, i)\)\(\sum_{i=1}^n x^i\)\({}_{0}{\mathbf{Z}}\times \prod_{i=1}^{n/2}{}_{2}{\mathbf{Z}}/2{\mathbf{Z}}\)
\({\mathbf{RP}}^n, n \geq 4\) odd\(\pi_*K({\mathbf{Z}}/2{\mathbf{Z}}, 1)+ \pi_*S^n\)\(\prod_{\text{odd}~ i \leq n-2} H_*M({\mathbf{Z}}/2{\mathbf{Z}}, i) \times H_* S^n\)\(\sum_{i=1}^n x^i\)\(H^*({\mathbf{RP}}^{n-1}) \times {}_{n}{\mathbf{Z}}\)
\({\mathbf{CP}}^1\)\(\pi_*K({\mathbf{Z}}, 2) + \pi_* S^3\)\(H_* S^2\)\(x^0 + x^2\)\({\mathbf{Z}}[{}_{2}x]/({}_2x^{2})\)
\({\mathbf{CP}}^2\)\(\pi_*K({\mathbf{Z}}, 2) + \pi_* S^5\)\(H_*S^2 \times H_* S^4\)\(x^0 + x^2 + x^4\)\({\mathbf{Z}}[{}_{2}x]/({}_2x^{3})\)
\({\mathbf{CP}}^n, n \geq 2\)\(\pi_*K({\mathbf{Z}}, 2) + \pi_*S^{2n+1}\)\(\prod_{i=1}^n H_* S^{2i}\)\(\sum_{i=1}^n x^{2i}\)\({\mathbf{Z}}[{}_{2}x]/({}_2x^{n+1})\)
Mobius Band\(\pi_* S^1\)\(H_* S^1\)\(1 + x\)?
Klein Bottle\(K({\mathbf{Z}}\rtimes_{-1} {\mathbf{Z}}, 1)\)\(H_*S^1 \times H_* {\mathbf{RP}}^\infty\)\(1 + 2x + x^2\)?

  • \({\mathbf{R}}^n\) is a contractible space, and so \([S^m, {\mathbf{R}}^n] = 0\) for all \(n, m\) which makes its homotopy groups all zero.

  • \(D(k, {\mathbf{R}}^n) = {\mathbf{R}}^n - \left\{{x_{1} \ldots x_{k}}\right\} \simeq\bigvee_{i=1}^k S^1\) by a deformation retract.

  • \(S^n \cong B^n / {\partial}B^n\) and employs an attaching map

\begin{align*} \phi: (D^n, {\partial}D^n) &\to S^n \\ (D^n, {\partial}D^n) &\mapsto (e^n, e^0) .\end{align*}

  • \(B^n \simeq{\mathbf{R}}^n\) by normalizing vectors.

  • Use the inclusion \(S^n \hookrightarrow B^{n+1}\) as the attaching map.

  • \({\mathbf{CP}}^1 \cong S^2\).

  • \({\mathbf{RP}}^1 \cong S^1\).

  • Use \(\left[ \pi_{1}, \prod \right]= 0\) and the universal cover \({\mathbf{R}}^1 \twoheadrightarrow S^1\) to yield the cover \({\mathbf{R}}^n \twoheadrightarrow T^n\).

  • Take the universal double cover \(S^n \twoheadrightarrow^{\times 2} {\mathbf{RP}}^n\) to get equality in \(\pi_{i\geq 2}\).

  • Use \({\mathbf{CP}}^n = S^{2n+1} / S^1\)

  • Alternatively, the fundamental group is \({\mathbf{Z}}\ast{\mathbf{Z}}/ bab^{-1}a\). Use the fact the \(\tilde K = {\mathbf{R}}^2\).

  • \(M \simeq S^1\) by deformation-retracting onto the center circle.

  • \(D(1, S^n) \cong {\mathbf{R}}^n\) and thus \(D(k, S^n) \cong D(k-1, {\mathbf{R}}^n) \cong \bigvee^{k-1} S^1\)