# Definitions

## Point-Set Topology

The prefix “locally blah” almost always means that for every $$x\in X$$, there exists some neighborhood $$N_x\ni x$$ which has property “blah”.

See limit point.

A set $$\mathcal{B}$$ is a basis for a topology iff

• $$\mathcal{B}$$ is closed under intersections,
• Every $$x\in X$$ is in some basic set,
• If $$x$$ is in the intersection of two basis sets $$B_1 \cap B_2$$, there is a third basic open $$B_3 \ni x$$ with $$B_3 \subset B_1 \cap B_2$$.

The topology generated by $$\mathcal{B}$$ is the following: $$U\subseteq X$$ is open iff for each $$x\in U$$ there is a basic open $$B$$ with $$x\in B \subset U$$. Equivalently, every open set is a union of basic open sets.

The boundary of a subset $$A\subseteq X$$ is defined as $${{\partial}}A \coloneqq{ \operatorname{cl}}_X(A) \setminus A^\circ$$. Equivalently, every point $$p\in A$$ intersects both $$A$$ and $$X\setminus A$$.

Given two topologies $$\tau_1, \tau_2$$,

• $$\tau_1$$ is finer/stronger/larger than $$\tau_2$$ iff $$\tau_1 \supseteq\tau_2$$ (idea: finer resolutions).
• $$\tau_1$$ is coarser/weaker/smaller than $$\tau_2$$ iff $$\tau_1 \supseteq\tau_2$$.

Two topologies are comparable if either $$\tau_1 \subseteq \tau_2$$ or $$\tau_2 \subseteq \tau_1$$.

The set of all topologies on a given set $$X$$ forms a complete lattice bounded under inclusion:

• $$\sup(\tau_1, \tau_2) = \tau_1 \cup\tau_2$$
• The finest topology is the discrete topology $$\tau_{{ 0_{\scriptscriptstyle \uparrow}}} \coloneqq 2^X$$, where every set is open.
• $$\inf(\tau_1, \tau_2) = \left\langle{\tau_1 \cap\tau_2}\right\rangle$$, the topology generated by the intersection.
• The coarsest topology is the indiscrete topology $$\tau_{{ \mathscr \emptyset^{\scriptscriptstyle \downarrow}}} \coloneqq\left\{{\emptyset, X}\right\}$$.

If $$f:X\to Y$$, then

• Increasing $$\tau(X)$$ or decreasing $$\tau(Y)$$ makes it easier for $$f$$ to be continuous, i.e. every map continuous with respect to $$\tau_1(X)$$ will remain continuous with respect to $$\tau_2(X)$$. Writing $$\tau_1(X) \to \tau_2(X) \iff \tau_1(X) \leq \tau_2(X)$$,

• Decreasing $$\tau(X)$$ or increasing $$\tau(Y)$$ makes it easier for $$f$$ to be an open map.

• For a fixed $$X$$, decreasing $$\tau(X)$$ makes it easier for sequences to converge in $$X$$.

Write $$\tau_{\mathrm{zar}}(X)$$ for the Zariski topology on a space and $$\tau_{{\mathrm{an}}}(X)$$ for the classical/Euclidean topology. Then $$\tau_{\mathrm{zar}}({\mathbf{C}}^n) < \tau_{{\mathrm{an}}}({\mathbf{C}}^n)$$, i.e. the Zariski topology is strictly weaker than the Euclidean topology and has fewer open sets.

A space $$X$$ is disconnected iff

• There exists a separation of $$X$$: a decomposition $$X = U\coprod V$$ with $$U, V$$ disjoint, open, and nonempty.
• I.e. $$X$$ can not be decomposed as the disjoint union of two proper nonempty sets.
• The only clopen sets of $$X$$ are $$\emptyset, X$$ -I.e. $$X$$ contains no proper nonempty clopen sets.
• For $$Y \subseteq X$$ a subspace, $$Y$$ is disconnected iff $$Y$$ is disconnected in the subspace topology. Equivalently, a separation of $$Y$$ in $$X$$ is a decomposition $$Y = U \coprod V$$ with $$U, V$$ open in $$Y$$ and \begin{align*} { \operatorname{cl}}_{Y}(U) \cap V = \emptyset,\qquad U \cap{ \operatorname{cl}}_{Y}(V) = \emptyset ,\end{align*} so neither set contains a limit point of the other.
• $$\mathop{\mathrm{Hom}}_{\mathsf{Top}}(X, \left\{{0, 1}\right\}) \cong \left\{{0, 1}\right\}$$, i.e. all such continuous functions are constant.

Some examples:

• $${\mathbf{Q}}$$ is disconnected, and $$\pi_0({\mathbf{Q}}) \cong {\mathbf{Q}}$$: the only connected components are singletons.

Set $$x\sim y$$ iff there exists a connected set $$U\ni x, y$$ and take equivalence classes. These classes are the connected components of $$X$$.

• A set $$U \subseteq X$$ is closed in $$X$$ if and only if its complement $$X\setminus U$$ is open.
• A set $$U$$ is closed in $$X$$ iff every limit point of $$U$$ in $$X$$ is contained in $$U$$.
• A set $$U$$ in $$X$$ is closed in $$X$$ iff $${ \operatorname{cl}}_X(U) = U$$.
• If $$Y \subseteq X$$ is a subspace and $$U \subseteq Y$$, then $$U$$ is closed in the subspace $$Y$$ iff $$U = Y \cap V$$ where $$V$$ is closed in $$X$$.

A map $$f:X\to Y$$ is closed if whenever $$U \subseteq X$$ is closed in $$X$$, $$f(U) \subseteq Y$$ is again closed in $$Y$$.

For $$U \subseteq X$$, the closure of $$U$$ in $$X$$ is given by \begin{align*} { \operatorname{cl}}_X(U) = \bigcap_{\substack{ B\supseteq U \\ \text{ closed} }} B ,\end{align*} the intersection of all closed sets in $$X$$ containing $$U$$. For $$Y\subseteq X$$ a subspace containing $$U$$, the closure of $$U$$ in $$Y$$ is \begin{align*} { \operatorname{cl}}_Y(U) = { \operatorname{cl}}_X(U) \cap Y .\end{align*} . 1 In general, we write $$\overline{U} \coloneqq{ \operatorname{cl}}_X(U)$$.

An equivalent condition: $$x\in { \operatorname{cl}}_X(U) \iff$$ every neighborhood of $$x$$ in $$X$$ intersects $$U$$. 2

A topological space $$(X, \tau)$$ is compact iff $$X$$ is Hausdorff and quasicompact: every open cover has a finite subcover. That is, if $$\left\{{U_{j}}\right\}_{j\in J} \subseteq \tau$$ is a collection of open sets such that $$X = \bigcup_{j\in J} U_{j}$$, then there exists a finite subset $$J' \subset J$$ such that $$X \subseteq \bigcup_{j\in J'} U_{j}$$.

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 subspace $$Q\subset X$$ is dense iff every neighborhood of every point in $$x$$ intersects $$Q$$. Equivalently, $${ \operatorname{cl}}_X(Q) = X$$.

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

A topological space $$X$$ is Hausdorff or $$T_2$$ iff points can be separated by disjoint neighborhoods: for every $$p\neq q \in X$$ there exist disjoint open sets $$U\ni p$$ and $$V\ni q$$.

A map $$\iota:A\to B$$ is injective if it admits a left inverse $$p:B\to A$$ satisfying $$p\circ \iota = \operatorname{id}_A$$. Equivalently, \begin{align*} \iota(x) = \iota(y) \in B \implies x = y \in A .\end{align*}

A point $$p\in A$$ is interior to $$A$$ if there exists a neighborhood $$U\ni p$$ that is entirely contained in $$A$$.

A point $$p\in A$$ is isolated if $$p$$ is not a limit point of $$A$$. Equivalently, there exists a punctured neighborhood of $$p$$ that does not intersect $$A$$.

For $$A\subset X$$, $$x$$ is a limit point of $$A$$ if every punctured neighborhood $$P_{x}$$ of $$x$$ intersects $$A$$. I.e., every neighborhood of $$x$$ intersects $$A$$ at a point other than $$x$$. Equivalently, $$x\in { \operatorname{cl}}_{X}(A\setminus\left\{{x}\right\})$$.

A space is locally connected iff every neighborhood of every point admits a smaller connected neighborhood. I.e. for all $$x\in X$$, for all $$N_x \ni x$$, there exists a connected set $$U \subset X$$ with $$x\in U$$.

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 $$X$$ is locally path-connected iff every point in $$X$$ admits some path-connected neighborhood. Equivalently, $$X$$ 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 refinement of an open cover $${\mathcal{U}}\rightrightarrows X$$ is an open cover $${\mathcal{V}}\rightrightarrows X$$ such that for every $$V_\beta \in {\mathcal{V}}$$, there exists a $$U_\alpha \in {\mathcal{U}}$$ such that $$V_\beta \subseteq U_\alpha$$ – setting $${\mathcal{V}}\leq {\mathcal{U}}$$ iff $${\mathcal{V}}$$ refines $${\mathcal{U}}$$ yields a preorder on all open covers of $$X$$.

A topological space $$X$$ is paracompact iff every open cover $${\mathcal{U}}\rightrightarrows X$$ admits an locally finite refinement – a division into (potentially more) open subsets $${\mathcal{V}}\rightrightarrows X$$ such that each $$x\in X$$ is contained in only finitely many $$V_\beta$$.

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

• $$q$$ is surjective, and
• $$U$$ is open in $$Y\iff q ^{-1} (U)$$ is open in $$X$$

Note that $$\implies$$ comes from the definition of continuity of $$q$$, but $$\impliedby$$ is a stronger condition.

Equivalently:

• $$p$$ maps saturated subsets of $$X$$ to open subsets of $$Y$$, or
• If $$U$$ is open in $$X$$, then $$(q^{-1}\circ q)(U)$$ is again open in $$X$$.

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 $$U\ni x, y$$, then the equivalence classes are the path components of $$X$$.

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

Given a collection of spaces $$\left\{{ (X_i, \tau(X_i) ) }\right\}_{i\in I}$$, the box topology is defined by \begin{align*} \tau^{\Box}\qty{ \prod_{i\in I} X_i } &\coloneqq\left\langle{\left\{{\prod_{i\in I} U_i {~\mathrel{\Big\vert}~}U_i \in \tau(X_i) }\right\}}\right\rangle ,\end{align*} the topology generated by products of open sets in the $$X_i$$, with no restrictions.

The product topology is defined by \begin{align*} \tau^{\prod}\qty{ \prod_{i\in I} X_i } &\coloneqq\left\{{\prod_{i\in I} U_i {~\mathrel{\Big\vert}~}U_i \in \tau(X_i),\, U_i \neq X_i \text{ for finitely many }i }\right\} ,\end{align*} whose open sets are products of open sets in the $$X_i$$ where all but finitely many components are not the entire space $$X_i$$. Equivalently, \begin{align*} \tau^\prod\qty{ \prod_{i\in I} X_i} = \inf\left\{{\tau\qty{\prod_{i\in I} X_i } {~\mathrel{\Big\vert}~}p_j: \prod_{i\in I} X_i \to X \text{ are $\tau{\hbox{-}}$continuous for all } j}\right\} ,\end{align*} the smallest/coarsest/initial topology such that the projections $$p_j: \prod_i X_i \to X_j$$ are continuous. Equivalently, \begin{align*} \tau^\prod\qty{\prod_{i\in I} X_i} = \left\langle{\left\{{ p_i^{-1}(U_i) {~\mathrel{\Big\vert}~}U_i\in \tau(X_i)}\right\}_{i\in I} }\right\rangle ,\end{align*} the topology generated by basic open sets which are preimages of opens in the components under the canonical projections.

Equivalently, it is the terminal cone over all of the $$X_i$$ in $${\mathsf{Top}}$$:

Note that \begin{align*} \tau^{\Box}(X) \geq \tau^{\prod}(X) ,\end{align*} i.e. the box topology is finer and has more open sets, making convergence harder in the box topology. The product topology is preferred since continuous maps $$f: Y\to \prod X_i$$ into the product can be given by continuous component maps $$f_i: Y\to X_i$$.

Several equivalent definitions. Let $$f: X\to Y$$ be continuous, then $$f$$ is proper iff

• Most general: preimages of compact sets are compact: if $$K \subseteq Y$$ is compact, then $$f^{-1}(K) \subseteq X$$ is compact.

• For $$Y$$ Hausdorff and locally compact, $$f$$ is a closed map with compact fibers: $$f^{-1}(\left\{{y}\right\})$$ is compact for every $$y\in Y$$.

• For $$X$$ Hausdorff and $$Y$$ locally compact, $$f$$ is universally closed: the map $$f\times \operatorname{id}_Z: X\times Z\to Y\times Z$$ is a closed map for every space $$Z$$.

• For $$X, Y$$ metric spaces, if $$\left\{{x_i}\right\}$$ is a sequence that eventually escapes every compact set in $$X$$, $$\left\{{f(x_i)}\right\}$$ eventually escapes every compact set in $$Y$$.

A topological space $$X$$ (possible non-Hausdorff) is quasicompact iff every open cover admits a finite subcover. If $$X$$ is additionally Hausdorff, $$X$$ is said to be compact.

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 retract $$r$$ of $$B$$ onto a subspace $$A$$ is a map $$r:B\to A$$ that is a left-inverse for the inclusion $$f:A\hookrightarrow B$$, so $$r \circ f = \operatorname{id}_A$$:

Equivalently, a continuous map $$r:B\to A$$ with $${ \left.{{r}} \right|_{{A}} } = \operatorname{id}_A$$ restricting to the identity on $$A$$, i.e. fixing $$A$$ pointwise. Note that $$r$$ is necessarily a surjection.

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.

If $$X$$ retracts onto $$A$$ with $$\iota:A\hookrightarrow X$$, then $$i_*$$ is injective. Any nonempty space retracts to a point via a constant map.

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 \coloneqq\left\{{Y \cap U {~\mathrel{\Big\vert}~}U \in \tau}\right\} .\end{align*}

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

• $$T_0$$: points are topologically distinguishable, i.e. 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. Equivalently, points are closed.

• $$T_2$$: 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.

• Not $$T_0$$: the space $$\left\{{ f:{\mathbf{R}}\to {\mathbf{C}}{~\mathrel{\Big\vert}~}\int_{\mathbf{R}}{\left\lvert {f} \right\rvert}^2 < \infty }\right\}$$, since two a.e. equal functions aren’t distinguishable (they have precisely the same set of neighborhoods).
• $$T_1$$ but not $$T_0$$: $$\operatorname{Spec}R$$ for $$R\in \mathsf{CRing}$$ with the Zariski topology. There are points that aren’t closed: $$\operatorname{Spec}R \setminus\operatorname{mSpec}R$$.

• Using open sets: closed under arbitrary unions and finite intersections.
• Using closed sets: closed under arbitrary intersections and finite unions.

A mnemonic: in $${\mathbf{R}}$$, $$\cap_{n\in {\mathbb{N}}} (-1/n, 1/n) = \left\{{0}\right\}$$ which is closed in $${\mathbf{R}}$$.

A topological embedding is a continuous map $$f:X\to Y$$ which is a homeomorphism onto its image, i.e. $$X\cong_{{\mathsf{Top}}} f(X)$$.

## Analysis and Metric Spaces

For a subset $$A$$ of a metric space $$(X, d)$$, the diameter of $$A$$ is defined as \begin{align*} {\operatorname{diam}}(A) \coloneqq\sup_{p, q\in A}d(p, q) .\end{align*}

For $$(X, d)$$ a metric space, $$S \subset X$$, and $$f\in X$$, the distance from $$f$$ to $$S$$ is \begin{align*} \operatorname{dist}(f, S) \coloneqq\inf_{s\in S} d(f, s) .\end{align*}

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

For $$f: (X, d_{x}) \to (Y, d_{Y})$$ metric spaces, $$f$$ is uniformly continuous iff \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*}

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

## Algebraic Topology

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Points $$x\in M^n$$ defined by \begin{align*} {\partial}M = \left\{{x\in M: H_{n}(M, M-\left\{{x}\right\}; {\mathbf{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. $$F_{i}^j(\sigma) = {\left.{{\sigma}} \right|_{{[v_{i} \ldots v_{j}]}} }$$.

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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.

A manifold that is compact, with or without boundary.

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 $${\mathbf{Z}}_{p}$$.

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 $$g\circ f \simeq\operatorname{id}_X$$. 3

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

• 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 $$U$$),
• $${ \left.{{p}} \right|_{{V_i}} }: V_i \to U$$ 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 $$T^2 \not\simeq S^2 \vee S^1 \vee S^1$$.

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

Letting $$\mathsf{C} \coloneqq\mathsf{Cov}(X) \leq {\mathsf{Top}}_{/ {X}}$$ be the subcategory of the slice category over $$X$$ of covering maps $$\tilde X\to X$$, the group of deck transformations is given by \begin{align*} \mathrm{Deck}(\tilde X\to X) \coloneqq\mathop{\mathrm{Aut}}_{\mathsf{C}}(\tilde X\to X) ,\end{align*} i.e topological automorphisms of $$\tilde X$$ which fix $$X$$ pointwise.

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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 $$\operatorname{id}_X$$ to $$\operatorname{id}_A$$ where $${ \left.{{H}} \right|_{{A}} } = \operatorname{id}_A$$ 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*}

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 $$H^n(S^n) \cong {\mathbf{Z}}$$, we have $$f^*: {\mathbf{Z}}\to{\mathbf{Z}}$$. 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 $$f$$, which only depends on the homotopy class $$f\in [S^n, S^n]$$.

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

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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 $$G$$.

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

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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 $$[X,Y] \coloneqq\hom (X,Y)/\simeq$$.

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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.

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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 $$\left< \alpha, \beta \right> \in H_{0} M = {\mathbf{Z}}$$ 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 {\mathbf{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 {\mathbf{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 $$n \equiv 0 \operatorname{mod}4$$) 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.

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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.

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$$H_{n}(X, X-A; {\mathbf{Z}})$$ 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 $$H_{n}(M, M - \left\{{x}\right\}) = {\mathbf{Z}}$$.

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

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

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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 $$W: (M, x) \to {\mathbf{Z}}_{2}$$ is trivial.
• $$\tilde M_{o} = M \coprod {\mathbf{Z}}_{2}$$ (two sheets).
• $$\tilde M_{o}$$ is disconnected
• The projection $$\tilde M_{o} \to M$$ admits a section.
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For a closed, orientable $$n{\hbox{-}}$$manifold, following map $$[M] \frown {-}$$ is an isomorphism: \begin{align*} D: H^k(M; R) \to H_{n-k}(M; R) \\ D(\alpha) = [M] \frown \alpha\end{align*}

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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 $$\pi_1(U;x) \to \pi_1(X, x)$$.

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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

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

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

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 $${\mathbf{R}}^n$$.

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

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*}

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## Homotopy

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 $$CX$$.

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*}

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## Homological Algebra

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

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$$S = \left\{{s_{i}}\right\}$$ 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).

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

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)

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

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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 $$A$$.

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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*}

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

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

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 $$M \cong R^S / \mathcal{R}$$.

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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 $$s_{i}\in S, r_{i} \in R$$.

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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.

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Footnotes
1.
This is theorem 17.4 in Munkres
2.
Munkres 17.5
3.
This is a useful property because it supplies you with a homotopy.