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# 数学代写|拓扑学代写TOPOLOGY代考|Homotopies. Homotopy type

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## 数学代写|拓扑学代写TOPOLOGY代考|Homotopies. Homotopy type

A continuous homotopy (or briefly homotopy or deformation) of a map $f: X \longrightarrow Y$, is a continuous map of the cylinder $X \times I$ to $Y$ :
$$F=F(x, t): X \times I \rightarrow Y, \quad x \in X, \quad a \leq t \leq b,$$
$(I$ an interval $[a, b])$ for which
$$F(x, a)=f(x) \text { for all } x \in X .$$
Two maps $f, g: X \rightarrow Y$ are homotopic if there is a continuous homotopy $F$ such that
$$F(x, a)=f(x), \quad F(x, b)=g(x), \quad x \in X .$$
One often needs to consider in this context pointed spaces $X, Y$, i.e. with particular points $x_0 \in X, y_0 \in Y$ specified. For such spaces maps $f: X \rightarrow Y$ are usually also required to be “pointed”, i.e. to satisfy $f\left(x_0\right)=y_0$, and homotopies between “pointed” maps are then also normally “pointed”, in the sense that one requires $F\left(x_0, t\right)=y_0$ for all $t$.

Each equivalence class of homotopic maps $f: X \rightarrow Y$ constitutes a pathcomponent of the function space $Y^X$, and is called a homotopy class of maps $X \rightarrow Y$ (or of pointed maps, as the case may be). Thus the set $\pi_0\left(Y^X\right)$ is comprised of homotopy classes.

## 数学代写|拓扑学代写TOPOLOGY代考|Covering homotopies. Fibrations

Consider a (continuous) map $p: X \rightarrow Y$. We say that an arbitrary mapping $f: Z \longrightarrow Y$ is covered (via $p$ ) if there is a mapping $g: Z \longrightarrow X$ such that $f=p \circ g$.

Suppose now that we have a homotopy $F: Z \times I \longrightarrow Y$, where $I=[a, b]$, and that at the initial time $t=a$ the map $f(z)=F(z, a)$ is covered by some $\operatorname{map} g: Z \rightarrow X$

Definition 3.1 The map $p: X \rightarrow Y$ is called a fibration if given any space $Z$ and any homotopy $F: Z \times I \rightarrow Y$ whose initial map $f(z)=F(z, a)$ : $Z \longrightarrow Y$ is covered (by $g(z)$, say), the whole homotopy $F$ “down below” in $Y$ is covered “up above” in $X$ by some homotopy $G: Z \times I \rightarrow X$, i.e.

$p \circ G(z, t)=F(z, t)$. The homotopy $G$ is called a covering homotopy for $F$ with initial map $g$.

For various technical reasons a weakened form of this definition is often employed in situations where the space $Z$ has one or another condition imposed on it (for example, cellularity – see Chapter 3). However the essential character of the concept of fibration is unaffected by such changes.

Usually the following additional condition is imposed in the above definition, namely that each point $z_1 \in Z$ remaining fixed under the homotopy $F(z, t)$ for all $t$ in any subinterval of $[a, b]$, should likewise remain fixed on that subinterval under $G(z, t)$.

In the most important situations the construction of a covering homotopy is carried out by means of a “homotopy connexion”. Roughly speaking a homotopy connexion is a recipe for obtaining from a given path in $Y$ beginning at $y_0 \in Y$ and any prescribed point $x_0 \in X$ above $y_0$, a unique covering path in $X$ beginning at $x_0$. Furthermore this covering path should depend continuously on both the given path in $Y$ and the initial point $x_0 \in X$ at which the covering path is to begin; this secures the covering-homotopy property for all reasonably well-behaved spaces $Z$.

## 数学代写|拓扑学代写TOPOLOGY代考|Homotopies. Homotopy type

$$F=F(x, t): X \times I \rightarrow Y, \quad x \in X, \quad a \leq t \leq b,$$
$(I$一个间隔$[a, b])$
$$F(x, a)=f(x) \text { for all } x \in X .$$

$$F(x, a)=f(x), \quad F(x, b)=g(x), \quad x \in X .$$

## 数学代写|拓扑学代写TOPOLOGY代考|Covering homotopies. Fibrations

$p \circ G(z, t)=F(z, t)$． 对于具有初始映射$g$的$F$，其同伦$G$称为覆盖同伦。

## MATLAB代写

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