# 1 Preface

References:

• Munkres [1]
• Hatcher [2]

Some fun resources:

## 1.1 Notation

• $$A\times B, \prod X_j$$ are direct products.

• $$A\oplus B, \bigoplus_j X_j$$ are direct sums, the subset of $$A\times B$$ where only finitely many terms are nonzero.

• $${{\mathbb{Z}}}^n$$ denotes the direct sum of $$n$$ copies of the group $${{\mathbb{Z}}}$$.
• Note that $$A \oplus B \hookrightarrow A\times B$$.
• $$A\ast B, \ast_j X_J$$ are free products, $$F_n \mathrel{\vcenter{:}}={\mathbb{Z}}^{\ast n}$$ is the free group on $$n$$ generators.

• Note that the abelianization yields $$\qty{\ast_j X_j} = \bigoplus_j X_j$$.
Notation Definition
$$X\times Y, \prod_{j\in J} X_j, X^{\times n}$$ Products
$$X\oplus Y, \bigoplus_{j\in J} X_j, X^{\oplus n}$$ Direct sums
$$X\ast Y, \ast_{j\in J} X_j, X^{\ast n}$$ Free products
$$X\otimes Y, \bigotimes_{j\in J} X_j, X^{\otimes n}$$ Tensor products
$${\mathbb{Z}}^n$$ The free abelian group of rank $$n$$
$${\mathbb{Z}}^{\ast n}$$ The free group on $$n$$ generators
$$\pi_0(X)$$ The set of path components of $$X$$
$$G=1$$ The trivial abelian group
$$G=0$$ The trivial nonabelian group

Both the product and direct sum have coordinate-wise operations. For finite index sets $${\left\lvert {J} \right\rvert}< \infty$$, the direct sum and product coincide, but in general there is only an injection $$\bigoplus_j X_i \hookrightarrow\prod_j X_j$$. In the direct sum $$\bigoplus_j X_j$$ have only finitely many nonzero entries, while the product allows infinitely many nonzero entries. So in general, I always use the product notation.

The free group on $$n$$ generators is the free product of $$n$$ free abelian groups, but is not generally abelian! So we use multiplicative notation, and elements \begin{align*} x \in {\mathbb{Z}}^{\ast n} = \left< a_1, \ldots, a_n\right> \end{align*} are finite words in the noncommuting symbols $$a_i^k$$ for $$k\in {\mathbb{Z}}$$. E.g. an element may look like \begin{align*} x = a_1^2 a_2^4 a_1 a_2^{-2} .\end{align*}

The free abelian group of rank $$n$$ is the abelianization of $${\mathbb{Z}}^{\ast n}$$, and its elements are characterized by \begin{align*} x\in {\mathbb{Z}}^{\ast n} = \left\langle{ a_1, \cdots, a_n }\right\rangle \implies x = \sum_n c_i a_i \text{ for some } c_i \in {\mathbb{Z}} \end{align*} where the $$a_i$$ are some generating set of $$n$$ elements and we used additive notation since the group is abelian. E.g. such an element may look like \begin{align*} x = 2a_1 + 4a_2 + a_1 - a_2 = 3a_1 + 3a_2 .\end{align*}

## 1.2 Conventions

• Spaces are assumed to be connected and path connected, so $$\pi_0(X) = H_0(X) = {\mathbb{Z}}$$.

• Graded objects like $$\pi_*, H_*, H^*$$ are sometimes represented as lists. In this case, all list entries start indexing at 1. Examples: \begin{align*} \pi_*(X) &= [\pi_1(X), \pi_2(X), \pi_3(X), \cdots] \\ H_*(X) &= [H_1(X), H_2(X), H_3(X), \cdots] .\end{align*}

## 1.3 Some Prerequisite Algebra Facts

A group morphism $$f:X \to Y$$ can not be injective if $$Y$$ is trivial unless $$X$$ is also trivial.

There are no nontrivial homomorphisms from finite groups into free groups. In particular, any homomorphism $${\mathbb{Z}}_n \to {\mathbb{Z}}$$ is trivial.

(Click to expand)

Homomorphisms preserve torsion; the former has $$n{\hbox{-}}$$torsion while the latter does not.

This is especially useful if you have some $$f: A\to B$$ and you look at the induced homomorphism $$f_*: \pi_1(A) \to\pi_1(B)$$. If the former is finite and the latter contains a copy of $${\mathbb{Z}}$$, then $$f_*$$ has to be the trivial map $$f_*([\alpha]) = e \in \pi_1(B)$$ for every $$[\alpha] \in \pi_1(A)$$.

# 2 Summary and Topics: Point-Set Topology

• Connectedness
• Compactness
• Hausdorff Spaces
• Path-Connectedness

# 3 Definitions

## 3.1 Point-Set Topology

A set $$S$$ in a metric space $$(X, d)$$ is bounded iff there exists an $$m\in {\mathbb{R}}$$ such that $$d(x, y) < m$$ for every $$x, y\in S$$.

There does not exist a disconnecting set $$X = A{\coprod}B$$ such that $$\emptyset \neq A, B \subsetneq$$, i.e. $$X$$ is the union of two proper disjoint nonempty sets. Additional condition for a subspace $$Y\subset X$$: $$\operatorname{cl}_{Y}(A) \cap V = A \cap\operatorname{cl}_{Y}(B) = \emptyset$$.

Equivalently, $$X$$ contains no proper nonempty clopen sets.

Set $$x\sim y$$ iff there exists a connected set $$U\ni x, y$$ and take equivalence classes.

• A set is closed if and only if its complement is open.
• A set is closed iff it contains all of its limit points.
• A closet set in a subspace: $$Y\subset X \implies \operatorname{cl}_{Y}(A) \mathrel{\vcenter{:}}=\operatorname{cl}_{X}(A)\cap Y$$.

See .

A topological space $$(X, \tau)$$ is compact if every open cover has a finite subcover.

That is, if $$\left\{{U_{j} {~\mathrel{\Big|}~}j\in J}\right\} \subset \tau$$ is a collection of open sets such that $$X = \cup_{j\in J} U_{j}$$, then there exists a finite subset $$J' \subset J$$ such that $$X \subseteq \cup_{j\in J'} U_{j}$$.

A map $$f:X\to Y$$ between topological spaces is continuous if and only if whenever $$U \subseteq Y$$ is open, $$f ^{-1} (U) \subseteq X$$ is open.

A collection of subsets $$\left\{{U_\alpha}\right\}$$ of $$X$$ is said to cover $$X$$ iff $$X = \cup_{\alpha} U_\alpha$$. If $$A\subseteq X$$ is a subspace, then this collection covers $$A$$ iff $$A\subseteq \cup_{\alpha} U_\alpha$$.

A subset $$Q\subset X$$ is dense iff $$y\in N_{y} \subset X \implies N_{y} \cap Q \neq \emptyset$$ iff $$\mkern 1.5mu\overline{\mkern-1.5muQ\mkern-1.5mu}\mkern 1.5mu = X$$.

A space is first-countable iff every point admits a countable neighborhood basis.

A topological space $$X$$ is Hausdorff iff for every $$p\neq q \in X$$ there exist disjoint open sets $$U\ni p$$ and $$V\ni q$$.

A map $$\iota$$ with a left inverse $$f$$ satisfying $$f\circ \iota = \operatorname{id}$$

For $$(X, d)$$ a compact metric space and $$\left\{{U_\alpha}\right\}\rightrightarrows X$$, there exist $$\delta_{L} > 0$$ such that \begin{align*} A\subset X, ~ {\operatorname{diam}}(A) < \delta_{L} \implies A\subseteq U_\alpha \text{ for some } \alpha .\end{align*}

For $$A\subset X$$, $$x$$ is a limit point of $$A$$ if every punctured neighborhood $$P_{x}$$ of $$x$$ satisfies $$P_{x} \cap A \neq \emptyset$$, i.e. every neighborhood of $$x$$ intersects $$A$$ in some point other than $$x$$ itself.

Equivalently, $$x$$ is a limit point of $$A$$ iff $$x\in \operatorname{cl}_{X}(A\setminus\left\{{x}\right\})$$.

A space is locally connected at a point $$x$$ iff $$\forall N_{x} \ni x$$, there exists a $$U\subset N_{x}$$ containing $$x$$ that is connected.

A space $$X$$ is locally compact iff every $$x\in X$$ has a neighborhood contained in a compact subset of $$X$$.

A collection of subsets $${\mathcal{S}}$$ of $$X$$ is locally finite iff each point of $$M$$ has a neighborhood that intersects at most finitely many elements of $${\mathcal{S}}$$.

A space is locally path-connected if it admits a basis of path-connected open subsets.

A neighborhood of a point $$x$$ is any open set containing $$x$$.

A space is normal if any two disjoint closed subsets can be separated by neighborhoods.

If $$p\in X$$, a neighborhood basis at $$p$$ is a collection $${\mathcal{B}}_{p}$$ of neighborhoods of $$p$$ such that if $$N_{p}$$ is a neighborhood of $$p$$, then $$N_{p} \supseteq B$$ for at least one $$B\in {\mathcal{B}}_{p}$$.

A map $$f:X\to Y$$ is an open map (respectively a closed map) if and only if whenever $$U \subseteq X$$ is open (resp. closed), $$f(U)$$ is again open (resp. closed)>

A topological space $$X$$ is paracompact iff every open cover of $$X$$ admits an open locally finite refinement.

A map $$q:X\to Y$$ is a quotient map if and only if

1. $$q$$ is surjective, and
2. $$U \subseteq Y$$ is open if and only if $$q ^{-1} (U)$$ is open.

A space $$X$$ is path connected if and only if for every pair of points $$x\neq y$$ there exists a continuous map $$f:I \to X$$ such that $$f(0) = x$$ and $$f(1) = y$$.

Set $$x\sim y$$ iff there exists a path-connected set $$U\ni x, y$$ and take equivalence classes.

A subset $$A\subseteq X$$ is precompact iff $$\operatorname{cl}_{X}(A)$$ is compact.

For $$(X, \tau_X)$$ and $$(Y, \tau_Y)$$ topological spaces, defining \begin{align*} \tau_{X \times Y} \mathrel{\vcenter{:}}=\left\{{U \times V {~\mathrel{\Big|}~}U\in \tau_X,\, V\in \tau_Y}\right\} \end{align*} yields the product topology on $$X \times Y$$.

A cover $${\mathcal{V}}\rightrightarrows X$$ is a refinement of $${\mathcal{U}}\rightrightarrows X$$ iff for each $$V\in {\mathcal{V}}$$ there exists a $$U\in{\mathcal{U}}$$ such that $$V\subseteq U$$.

A space $$X$$ is regular if whenever $$x\in X$$ and $$F\not\ni x$$ is closed, $$F$$ and $$x$$ are separated by neighborhoods.

A map $$r$$ in $$A\mathrel{\textstyle\substack{\hookrightarrow^{\iota}\\\textstyle\dashleftarrow_{r}}} X$$ satisfying \begin{align*}r\circ\iota = \operatorname{id}_{A}.\end{align*} Equivalently $$X \twoheadrightarrow_{r} A$$ and $${\left.{{r}} \right|_{{A}} } = \operatorname{id}_{A}$$. If $$X$$ retracts onto $$A$$, then $$i_*$$ is injective.

Alt: Let $$X$$ be a topological space and $$A \subset X$$ be a subspace, then a retraction of $$X$$ onto $$A$$ is a map $$r: X\to X$$ such that the image of $$X$$ is $$A$$ and $$r$$ restricted to $$A$$ is the identity map on $$A$$.

A subset $$U \subseteq X$$ is saturated with respect to a surjective map $$p: X\twoheadrightarrow Y$$ if and only if whenever $$U \cap p ^{-1} (y) = V \neq \emptyset$$, we have $$V \subseteq U$$, i.e. $$U$$ contains every set $$p ^{-1} (y)$$ that it intersects. Equivalently, $$U$$ is the complete inverse image of a subset of $$Y$$.

A space $$X$$ is separable iff $$X$$ contains a countable dense subset.

A space is second-countable iff it admits a countable basis.

For $$(X, \tau)$$ a topological space and $$U \subseteq X$$ an arbitrary subset, the space $$(U, \tau_U)$$ is a topological space with a subspace topology defined by \begin{align*} \tau_U \mathrel{\vcenter{:}}=\left\{{Y \cap U {~\mathrel{\Big|}~}U \in \tau}\right\} .\end{align*}

A map $$\pi$$ with a right inverse $$f$$ satisfying \begin{align*}\pi \circ f = \operatorname{id}\end{align*}

• $$T_0$$: For any 2 points $$x_1\neq x_2$$, at least one $$x_i$$ (say $$x_1$$) admits a neighborhood not containing $$x_2$$.

• $$T_1$$: For any 2 points, both admit neighborhoods not containing the other.

• $$T_2$$: For any 2 points, both admit disjoint separating neighborhoods.

• $$T_{2.5}$$: For any 2 points, both admit disjoint closed separating neighborhoods.

• $$T_3$$: $$T_0$$ & regular. Given any point $$x$$ and any closed $$F\not\ni x$$, there are neighborhoods separating $$F$$ and $$x$$.

• $$T_{3.5}$$: $$T_0$$ & completely regular. Any point $$x$$ and closed $$F\not\ni x$$ can be separated by a continuous function.

• $$T_4$$: $$T_1$$ & normal. Any two disjoint closed subsets can be separated by neighborhoods.

Closed under arbitrary unions and finite intersections.

For $$X$$ an arbitrary set, a collection of subsets $${\mathcal{B}}$$ is a basis for $$X$$ iff $${\mathcal{B}}$$ is closed under intersections, and every intersection of basis elements contains another basis element. The set of unions of elements in $$B$$ is a topology and is denoted the topology generated by $${\mathcal{B}}$$.

A continuous map $$f:X \to Y$$ for which $$X\cong f(X)$$ are homeomorphic is called a topological embedding.

For $$f: (X, d_{x}) \to (Y, d_{Y})$$ metric spaces, \begin{align*} \forall \varepsilon> 0, ~\exists \delta > 0 \text{ such that } \quad d_{X}(x_{1}, x_{2}) < \delta \implies d_{Y}(f(x_{1}), f(x_{2})) < \varepsilon .\end{align*}

## 3.2 Algebraic Topology

For an $$R{\hbox{-}}$$module $$M$$, a basis $$B$$ is a linearly independent generating set.

Points $$x\in M^n$$ defined by \begin{align*} {\partial}M = \left\{{x\in M: H_{n}(M, M-\left\{{x}\right\}; {\mathbb{Z}}) = 0}\right\} \end{align*}

Denoting $$\Delta^p \xrightarrow{\sigma} X \in C_{p}(X; G)$$, a map that sends pairs ($$p{\hbox{-}}$$chains, $$q{\hbox{-}}$$cochains) to $$(p-q){\hbox{-}}$$chains $$\Delta^{p-q} \to X$$ by \begin{align*} H_{p}(X; R)\times H^q(X; R) \xrightarrow{\frown} H_{p-q}(X; R)\\ \sigma \frown \psi = \psi(F_{0}^q(\sigma))F_{q}^p(\sigma) \end{align*} where $$F_{i}^j$$ is the face operator, which acts on a simplicial map $$\sigma$$ by restriction to the face spanned by $$[v_{i} \ldots v_{j}]$$, i.e. $$F_{i}^j(\sigma) = {\left.{{\sigma}} \right|_{{[v_{i} \ldots v_{j}]}} }$$.

A map $$X \xrightarrow{f} Y$$ is said to be cellular if $$f(X^{(n)}) \subseteq Y^{(n)}$$ where $$X^{(n)}$$ denotes the $$n{\hbox{-}}$$ skeleton.

An element $$c \in C_{p}(X; R)$$ can be represented as the singular $$p$$ simplex $$\Delta^p \to X$$.

Given two maps between chain complexes $$(C_*, {\partial}_{C}) \xrightarrow{f, ~g} (D_*, {\partial}_{D})$$, a chain homotopy is a family $$h_{i}: C_{i} \to B_{i+1}$$ satisfying \begin{align*}f_{i}-g_{i} = {\partial}_{B, i-1}\circ h_{n} + h_{i+1}\circ {\partial}_{A, i}\end{align*} .

A map between chain complexes $$(C_*, {\partial}_{C}) \xrightarrow{f} (D_*, {\partial}_{D})$$ is a chain map iff each component $$C_{i} \xrightarrow{f_{i}} D_{i}$$ satisfies \begin{align*} f_{i-1}\circ{\partial}_{C, i} = {\partial}_{D,i} \circ f_{i} \end{align*} (i.e this forms a commuting ladder)

A manifold that is compact, with or without boundary.

An cochain $$c \in C^p(X; R)$$ is a map $$c \in \hom(C_{p}(X; R), R)$$ on chains.

A constant map $$f: X\to Y$$ iff $$f(X) = y_{0}$$ for some $$y_{0}\in Y$$, i.e. for every $$x\in X$$ the output value $$f(x) = y_{0}$$ is the same.

For a directed system $$(X_{i}, f_{ij}$$, the colimit is an object $$X$$ with a sequence of projections $$\pi_{i}:X\to X_{i}$$ such that for any $$Y$$ mapping into the system, the following diagram commutes:

• Products
• Pullbacks
• Inverse / projective limits
• The $$p{\hbox{-}}$$adic integers $${\mathbb{Z}}_{p}$$.

For a space $$X$$, defined as \begin{align*} CX = \frac{X\times I} {X \times\left\{{0}\right\}} .\end{align*} Example: The cone on the circle $$CS^1$$

Note that the cone embeds $$X$$ in a contractible space $$CX$$.

A space $$X$$ is contractible if $$\operatorname{id}_X$$ is nullhomotopic. i.e. the identity is homotopic to a constant map $$c(x) = x_0$$.

Equivalently, $$X$$ is contractible if $$X \simeq\left\{{x_0}\right\}$$ is homotopy equivalent to a point. This means that there exists a mutually inverse pair of maps $$f: X \to\left\{{x_0}\right\}$$ and $$g:\left\{{x_0}\right\} \to X$$ such that $$f\circ g \simeq\operatorname{id}_{\left\{{x_0}\right\}}$$ and $$g\circ f \simeq\operatorname{id}_X$$.1

A covering space of $$X$$ is the data $$p: \tilde X \to X$$ such that

1. Each $$x\in X$$ admits a neighborhood $$U$$ such that $$p ^{-1} (U)$$ is a union of disjoint open sets in $$\tilde V_i \subseteq X$$ (the sheets of $$\tilde X$$ over $$U$$),
2. $${ \left.{{p}} \right|_{{V_i}} }: V_i \to U$$ is a homeomorphism for each sheet.

An isomorphism of covering spaces $$\tilde X_1 \cong \tilde X_2$$ is a commutative diagram

A map taking pairs ($$p{\hbox{-}}$$cocycles, $$q{\hbox{-}}$$cocycles) to $$(p+q){\hbox{-}}$$cocyles by \begin{align*} H^p(X; R) \times H^q(X; R) \xrightarrow{\smile} H^{p+q}(X; R)\\ (a \cup b)(\sigma) = a(\sigma \circ I_{0}^p)~b(\sigma \circ I_{p}^{p+q}) \end{align*} where $$\Delta^{p+q} \xrightarrow{\sigma} X$$ is a singular $$p+q$$ simplex and

\begin{align*}I_{i}^j: [i, \cdots, j] \hookrightarrow\Delta^{p+q} .\end{align*}

is an embedding of the $$(j-i){\hbox{-}}$$simplex into a $$(p+q){\hbox{-}}$$simplex.

On a manifold, the cup product is Poincaré dual to the intersection of submanifolds. Also used to show $$T^2 \not\simeq S^2 \vee S^1 \vee S^1$$.

An $$n{\hbox{-}}$$cell of $$X$$, say $$e^n$$, is the image of a map $$\Phi: B^n \to X$$. That is, $$e^n = \Phi(B^n)$$. Attaching an $$n{\hbox{-}}$$cell to $$X$$ is equivalent to forming the space $$B^n \coprod_{f} X$$ where $$f: {\partial}B^n \to X$$.

• A $$0{\hbox{-}}$$cell is a point.
• A $$1{\hbox{-}}$$cell is an interval $$[-1, 1] = B^1 \subset {\mathbb{R}}^1$$. Attaching requires a map from $$S^0 =\left\{{-1, +1}\right\} \to X$$
• A $$2{\hbox{-}}$$cell is a solid disk $$B^2 \subset {\mathbb{R}}^2$$ in the plane. Attaching requires a map $$S^1 \to X$$.
• A $$3{\hbox{-}}$$cell is a solid ball $$B^3 \subset {\mathbb{R}}^3$$. Attaching requires a map from the sphere $$S^2 \to X$$.

For a covering space $$\tilde X \xrightarrow{p} X$$, self-isomorphisms $$f:\tilde X \to \tilde X$$ of covering spaces are referred to as deck transformations.

A map $$r$$ in $$A\mathrel{\textstyle\substack{\hookrightarrow^{\iota}\\\textstyle\dashleftarrow_{r}}} X$$ that is a retraction (so $$r\circ \iota = \operatorname{id}_{A}$$) that also satisfies $$\iota \circ r \simeq\operatorname{id}_{X}$$.

Note that this is equality in one direction, but only homotopy equivalence in the other.

Equivalently, a map $$F:I\times X\to X$$ such that \begin{align*} F_{0}(x) &= \operatorname{id}_{X} F_{t}(x)\mathrel{\Big|}_{A} &= \operatorname{id}_{A} F_{1}(X) &= A .\end{align*}

Alt:

A deformation retract is a homotopy $$H:X\times I \to X$$ from the identity on $$X$$ to the identity on $$A$$ that fixes $$A$$ at all times: \begin{align*} H: X\times I \to X \\ H(x, 0) = \operatorname{id}_X \\ H(x, 1) = \operatorname{id}_A \\ x\in A \implies H(x, t) \in A \quad \forall t \end{align*}

Equivalently, this requires that $${\left.{{H}} \right|_{{A}} } = \operatorname{id}_A$$

A deformation retract between a space and a subspace is a homotopy equivalence, and further $$X\simeq Y$$ iff there is a $$Z$$ such that both $$X$$ and $$Y$$ are deformation retracts of $$Z$$. Moreover, if $$A$$ and $$B$$ both have deformation retracts onto a common space $$X$$, then $$A \simeq B$$.

Given any $$f: S^n \to S^n$$, there are induced maps on homotopy and homology groups. Taking $$f^*: H^n(S^n) \to H^n(S^n)$$ and identifying $$H^n(S^n) \cong {\mathbb{Z}}$$, we have $$f^*: {\mathbb{Z}}\to{\mathbb{Z}}$$. But homomorphisms of free groups are entirely determined by their action on generators. So if $$f^*(1) = n$$, define $$n$$ to be the degree of $$f$$.

For a functor $$T$$ and an $$R{\hbox{-}}$$module $$A$$, a left derived functor $$(L_{nT})$$ is defined as $$h_{n}(TP_{A})$$, where $$P_{A}$$ is a projective resolution of $$A$$.

For $$x\in M$$, the only nonvanishing homology group $$H_{i}(M, M - \left\{{x}\right\}; {\mathbb{Z}})$$

A functor $$T$$ is right exact if a short exact sequence

\begin{align*}0 \to A \to B \to C \to 0 \end{align*} yields an exact sequence

\begin{align*}\ldots TA \to TB \to TC \to 0 \end{align*} and is left exact if it yields

\begin{align*}0 \to TA \to TB \to TC \to \ldots \end{align*} Thus a functor is exact iff it is both left and right exact, yielding

\begin{align*}0 \to TA \to TB \to TC \to 0 .\end{align*}

$${\,\cdot\,}\otimes_{R} {\,\cdot\,}$$ is a right exact bifunctor.

An $$R{\hbox{-}}$$module is flat if $$A\otimes_{R} {\,\cdot\,}$$ is an exact functor.

An action $$G\curvearrowright X$$ is properly discontinuous if each $$x\in X$$ has a neighborhood $$U$$ such that all of the images $$g(U)$$ for $$g\in G$$ are disjoint, i.e. $$g_1(U) \cap g_2(U) \neq \emptyset \implies g_1 = g_2$$. The action is free if there are no fixed points.

Sometimes a slightly weaker condition is used: every point $$x\in X$$ has a neighborhood $$U$$ such that $$U \cap G(U) \neq \emptyset$$ for only finitely many $$G$$.

A $${\hbox{-}}$$module $$M$$ with a basis $$S = \left\{{s_{i}}\right\}$$ of generating elements. Every such module is the image of a unique map $$\mathcal{F}(S) = R^S \twoheadrightarrow M$$, and if $$M = \left< S \mathrel{\Big|}\mathcal{R} \right>$$ for some set of relations $$\mathcal{R}$$, then $$M \cong R^S / \mathcal{R}$$.

For a connected, closed, orientable manifold, $$[M]$$ is a generator of $$H_{n}(M; {\mathbb{Z}}) = {\mathbb{Z}}$$.

$$S = \left\{{s_{i}}\right\}$$ is a generating set for an $$R{\hbox{-}}$$ module $$M$$ iff \begin{align*}x\in M \implies x = \sum r_{i} s_{i}\end{align*} for some coefficients $$r_{i} \in R$$ (where this sum may be infinite).

Let $$X, Y$$ be topological spaces and $$f,g: X \to Y$$ continuous maps. Then a homotopy from $$f$$ to $$g$$ is a continuous function

$$F: X \times I \to Y$$

such that

$$F(x, 0) = f(x)$$ and $$F(x,1) = g(x)$$

for all $$x\in X$$. If such a homotopy exists, we write $$f\simeq g$$. This is an equivalence relation on $$\text{Hom}(X,Y)$$, and the set of such classes is denoted $$[X,Y] \mathrel{\vcenter{:}}=\hom (X,Y)/\simeq$$.

Let $$f: X \to Y$$ be a continuous map, then $$f$$ is said to be a homotopy equivalence if there exists a continuous map $$g: X \to Y$$ such that

$$f\circ g \simeq\operatorname{id}_Y$$ and $$g\circ f \simeq\operatorname{id}_X$$.

Such a map $$g$$ is called a homotopy inverse of $$f$$, the pair of maps is a homotopy equivalence.

If such an $$f$$ exists, we write $$X \simeq Y$$ and say $$X$$ and $$Y$$ have the same homotopy type, or that they are homotopy equivalent.

Note that homotopy equivalence is strictly weaker than homeomorphic equivalence, i.e., $$X\cong Y$$ implies $$X \simeq Y$$ but not necessarily the converse.

For a manifold $$M$$, a map on homology defined by \begin{align*} H_{\widehat{i}}M \otimes H_{\widehat{j}}M \to H_{\widehat{i+j}}X\\ \alpha\otimes\beta \mapsto \left< \alpha, \beta \right> \end{align*} obtained by conjugating the cup product with Poincaré Duality, i.e.

\begin{align*}\left< \alpha, \beta \right> = [M] \frown ([\alpha]^\vee\smile [\beta]^\vee) .\end{align*}

Then, if $$[A], [B]$$ are transversely intersecting submanifolds representing $$\alpha, \beta$$, then \begin{align*}\left<\alpha, \beta\right> = [A\cap B]\end{align*} . If $$\widehat{i} = j$$ then $$\left< \alpha, \beta \right> \in H_{0} M = {\mathbb{Z}}$$ is the signed number of intersection points.

Alt: The pairing obtained from dualizing Poincare Duality to obtain \begin{align*}\mathrm{F}(H_{i} M) \otimes\mathrm{F}(H_{n-i}M) \to {\mathbb{Z}}\end{align*} Computed as an oriented intersection number between two homology classes (perturbed to be transverse).

The nondegenerate bilinear form cohomology induced by the Kronecker Pairing: \begin{align*}I: H^k(M_{n}) \times H^{n-k}(M^n) \to {\mathbb{Z}}\end{align*} where $$n=2k$$.

• When $$k$$ is odd, $$I$$ is skew-symmetric and thus a symplectic form.
• When $$k$$ is even (and thus $$n \equiv 0 \pmod 4$$) this is a symmetric form.
• Satisfies $$I(x,y) = (-1)^{k(n-k)} I(y, x)$$

A map pairing a chain with a cochain, given by \begin{align*} H^n(X; R) \times H_{n}(X; R) \to R \\ ([\psi, \alpha]) \mapsto \psi(\alpha) \end{align*} which is a nondegenerate bilinear form.

At a point $$x \in V \subset M$$, a generator of $$H_{n}(V, V-\left\{{x}\right\})$$. The degree of a map $$S^n \to S^n$$ is the sum of its local degrees.

A generating $$S$$ for a module $$M$$ is linearly independent if $$\sum r_{i} s_{i} = 0_M \implies \forall i,~r_{i} = 0$$ where $$s_{i}\in S, r_{i} \in R$$.

$$H_{n}(X, X-A; {\mathbb{Z}})$$ is the local homology at $$A$$, also denoted $$H_{n}(X \mathrel{\Big|}A)$$

At a point $$x\in M^n$$, a choice of a generator $$\mu_{x}$$ of $$H_{n}(M, M - \left\{{x}\right\}) = {\mathbb{Z}}$$.

An $$n{\hbox{-}}$$manifold is a Hausdorff space in which each neighborhood has an open neighborhood homeomorphic to $${\mathbb{R}}^n$$.

A manifold in which open neighborhoods may be isomorphic to either $${\mathbb{R}}^n$$ or a half-space $$\left\{{\mathbf{x} \in {\mathbb{R}}^n \mathrel{\Big|}x_{i} > 0}\right\}$$.

A covering space is normal if and only if for every $$x\in X$$ and every pair of lifts $$\tilde x_1, \tilde x_2$$, there is a deck transformation $$f$$ such that $$f(\tilde x_1) = \tilde x_2$$.

A map $$X\xrightarrow{f} Y$$ is nullhomotopic if it is homotopic to a constant map $$X \xrightarrow{g} \left\{{y_{0}}\right\}$$; that is, there exists a homotopy \begin{align*} F: X\times I &\to Y \\ {\left.{{F}} \right|_{{X\times\left\{{0}\right\}}} } &= f \quad F(x, 0) = f(x) \\ {\left.{{F}} \right|_{{X\times\left\{{1}\right\}}} } &= g \quad F(x, 1) = g(x) = y_{0}\\ .\end{align*}

Alt:

If $$f$$ is homotopic to a constant map, say $$f: x \mapsto y_0$$ for some fixed $$y_0 \in Y$$, then $$f$$ is said to be nullhomotopic. In other words, if $$f:X\to Y$$ is nullhomotopic, then there exists a homotopy $$H: X\times I \to Y$$ such that $$H(x, 0) = f(x)$$ and $$H(x, 1) = y_0$$.

Note that constant maps (or anything homotopic) induce zero homomorphisms.

For a group action $$G\curvearrowright X$$, the orbit space $$X/G$$ is defined as $$X/\sim$$ where $$\forall x,y\in X, x\sim y \iff \exists g\in G \mathrel{\Big|}g.x = y$$.

A manifold for which an orientation exists, see “Orientation of a Manifold.”

For any manifold $$M$$, a two sheeted orientable covering space $$\tilde M_{o}$$. $$M$$ is orientable iff $$\tilde M$$ is disconnected. Constructed as \begin{align*} \tilde M = \coprod_{x\in M}\left\{{\mu_{x} \mathrel{\Big|}\mu_{x}~ \text{is a local orientation}}\right\} .\end{align*}

A family of $$\left\{{\mu_{x}}\right\}_{x\in M}$$ with local consistency: if $$x,y \in U$$ then $$\mu_{x}, \mu_{y}$$ are related via a propagation.

Formally, a function \begin{align*}M^n \to \coprod_{x\in M} H(X \mathrel{\Big|}\left\{{x}\right\})\\ x \mapsto \mu_{x}\end{align*} such that $$\forall x \exists N_{x}$$ in which $$\forall y\in N_{x}$$, the preimage of each $$\mu_{y}$$ under the map $$H_{n}(M\mathrel{\Big|}N_{x}) \twoheadrightarrow H_{n}(M\mathrel{\Big|}y)$$ is a single generator $$\mu_{N_{x}}$$.

TFAE:

• $$M$$ is orientable.
• The map $$W: (M, x) \to {\mathbb{Z}}_{2}$$ is trivial.
• $$\tilde M_{o} = M \coprod {\mathbb{Z}}_{2}$$ (two sheets).
• $$\tilde M_{o}$$ is disconnected
• The projection $$\tilde M_{o} \to M$$ admits a section.

A pairing alone is an $$R{\hbox{-}}$$bilinear module map, or equivalently a map out of a tensor product since $$p: M\otimes_{R} N \to L$$ can be partially applied to yield $$\phi: M \to L^N = \hom_{R}(N, L)$$. A pairing is perfect when $$\phi$$ is an isomorphism.

For a closed, orientable $$n{\hbox{-}}$$manifold, following map $$[M] \frown {\,\cdot\,}$$ is an isomorphism: \begin{align*} D: H^k(M; R) \to H_{n-k}(M; R) \\ D(\alpha) = [M] \frown \alpha\end{align*}

A space $$X$$ is semilocally simply connected if every $$x\in X$$ has a neighborhood $$U$$ such that $$U\hookrightarrow X$$ induces the trivial map $$\pi_1(U;x) \to \pi_1(X, x)$$.

Given a simplex $$\sigma = [v_1 \cdots v_n]$$, define the face map \begin{align*} {\partial}_i:\Delta^n &\to \Delta^{n-1} \\ \sigma &\mapsto [v_1 \cdots \widehat{v}_i \cdots v_n] \end{align*}

A simplicial complex is a set $$K$$ satisfying

1. $$\sigma \in K \implies {\partial}_i\sigma \in K$$.

2. $$\sigma,\tau\in K \implies \sigma\cap\tau = \emptyset,~ {\partial}_i\sigma,~\text{or}~{\partial}_i\tau$$.

This amounts to saying that any collection of $$(n-1)$$-simplices uniquely determines an $$n$$-simplex (or its lack thereof), or that that map $$\Delta^k \to X$$ is a continuous injection from the standard simplex in $${\mathbb{R}}^n$$.

1. $${\left\lvert {K\cap B_\varepsilon(\sigma)} \right\rvert} < \infty$$ for every $$\sigma\in K$$, identifying $$\sigma \subseteq {\mathbb{R}}^n$$.

For a map \begin{align*}K\xrightarrow{f} L\end{align*} between simplicial complexes, $$f$$ is a simplicial map if for any set of vertices $$\left\{{v_{i}}\right\}$$ spanning a simplex in $$K$$, the set $$\left\{{f(v_{i})}\right\}$$ are the vertices of a simplex in $$L$$.

A space $$X$$ is simply connected if and only if $$X$$ is path-connected and every loop $$\gamma : S^1 \xrightarrow{} X$$ can be contracted to a point.

Equivalently, there exists a lift $$\widehat{\gamma }: D^2 \xrightarrow{} X$$ such that $${ \left.{{\widehat{\gamma}}} \right|_{{{{\partial}}D^2}} } = \gamma$$.

Equivalently, for any two paths $$p_1, p_2: I \xrightarrow{} X$$ such that $$p_1(0) = p_2(0)$$ and $$p_1(1) = p_2(1)$$, there exists a homotopy $$F: I^2 \xrightarrow{} X$$ such that $${ \left.{{F}} \right|_{{0}} } = p_1,\, { \left.{{F}} \right|_{{0}} } = p_2$$.

Equivalently, $$\pi _1 X = 1$$ is trivial.

\begin{align*}x \in C_{n}(x) \implies X = \sum_{i} n_{i} \sigma_{i} = \sum_{i} n_{i} (\Delta^n \xrightarrow{\sigma_{i}} X) .\end{align*}

\begin{align*}x \in C^n(x) \implies X = \sum_{i} n_{i} \psi_{i} = \sum_{i} n_{i} (\sigma_{i} \xrightarrow{\psi_{i}} X) .\end{align*}

Compact represented as $$\Sigma X = CX \coprod_{\operatorname{id}_{X}} CX$$, two cones on $$X$$ glued along $$X$$. Explicitly given by

\begin{align*}\Sigma X = \frac{X\times I}{(X\times\left\{{0}\right\}) \cup(X\times\left\{{1}\right\}) \cup(\left\{{x_{0}}\right\} \times I)} .\end{align*}

For an $$R{\hbox{-}}$$module \begin{align*} \operatorname{Tor}_{R}^n({\,\cdot\,}, B) = L_{n}({\,\cdot\,}\otimes_{R} B) ,\end{align*} where $$L_{n}$$ denotes the $$n$$th left derived functor.

# 4 Examples

## 4.1 Point-Set

### 4.1.1 Common Spaces and Operations

The following are some standard “nice” spaces: \begin{align*} S^n, {\mathbb{D}}^n, T^n, {\mathbb{RP}}^n, {\mathbb{CP}}^n, \mathbb{M}, \mathbb{K}, \Sigma_{g}, {\mathbb{RP}}^\infty, {\mathbb{CP}}^\infty .\end{align*}

The following are useful spaces to keep in mind to furnish counterexamples:

• Finite discrete sets with the discrete topology.
• Subspaces of $${\mathbb{R}}$$: $$(a, b), (a, b], (a, \infty)$$, etc.
• Sets given by real sequences, such as $$\left\{{0}\right\} \cup\left\{{{1 \over n}{~\mathrel{\Big|}~}n\in {\mathbb{Z}}^{\geq 1}}\right\}$$
• $${\mathbb{Q}}$$
• The topologist’s sine curve
• One-point compactifications
• $${\mathbb{R}}^\omega$$ for $$\omega$$ the least uncountable ordinal (?)
• The Hawaiian earring
• The Cantor set

Examples of some more exotic spaces that show up less frequently:

• $${\mathbb{HP}}^n$$, quaternionic projective space
• The Dunce Cap
• The Alexander Horned sphere

The following spaces are non-Hausdorff:

• The cofinite topology on any infinite set.
• $${\mathbb{R}}/{\mathbb{Q}}$$
• The line with two origins.
• Any variety $$V(J) \subseteq {\mathbb{A}}^n_{/k}$$ for $$k$$ a field and $$J{~\trianglelefteq~}k[x_1, \cdots, x_{n}]$$.

The following are some examples of ways to construct specific spaces for examples or counterexamples:

• Knot complements in $$S^3$$

• Covering spaces (hyperbolic geometry)

• Lens spaces

• Matrix groups

• Prism spaces

• Pair of pants

• Seifert surfaces

• Surgery

• Simplicial Complexes

• Nice minimal example:

Operations

• Cartesian product $$A\times B$$
• Wedge product $$A \vee B$$
• Connect Sum $$A \# B$$
• Quotienting $$A/B$$
• Puncturing $$A\setminus \left\{{a_{i}}\right\}$$
• Smash product
• Join
• Cones
• Suspension
• Loop space
• Identifying a finite number of points

### 4.1.2 Alternative Topologies

The following are some nice examples of topologies to put on familiar spaces to produce counterexamples:

• Discrete
• Cofinite
• Discrete and Indiscrete
• Uniform

The cofinite topology on any space $$X$$ is always

• Non-Hausdorff
• Compact

A topology $$(X, \tau)$$ is the discrete topology iff points $$x\in X$$ are open.

(Click to expand)

If $$\left\{{x}\right\}_i$$ is open for each $$x_i \in X$$, then

• Any set $$U$$ can be written as $$U = \cup_{i\in I} x_I$$ (for some $$I$$ depending on $$U$$), and
• Unions of open sets are open.

Thus $$U$$ is open.

Some facts about the discrete topology:

• Definition: every subset is open.
• Always Hausdorff
• Compact iff finite
• Totally disconnected
• If $$X$$ is discrete, every map $$f:X\to Y$$ for any $$Y$$ is continuous (obvious!)

Some facts about the indiscrete topology:

• Definition: the only open sets are $$\emptyset, X$$
• Never Hausdorff
• If $$Y$$ is indiscrete, every map $$f:X\to Y$$ is continuous (obvious!)
• Always compact

# 5 Theorems

The following properties are “pushed forward” through continuous maps, in the sense that if property $$P$$ holds for $$X$$ and $$f:X\to Y$$, then $$f(X)$$ also satisfies $$P$$:

• Compactness
• Separability
• If $$f$$ is surjective:
• Connectedness
• Density

The following are not preserved:

• Openness
• Closedness

See more here.

## 5.1 Metric Spaces and Analysis

A bounded collection of nested closed sets $$C_1 \supset C_2 \supset \cdots$$ in a metric space $$X$$ is nonempty $$\iff X$$ is complete.

If $$\left\{{[a_n, b_n] {~\mathrel{\Big|}~}n\in {\mathbb{Z}}^{\geq 0}}\right\}$$ is a nested sequence of closed and bounded intervals, then their intersection is nonempty. If $${\operatorname{diam}}([a_n, b_n]) \overset{n\to\infty}0$$, then the intersection contains exactly one point.

A continuous function on a compact set is uniformly continuous.

(Click to expand)

Take $$\left\{{B_{\varepsilon\over 2}(y) {~\mathrel{\Big|}~}y\in Y}\right\}\rightrightarrows Y$$, pull back to an open cover of $$X$$, has Lebesgue number $$\delta_L > 0$$, then $$x' \in B_{\delta_L}(x) \implies f(x), f(x') \in B_{\varepsilon\over 2}(y)$$ for some $$y$$.

Lipschitz continuity implies uniform continuity (take $$\delta = \varepsilon/C$$)

Counterexample to the converse: $$f(x) = \sqrt x$$ on $$[0, 1]$$ has unbounded derivative.

For $$f:X \to Y$$ continuous with $$X$$ compact and $$Y$$ ordered in the order topology, there exist points $$c, d\in X$$ such that $$f(x) \in [f(c), f(d)]$$ for every $$x$$.

A metric space $$X$$ is sequentially compact iff it is complete and totally bounded.

A metric space is totally bounded iff every sequence has a Cauchy subsequence.

A metric space is compact iff it is complete and totally bounded.

If $$X$$ is a complete metric space, $$X$$ is a Baire space: the intersection of countably many dense open sets in $$X$$ is again dense in $$X$$.

## 5.2 Compactness

$$U\subset X$$ a Hausdorff spaces is closed $$\iff$$ it is compact.

A closed subset $$A$$ of a compact set $$B$$ is compact.

(Click to expand)
• Let $$\left\{{A_i}\right\} \rightrightarrows A$$ be a covering of $$A$$ by sets open in $$A$$.
• Each $$A_i = B_i \cap A$$ for some $$B_i$$ open in $$B$$ (definition of subspace topology)
• Define $$V = \left\{{B_i}\right\}$$, then $$V \rightrightarrows A$$ is an open cover.
• Since $$A$$ is closed, $$W\mathrel{\vcenter{:}}= B\setminus A$$ is open
• Then $$V\cup W$$ is an open cover of $$B$$, and has a finite subcover $$\left\{{V_i}\right\}$$
• Then $$\left\{{V_i \cap A}\right\}$$ is a finite open cover of $$A$$.

The continuous image of a compact set is compact.

A closed subset of a Hausdorff space is compact.

## 5.3 Separability

A retract of a Hausdorff/connected/compact space is closed/connected/compact respectively.

Points are closed in $$T_1$$ spaces.

## 5.4 Maps and Homeomorphism

A continuous bijection $$f: X\to Y$$ where $$X$$ is compact and $$Y$$ is Hausdorff is an open map and hence a homeomorphism.

Every space has at least one retraction - for example, the constant map $$r:X \to\left\{{x_0}\right\}$$ for any $$x\_0 \in X$$.

A continuous bijective open map is a homeomorphism.

For $$f:X\to Y$$, TFAE:

• $$f$$ is continuous
• $$A\subset X \implies f(\operatorname{cl}_X(A)) \subset \operatorname{cl}_X(f(A))$$
• $$B$$ closed in $$Y \implies f^{-1}(B)$$ closed in $$X$$.
• For each $$x\in X$$ and each neighborhood $$V \ni f(x)$$, there is a neighborhood $$U\ni x$$ such that $$f(U) \subset V$$.
(Click to expand)

See Munkres page 104.

If $$f:X\to Y$$ is continuous where $$X$$ is compact and $$Y$$ is Hausdorff, then

• $$f$$ is a closed map.
• If $$f$$ is surjective, $$f$$ is a quotient map.
• If $$f$$ is injective, $$f$$ is a topological embedding.
• If $$f$$ is bijective, it is a homeomorphism.

## 5.5 The Tube Lemma

Let $$X, Y$$ be spaces with $$Y$$ compact. For each $$U \subseteq X \times Y$$ and each slice $$\left\{{x}\right\} \times Y \subseteq U$$, there is an open $$O \subseteq X$$ such that \begin{align*} \left\{{x}\right\} \times Y \subseteq O \times Y \subseteq U .\end{align*}

(Click to expand)
• For each $$y\in Y$$ choose neighborhoods $$A_y, B_y \subseteq Y$$ such that \begin{align*} (x, y) \in A_y \times B_y \subseteq U .\end{align*}
• By compactness of $$Y$$, reduce this to finitely many $$B_y \rightrightarrows Y$$ so $$Y = \bigcup_{j=1}^n B_{y_j}$$
• Set $$O\mathrel{\vcenter{:}}=\cap_{j=1}^n B_{y_j}$$; this works.

# 6 Summary of Standard Topics

• Algebraic topology topics:
• Classification of compact surfaces
• Euler characteristic
• Connect sum
• Homology and cohomology groups
• Fundamental group
• Singular/cellular/simplicial homology
• Mayer-Vietoris long exact sequences for homology and cohomology
• Diagram chasing
• Degree of maps from $$S^n \to S^n$$
• Orientability, compactness
• Top-level homology and cohomology
• Reduced homology and cohomology
• Relative homology
• Homotopy and homotopy invariance
• Deformation retract
• Retract
• Excision
• Kunneth formula
• Factoring maps
• Fundamental theorem of algebra
• Algebraic topology theorems:
• Brouwer fixed point theorem
• Poincaré lemma
• Poincaré duality
• de Rham theorem
• Seifert-van Kampen theorem
• Covering space theory topics:
• Covering maps
• Free actions
• Properly discontinuous action
• Universal cover
• Correspondence between covering spaces and subgroups of the fundamental group of the base.
• Lifting paths
• Homotopy lifting property
• Deck transformations
• The action of the fundamental group
• Normal/regular cover

# 7 Examples: Algebraic Topology

## 7.1 Standard Spaces and Modifications

\begin{align*} {\mathbb{D}}^n = \mathbb{B}^n &\mathrel{\vcenter{:}}=\left\{{ \mathbf{x} \in {\mathbb{R}}^{n} {~\mathrel{\Big|}~}{\left\lVert {\mathbf{x}} \right\rVert} \leq 1}\right\} {\mathbb{S}}^n &\mathrel{\vcenter{:}}=\left\{{ \mathbf{x} \in {\mathbb{R}}^{n+1} {~\mathrel{\Big|}~}{\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.
• $$S^n \cong D^n \displaystyle\coprod_{{{\partial}}D^n} D^n$$: 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 $$n+1$$.

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 $${\mathbb{R}}^{n+1}$$.

One can also define $${\mathbb{RP}}^ \infty \mathrel{\vcenter{:}}=\varinjlim_{n} {\mathbb{RP}}^n$$. Fits into a fiber bundle of the form

Defined in a similar ways,

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

Can similarly define $${\mathbb{CP}}^ \infty \mathrel{\vcenter{:}}=\varinjlim_n {\mathbb{CP}}^n$$. Fits into a fiber bundle of the form

The $$n{\hbox{-}}$$torus, defined as \begin{align*} T^n \mathrel{\vcenter{:}}=\prod_{j=1}^n S^1 = S^1 \times S^1 \times \cdots .\end{align*}

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

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

Lie Groups:

• The general linear group, $$\operatorname{GL}_{n}({\mathbb{R}})$$
• The special linear group $$SL_{n}({\mathbb{R}})$$
• The orthogonal group, $$O_{n}({\mathbb{R}})$$
• The special orthogonal group, $$SO_{n}({\mathbb{R}})$$
• The real unitary group, $$U_{n}({\mathbb{C}})$$
• The special unitary group, $$SU_{n}({\mathbb{R}})$$
• 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 $${\mathbb{A}}^n(k) = k^n \rtimes GL_{n}(k)$$
• The projective space $${\mathbb{P}}^n(k)$$
• 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 $$PSL_{n}({\mathbb{Z}})$$
• Specifically $$PSL_{2}({\mathbb{Z}})$$

$$K(G, n)$$ 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:

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

$$M(G, n)$$ 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:

• $$M({\mathbb{Z}}, n) = S^n$$
• $$M({\mathbb{Z}}/2{\mathbb{Z}}, 1) = {\mathbb{RP}}^2$$
• $$M({\mathbb{Z}}/p{\mathbb{Z}}, n)$$ 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.
• $${\mathbb{CP}}^n = {\mathbb{C}}^n \coprod {\mathbb{CP}}^{n-1} = \coprod_{i=0}^n {\mathbb{C}}^i$$
• $${\mathbb{CP}}^n = S^{2n+1} / S^n$$
• $$S^n / S^k \simeq S^n \vee \Sigma S^k$$.

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

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

## 7.2 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

# 8 Low Dimensional Homology Examples

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

# 9 Table of Homotopy and Homology Structures

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

$$X$$ $$\pi_*(X)$$ $$H_*(X)$$ CW Structure $$H^*(X)$$
$${\mathbb{R}}^1$$ $$0$$ $$0$$ $${\mathbb{Z}}\cdot 1 + {\mathbb{Z}}\cdot x$$ 0
$${\mathbb{R}}^n$$ $$0$$ $$0$$ $$({\mathbb{Z}}\cdot 1 + {\mathbb{Z}}\cdot x)^n$$ 0
$$D(k, {\mathbb{R}}^n)$$ $$\pi_*\bigvee^k S^1$$ $$\bigoplus_{k} H_* M({\mathbb{Z}}, 1)$$ $$1 + kx$$ ?
$$B^n$$ $$\pi_*({\mathbb{R}}^n)$$ $$H_*({\mathbb{R}}^n)$$ $$1 + x^n + x^{n+1}$$ 0
$$S^n$$ $$[0 \ldots , {\mathbb{Z}}, ? \ldots]$$ $$H_*M({\mathbb{Z}}, n)$$ $$1 + x^n$$ or $$\sum_{i=0}^n 2x^i$$ $${\mathbb{Z}}[{}_{n}x]/(x^2)$$
$$D(k, S^n)$$ $$\pi_*\bigvee^{k-1}S^1$$ $$\bigoplus_{k-1}H_*M({\mathbb{Z}}, 1)$$ $$1 + (k-1)x^1$$ ?
$$T^2$$ $$\pi_*S^1 \times \pi_* S^1$$ $$(H_* M({\mathbb{Z}}, 1))^2 \times H_* M({\mathbb{Z}}, 2)$$ $$1 + 2x + x^2$$ $$\Lambda({}_{1}x_{1}, {}_{1}x_{2})$$
$$T^n$$ $$\prod^n \pi_* S^1$$ $$\prod_{i=1}^n (H_* M({\mathbb{Z}}, i))^{n\choose i}$$ $$(1 + x)^n$$ $$\Lambda({}_{1}x_{1}, {}_{1}x_{2}, \ldots {}_{1}x_{n})$$
$$D(k, T^n)$$ $$[0, 0, 0, 0, \ldots]$$? $$[0, 0, 0, 0, \ldots]$$? $$1 + x$$ ?
$$S^1 \vee S^1$$ $$\pi_*S^1 \ast \pi_* S^1$$ $$(H_*M({\mathbb{Z}}, 1))^2$$ $$1 + 2x$$ ?
$$\bigvee^n S^1$$ $$\ast^n \pi_* S^1$$ $$\prod H_* M({\mathbb{Z}}, 1)$$ $$1 + x$$ ?
$${\mathbb{RP}}^1$$ $$\pi_* S^1$$ $$H_* M({\mathbb{Z}}, 1)$$ $$1 + x$$ $${}_{0}{\mathbb{Z}}\times {}_{1}{\mathbb{Z}}$$
$${\mathbb{RP}}^2$$ $$\pi_*K({\mathbb{Z}}/2{\mathbb{Z}}, 1)+ \pi_* S^2$$ $$H_*M({\mathbb{Z}}/2{\mathbb{Z}}, 1)$$ $$1 + x + x^2$$ $${}_{0}{\mathbb{Z}}\times {}_{2}{\mathbb{Z}}/2{\mathbb{Z}}$$
$${\mathbb{RP}}^3$$ $$\pi_*K({\mathbb{Z}}/2{\mathbb{Z}}, 1)+ \pi_* S^3$$ $$H_*M({\mathbb{Z}}/2{\mathbb{Z}}, 1) + H_*M({\mathbb{Z}}, 3)$$ $$1 + x + x^2 + x^3$$ $${}_{0}{\mathbb{Z}}\times {}_{2}{\mathbb{Z}}/2{\mathbb{Z}}\times {}_{3}{\mathbb{Z}}$$
$${\mathbb{RP}}^4$$ $$\pi_*K({\mathbb{Z}}/2{\mathbb{Z}}, 1)+ \pi_* S^4$$ $$H_*M({\mathbb{Z}}/2{\mathbb{Z}}, 1) + H_*M({\mathbb{Z}}/2{\mathbb{Z}}, 3)$$ $$1 + x + x^2 + x^3 + x^4$$ $${}_{0}{\mathbb{Z}}\times ({}_{2}{\mathbb{Z}}/2{\mathbb{Z}})^2$$
$${\mathbb{RP}}^n, n \geq 4$$ even $$\pi_*K({\mathbb{Z}}/2{\mathbb{Z}}, 1)+ \pi_*S^n$$ $$\prod_{\text{odd}~i < n} H_*M({\mathbb{Z}}/2{\mathbb{Z}}, i)$$ $$\sum_{i=1}^n x^i$$ $${}_{0}{\mathbb{Z}}\times \prod_{i=1}^{n/2}{}_{2}{\mathbb{Z}}/2{\mathbb{Z}}$$
$${\mathbb{RP}}^n, n \geq 4$$ odd $$\pi_*K({\mathbb{Z}}/2{\mathbb{Z}}, 1)+ \pi_*S^n$$ $$\prod_{\text{odd}~ i \leq n-2} H_*M({\mathbb{Z}}/2{\mathbb{Z}}, i) \times H_* S^n$$ $$\sum_{i=1}^n x^i$$ $$H^*({\mathbb{RP}}^{n-1}) \times {}_{n}{\mathbb{Z}}$$
$${\mathbb{CP}}^1$$ $$\pi_*K({\mathbb{Z}}, 2) + \pi_* S^3$$ $$H_* S^2$$ $$x^0 + x^2$$ $${\mathbb{Z}}[{}_{2}x]/({}_2x^{2})$$
$${\mathbb{CP}}^2$$ $$\pi_*K({\mathbb{Z}}, 2) + \pi_* S^5$$ $$H_*S^2 \times H_* S^4$$ $$x^0 + x^2 + x^4$$ $${\mathbb{Z}}[{}_{2}x]/({}_2x^{3})$$
$${\mathbb{CP}}^n, n \geq 2$$ $$\pi_*K({\mathbb{Z}}, 2) + \pi_*S^{2n+1}$$ $$\prod_{i=1}^n H_* S^{2i}$$ $$\sum_{i=1}^n x^{2i}$$ $${\mathbb{Z}}[{}_{2}x]/({}_2x^{n+1})$$
Mobius Band $$\pi_* S^1$$ $$H_* S^1$$ $$1 + x$$ ?
Klein Bottle $$K({\mathbb{Z}}\rtimes_{-1} {\mathbb{Z}}, 1)$$ $$H_*S^1 \times H_* {\mathbb{RP}}^\infty$$ $$1 + 2x + x^2$$ ?
• $${\mathbb{R}}^n$$ is a contractible space, and so $$[S^m, {\mathbb{R}}^n] = 0$$ for all $$n, m$$ which makes its homotopy groups all zero.

• $$D(k, {\mathbb{R}}^n) = {\mathbb{R}}^n - \left\{{x_{1} \ldots x_{k}}\right\} \simeq\bigvee_{i=1}^k S^1$$ 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*}

• $$B^n \simeq{\mathbb{R}}^n$$ by normalizing vectors.

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

• $${\mathbb{CP}}^1 \cong S^2$$.

• $${\mathbb{RP}}^1 \cong S^1$$.

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

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

• Use $${\mathbb{CP}}^n = S^{2n+1} / S^1$$

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

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

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

# 10 Theorems: Algebraic Topology

## 10.1 General Homotopies

\begin{align*} X\times{\mathbb{R}}^n \simeq X \times{\{\operatorname{pt}\}}\cong X .\end{align*}

The ranks of $$\pi_{0}$$ and $$H_{0}$$ are the number of path components.

Any two continuous functions into a convex set are homotopic.

(Click to expand)

The linear homotopy. Supposing $$X$$ is convex, for any two points $$x,y\in X$$, the line $$tx + (1-t)y$$ is contained in $$X$$ for every $$t\in[0,1]$$. So let $$f, g: Z \to X$$ be any continuous functions into $$X$$. Then define $$H: Z \times I \to X$$ by $$H(z,t) = tf(z) + (1-t)g(z)$$, the linear homotopy between $$f,g$$. By convexity, the image is contained in $$X$$ for every $$t,z$$, so this is a homotopy between $$f,g$$.

## 10.2 Fundamental Group

### 10.2.1 Definition

Given a pointed space $$(X,x_{0})$$, we define the fundamental group $$\pi_{1}(X)$$ as follows:

• Take the set \begin{align*} L \mathrel{\vcenter{:}}=\left\{{\alpha: S^1\to X \mathrel{\Big|}\alpha(0) = \alpha(1) = x_{0}}\right\} .\end{align*}

• Define an equivalence relation $$\alpha \sim \beta$$ iff $$\alpha \simeq\beta$$ in $$X$$, so there exists a homotopy

\begin{align*} H: &S^1 \times I \to X \\ & \begin{cases} H(s, 0) = \alpha(s)\\ H(s, 1) = \beta(s) , \end{cases} \end{align*} - Check that this relation is

• Symmetric: Follows from considering $$H(s, 1-t)$$.

• Reflexive: Take $$H(s, t) = \alpha (s)$$ for all $$t$$.

• Transitive: Follows from reparameterizing.

• Define $$L/\sim$$, which contains elements like $$[\alpha]$$ and $$[\operatorname{id}_{x_{0}}]$$, the equivalence classes of loops after quotienting by this relation.

• Define a product structure: for $$[\alpha], [\beta] \in L/\sim$$, define $$[\alpha][\beta] = [\alpha \cdot \beta]$$, where we just need to define a product structure on actual loops. Do this by reparameterizing: \begin{align*} (\alpha \cdot \beta )(s) \mathrel{\vcenter{:}}= \begin{cases} \alpha (2s) & s \in [0, 1/2] \\ \beta (2s-1) & s \in [1/2, 1] . \end{cases} \end{align*}

• Check that this map is:

• Continuous: by the pasting lemma and assumed continuity of $$f, g$$.

• Well-defined: ?

• Check that this is actually a group

• Identity element: The constant loop $$\operatorname{id}_{x_0}: I\to X$$ where $$\operatorname{id}_{x_0}(t) = x_0$$ for all $$t$$.

• Inverses: The reverse loop $$\mkern 1.5mu\overline{\mkern-1.5mu\alpha\mkern-1.5mu}\mkern 1.5mu(t) \mathrel{\vcenter{:}}=\alpha(1-t)$$.

• Closure: Follows from the fact that start/end points match after composing loops, and reparameterizing.

• Associativity: Follows from reparameterizing.

Elements of the fundamental group are homotopy classes of loops, and every continuous map between spaces induces a homomorphism on fundamental groups.

### 10.2.2 Conjugacy in $$\pi_{1}$$:

• See Hatcher 1.19, p.28
• See Hatcher’s proof that $$\pi_{1}$$ is a group
• See change of basepoint map

### 10.2.3 Calculating $$\pi_1$$

If $$\tilde X \to X$$ the universal cover of $$X$$ and $$G\curvearrowright\tilde X$$ with $$\tilde X/G = X$$ then $$\pi_1(X) = G$$.

$$\pi_1 X$$ for $$X$$ a CW-complex only depends on the 2-skeleton $$X^{2}$$, and in general $$\pi_k(X)$$ only depends on the $$k+2$$-skeleton. Thus attaching $$k+2$$ or higher cells does not change $$\pi_k$$.

Suppose $$X = U_{1} \cup U_{2}$$ where $$U_1, U_2$$, and $$U \mathrel{\vcenter{:}}= U_{1} \cap U_{2} \neq \emptyset$$ are open and path-connected2

, and let $$x_0 \in U$$.

Then the inclusion maps $$i_{1}: U_{1} \hookrightarrow X$$ and $$i_{2}: U_{2} \hookrightarrow X$$ induce the following group homomorphisms: \begin{align*} i_{1}^*: \pi_{1}(U_{1}, x_0) \to\pi_{1}(X, x_0) \\ i_{2}^*: \pi_{1}(U_{2}, x_0) \to\pi_{1}(X, x_0) \end{align*}

There is a natural isomorphism \begin{align*} \pi_{1}(X) \cong \pi_{1} U \ast_{\pi_{1}(U \cap V)} \pi_{1} V ,\end{align*}

where the amalgamated product can be computed as follows: A pushout is the colimit of the following diagram

For groups, the pushout is realized by the amalgamated free product: if \begin{align*} \begin{cases} \pi_1 U_1 = A = \left\langle{G_{A} {~\mathrel{\Big|}~}R_{A}}\right\rangle \\ \pi_1 U_2 = B = \left\langle{G_{B} {~\mathrel{\Big|}~}R_{B}}\right\rangle \end{cases} \implies A \ast_{Z} B \mathrel{\vcenter{:}}=\left\langle{ G_{A}, G_{B} {~\mathrel{\Big|}~}R_{A}, R_{B}, T}\right\rangle \end{align*} where $$T$$ is a set of relations given by \begin{align*} T = \left\{{\iota_{1}^*(z) \iota_{2}^* (z) ^{-1} {~\mathrel{\Big|}~}z\in \pi_1 (U_1 \cap U_2)}\right\} ,\end{align*} where $$\iota_2^*(z) ^{-1}$$ denotes the inverse group element. If we have presentations

\begin{align*} \pi_{1}(U, x_0) &= \left\langle u_{1}, \cdots, u_{k} {~\mathrel{\Big|}~}\alpha_{1}, \cdots, \alpha_{l}\right\rangle \\ \pi_{1}(V, w) &=\left\langle v_{1}, \cdots, v_{m} {~\mathrel{\Big|}~}\beta_{1}, \cdots, \beta_{n}\right\rangle \\ \pi_{1}(U \cap V, x_0) &=\left\langle w_{1}, \cdots, w_{p} {~\mathrel{\Big|}~}\gamma_{1}, \cdots, \gamma_{q}\right\rangle \end{align*}

then \begin{align*} \pi_{1}(X, w) &= \left\langle u_{1}, \cdots, u_{k}, v_{1}, \cdots, v_{m} \middle\vert \begin{cases} \alpha_{1}, \cdots, \alpha_{l} \\ \beta_{1}, \cdots, \beta_{n} \\ I\left(w_{1}\right) J\left(w_{1}\right)^{-1}, \cdots, I\left(w_{p}\right) J\left(w_{p}\right)^{-1} \\ \end{cases} \right\rangle \\ \\ &= \frac{ \pi_{1}(U_1) \ast \pi_{1}(U_2) } { \left\langle{ \left\{{\iota_1^*(w_{i}) \iota_2^*(w_{i})^{-1}{~\mathrel{\Big|}~}1\leq i \leq p}\right\} }\right\rangle } \end{align*}

(Click to expand)
• Construct a map going backwards
• Show it is surjective
• “There and back” paths
• Show it is injective
• Divide $$I\times I$$ into a grid

$$A = {\mathbb{Z}}/4{\mathbb{Z}}= \left\langle{x {~\mathrel{\Big|}~}x^4}\right\rangle, B = {\mathbb{Z}}/6{\mathbb{Z}}= \left\langle{y {~\mathrel{\Big|}~}x^6}\right\rangle, Z = {\mathbb{Z}}/2{\mathbb{Z}}= \left\langle{z {~\mathrel{\Big|}~}z^2}\right\rangle$$. Then we can identify $$Z$$ as a subgroup of $$A, B$$ using $$\iota_{A}(z) = x^2$$ and $$\iota_{B}(z) = y^3$$. So \begin{align*}A\ast_{Z} B = \left\langle{x, y {~\mathrel{\Big|}~}x^4, y^6, x^2y^{-3}}\right\rangle\end{align*} .

\begin{align*} \pi_1(X \vee Y) = \pi_1(X) \ast \pi_1(Y) .\end{align*}

(Click to expand)

By van Kampen, this is equivalent to the amalgamated product over $$\pi_1(x_0) = 1$$, which is just a free product.

### 10.2.4 Facts

$$H_{1}$$ is the abelianization of $$\pi_{1}$$.

If $$X, Y$$ are path-connected, then \begin{align*} \pi_1 (X \times Y) = \pi_1(X) \times\pi_2(Y) .\end{align*}

(Click to expand)
• A loop in $$X \times Y$$ is a continuous map $$\gamma : I \xrightarrow{} X \times Y$$ given by $$\gamma (t) = (f(t), g(t)$$ in components.
• $$\gamma$$ being continuous in the product topology is equivalent to $$f, g$$ being continuous maps to $$X, Y$$ respectively.
• Similarly a homotopy $$F: I^2 \to X \times Y$$ is equivalent to a pair of homotopies $$f_t, g_t$$ of the corresponding loops.
• So the map $$[ \gamma ] \mapsto ([f], [g])$$ is the desired bijection.

$$\pi_{1}(X) = 1$$ iff $$X$$ is simply connected.

(Click to expand)

$$\Rightarrow$$: Suppose $$X$$ is simply connected. Then every loop in $$X$$ contracts to a point, so if $$\alpha$$ is a loop in $$X$$, $$[\alpha] = [\operatorname{id}_{x_{0}}]$$, the identity element of $$\pi_{1}(X)$$. But then there is only one element in in this group.

$$\Leftarrow$$: Suppose $$\pi_{1}(X) = 0$$. Then there is just one element in the fundamental group, the identity element, so if $$\alpha$$ is a loop in $$X$$ then $$[\alpha] = [\operatorname{id}_{x_{0}}]$$. So there is a homotopy taking $$\alpha$$ to the constant map, which is a contraction of $$\alpha$$ to a point.

:::{.fact “Unsorted facts”}

• For a graph $$G$$, we always have $$\pi_{1}(G) \cong {\mathbb{Z}}^n$$ where $$n = |E(G - T)|$$, the complement of the set of edges in any maximal tree. Equivalently, $$n = 1-\chi(G)$$. Moreover, $$X \simeq\bigvee^n S^1$$ in this case.

:::

## 10.3 General Homotopy Theory

A map $$X \xrightarrow{f} Y$$ on CW complexes that is a weak homotopy equivalence (inducing isomorphisms in homotopy) is in fact a homotopy equivalence.

Individual maps may not work: take $$S^2 \times{\mathbb{RP}}^3$$ and $$S^3 \times{\mathbb{RP}}^2$$ which have isomorphic homotopy but not homology.

The Hurewicz map on an $$n-1{\hbox{-}}$$connected space $$X$$ is an isomorphism $$\pi_{k\leq n}X \to H_{k\leq n} X$$.

I.e. for the minimal $$i\geq 2$$ for which $$\pi_{iX} \neq 0$$ but $$\pi_{\leq i-1}X = 0$$, $$\pi_{iX} \cong H_{iX}$$.

Any continuous map between CW complexes is homotopy equivalent to a cellular map.

• $$\pi_{k\leq n}S^n = 0$$
• $$\pi_{n}(X) \cong \pi_{n}(X^{(n)})$$

:::{.fact title="Unsorted facts about higher homotopy groups}

• $$\pi_{i\geq 2}(X)$$ is always abelian.

• $$X$$ simply connected $$\implies \pi_{k}(X) \cong H_{k}(X)$$ up to and including the first nonvanishing $$H_{k}$$
• $$\pi_{k} \bigvee X \neq \prod \pi_{k} X$$ (counterexample: $$S^1 \vee S^2$$)

• Nice case: $$\pi_{1}\bigvee X = \ast \pi_{1} X$$ by Van Kampen.
• $$\pi_{i}(\widehat{X}) \cong \pi_{i}(X)$$ for $$i\geq 2$$ whenever $$\widehat{X} \twoheadrightarrow X$$ is a universal cover.

• $$\pi_{i}(S^n) = 0$$ for $$i < n$$, $$\pi_{n}(S^n) = {\mathbb{Z}}$$

• Not necessarily true that $$\pi_{i}(S^n) = 0$$ when $$i > n$$!!!
• E.g. $$\pi_{3}(S^2) = {\mathbb{Z}}$$ by Hopf fibration
• $$S^n / S^k \simeq S^n \vee \Sigma S^{k}$$

• $$\Sigma S^n = S^{n+1}$$
• General mantra: homotopy plays nicely with products, homology with wedge products.3

• $$\pi_{k}\prod X = \prod \pi_{k} X$$ by LES.4

• In general, homotopy groups behave nicely under homotopy pull-backs (e.g., fibrations and products), but not homotopy push-outs (e.g., cofibrations and wedges). Homology is the opposite.

• Constructing a $$K(\pi, 1)$$: since $$\pi = \left< S \mathrel{\Big|}R\right> = F(S)/R$$, take $$\bigvee^{|S|} S^1 \cup_{|R|} e^2$$. In English, wedge a circle for each generator and attach spheres for relations.

:::

# 11 Covering Spaces

Some pictures to keep in mind when it comes to covers and path lifting: