## MATH443 Algebraic topology课程简介

Topology is the branch of mathematics concerned with the study of properties that are preserved under continuous transformations. One of the main goals of topology is to classify topological spaces up to homeomorphism, which is a continuous transformation that can be reversed.

However, determining whether two topological spaces are homeomorphic can be a difficult task. Algebraic constructions known as invariants are used to help determine whether two differently presented topological spaces are indeed different. These invariants are algebraic objects that can be associated with a given topological space and remain unchanged under homeomorphism. This allows us to distinguish between topological spaces that are not homeomorphic.

Homotopy equivalence is a fundamental concept in algebraic topology. Two topological spaces are homotopy equivalent if they can be continuously deformed into each other. More precisely, if there exists a continuous map between the spaces that is invertible up to homotopy, then the spaces are said to be homotopy equivalent. Homotopy equivalence is an equivalence relation, which means that it is reflexive, symmetric, and transitive.

## PREREQUISITES

Group presentations and homomorphisms are also important concepts in algebraic topology. A group presentation is a way of describing a group using generators and relations. Homomorphisms are maps between groups that preserve their algebraic structure. In algebraic topology, groups can be associated with topological spaces, and homomorphisms between groups can be used to study the properties of the associated spaces.

Covering spaces are another important topic in algebraic topology. A covering space is a space that locally looks like a product of a space and a discrete set. Covering spaces can be used to study the topology of a space by relating it to the topology of a simpler space.

The fundamental group is an algebraic invariant that can be associated with a topological space. The fundamental group is a group that captures the essential features of the topology of the space. It is a measure of the number of ways that loops in the space can be continuously deformed. The fundamental group is an important tool for distinguishing between topological spaces.

Homology theories are another set of algebraic invariants that can be associated with a topological space. Homology theories assign algebraic objects to topological spaces that measure the number of holes in the space of a given dimension. Homology theories are more general than the fundamental group and can be used to study a wider class of topological spaces.

Finally, the cohomology ring of a space is a more advanced concept that can be used to study the topology of a space. The cohomology ring is an algebraic object that is associated with a space and captures information about the ways in which the space can be decomposed into simpler spaces. The cohomology ring is a more powerful invariant than the homology groups and can be used to study more complex topological spaces.

## MATH443 Algebraic topology（EXAM HELP， ONLINE TUTOR）

Lemma 1. Let $f_0, f_1$ and $f_2$ be maps $X \rightarrow Y$. If $f_0 \simeq f_1$ and $f_1 \simeq f_2$ then $f_0 \simeq f_2$.
Proof. Let $F_0: X \times I \rightarrow Y$ be a homotopy between $f_0$ and $f_1$, and $F_1: X \times I \rightarrow Y$ a homotopy between $f_1$ and $f_2$.
Define $F: X \times I \rightarrow Y$ by
$$F(x, t)= \begin{cases}F_0(x, 2 t), & t \in[0,1 / 2] \ F_1(x, 2 t-1), & t \in[1 / 2,1] .\end{cases}$$
If $t=1 / 2$ then $F_0(x, 2 t)=F_0(x, 1)=f_1(x)=F_1(x, 0)=F_1(x, 2 t-1)$, i.e. the map $F$ is well-defined. By the pasting lemma, $F$ is continuous. Since $F(x, 0)=F_0(x, 0)=f_0(x)$ and $F(x, 1)=F_1(x, 1)=f_2(x), F$ is a homotopy between $f_0$ and $f_2$.

To elaborate, the idea behind the proof is to “combine” the two given homotopies $F_0$ and $F_1$ to obtain a homotopy between $f_0$ and $f_2$. This is achieved by defining $F$ to be a combination of $F_0$ and $F_1$, where $F_0$ is used for the first half of the interval $[0,1]$ and $F_1$ is used for the second half. The key observation is that $F$ is well-defined at $t=1/2$ because $f_1 = F_0(\cdot,1) = F_1(\cdot,0)$.

To show that $F$ is continuous, the pasting lemma is used. Specifically, we note that $F$ is continuous on $X \times [0,1/2]$ because it is the restriction of the continuous function $F_0$. Similarly, $F$ is continuous on $X \times [1/2,1]$ because it is the restriction of the continuous function $F_1$. Since the two sets $X \times [0,1/2]$ and $X \times [1/2,1]$ intersect only at the point $(x,1/2)$, and $F$ agrees with both $F_0$ and $F_1$ at this point, the pasting lemma implies that $F$ is continuous on the entire interval $[0,1]$.

Finally, it is clear from the definition of $F$ that $F(x,0) = F_0(x,0) = f_0(x)$ and $F(x,1) = F_1(x,1) = f_2(x)$, so $F$ is a homotopy between $f_0$ and $f_2$.

Lemma 2. If $f_0, f_1: X \rightarrow Y$ are homotopic and $g_0, g_1: Y \rightarrow Z$ are homotopic then $g_0 f_0, g_1 f_1: X \rightarrow$ $Z$ are homotopic.

Proof. Let $F: X \times I \rightarrow Y$ be a homotopy between $f_0$ and $f_1$, and let $G: Y \times I \rightarrow Z$ be a homotopy between $g_0$ and $g_1$.

One proof: Now the composite $g_0 F: X \times I \rightarrow Z$ is a homotopy between $g_0 f_0$ and $g_0 f_1$, and the composite $G\left(f_1 \times \operatorname{id}_I\right): X \times I \rightarrow Z$ is a homotopy between $g_0 f_1$ and $g_1 f_1$. By lemma 1 , $g_0 f_0 \simeq g_1 f_1$.

The proof is correct.

To elaborate, the proof shows that the composite maps $g_0 f_0$ and $g_1 f_1$ are homotopic by constructing a homotopy between them. The idea is to use the given homotopies $F$ and $G$ to “combine” $g_0$ and $g_1$ with $f_0$ and $f_1$, respectively. Specifically, we define two maps $H_0$ and $H_1$ by

H_0(x, t) = (g_0 \circ F)(x, t) \quad \text{and} \quad H_1(x, t) = (g_1 \circ F)(x, t).H0​(x,t)=(g0​∘F)(x,t)andH1​(x,t)=(g1​∘F)(x,t).

Intuitively, $H_0$ and $H_1$ are obtained by “pushing” each point $(x, t)$ in $X \times I$ along $F$ to a point in $Y$, and then applying either $g_0$ or $g_1$ to that point. Since $F$ is a homotopy between $f_0$ and $f_1$, we have that $H_0(x, 0) = (g_0 \circ f_0)(x)$ and $H_1(x, 0) = (g_1 \circ f_0)(x)$, and $H_0(x, 1) = (g_0 \circ f_1)(x)$ and $H_1(x, 1) = (g_1 \circ f_1)(x)$.

To show that $H_0$ and $H_1$ are homotopic, we use the homotopy $G$ to “slide” each point $(y, t)$ in $Y \times I$ to a nearby point $(y’, t)$, where $y’$ is on the path from $g_0(y)$ to $g_1(y)$. Specifically, we define a map $H: X \times I \rightarrow Z$ by

H(x, t) = (G \circ (H_0 \times \operatorname{id}_I))(x, t) \quad \text{if } t \in [0, 1/2],H(x,t)=(G∘(H0​×idI​))(x,t)if t∈[0,1/2],

and

H(x, t) = (G \circ (H_1 \times \operatorname{id}_I))(x, 2t-1) \quad \text{if } t \in [1/2, 1].H(x,t)=(G∘(H1​×idI​))(x,2t−1)if t∈[1/2,1].

Intuitively, $H$ is obtained by first applying $H_0$ or $H_1$ to each point $(x, t)$ in $X \times I$, depending on whether $t$ is in the first or second half of the interval $[0,1]$, and then “sliding” each resulting point along $G$ to a nearby point in $Z$. Since $H_0$ is a homotopy between $g_0 f_0$ and $g_0 f_1$, and $H_1$ is a homotopy between $g_1 f_0$ and $g_1 f_1$, it follows that $H$ is a homotopy between $g_0 f_0$ and $g_1 f_1$.

The last step of the proof uses Lemma 1 to conclude that $g_0 f_0$ and $g_1 f_1$ are homotopic, since we have constructed a homotopy $H$ between them.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

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