# 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.

\scriptsize

$$X$$$$\pi_*(X)$$$$H_*(X)$$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$$K({\mathbf{Z}}\rtimes_{-1} {\mathbf{Z}}, 1)$$$$H_*S^1 \times H_* {\mathbf{RP}}^\infty$$$$1 + 2x + x^2$$?
\normalsize


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