Abstract Nonsense

Homtopy and the Homotopy Category (Pt. I)

Point of Post: In this post we motivate, define, and discuss the notion of homotopy. We then introduce the homotopy category as a quotient category of $\mathbf{Top}$.

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Motivation

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In topology we care about the geometry of a space–how we can describe all of the geometric properties that a space has. For example, we care if the space is compact, we care if it locally connected, we care if it $T_4$. All of the properties factor into whether or not two given topological spaces are to be considered “the same”.  That said, for large portions of mathematics some of the topological invariants of a space are less important than others. This can occur either because they literally matter less to us (for example, compactness is something that, while nice, isn’t an absolute necessity for a topological space to be nice). This can also be true because the subject area we are are working in contains spaces which necessarily already satisfy some of the properties (for example, metric spaces already satisfy all of the nice separation axioms).

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Thus, it seems that homeomorphism, which is sensitive to this “redundant” or “superfluous” information is to strong of an equivalence. If we really only care about a proper subset of the topological invariants associated to a set, it behooves us to consider a weaker equivalence. This will allow us to focus in on the attributes of a space that we care about. For example, if we cared about, say, compactness the fact that two topological spaces aren’t homeomorphic doesn’t tell us anything–they could be different because one is locally connected and the other isn’t, not because they both are compact or non-compact. Creating an equivalence which is insensitive to these other attributes of the space will allow us to circumvent issues like this.

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The other reason that a weaker equivalence is nicer, is that should be easier to prove equivalence/non-equivalence than with a stricter notion. Thus, not only are we able to theoretically focus in on things that are important to us, but we will able to more easily compute the amount of variance that two spaces have with respect to these important aspects of the space.

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Now that we understand why we would like a new, weaker form of equivalence between topological spaces, we need to decide which particular attributes are important to us so that we can work towards properly defining this new notion. So, which one is it? Perhaps one of most visually striking, obvious attributes of a topological space is its level of connectedness. When one looks at a space it’s not at all obvious if it is compact, or if it is $T_{3.5}$ but it should be immediately obvious how many pieces it is in. But, we do not only care about this very coarse measure of a space’s connectedness–we’d also like to measure things like whether or not the space has “holes”. Things picked up like when we consider the singular homology  of a space.

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So, we now seek to find a weaker equivalence of spaces that singles out this kind of connectedness. The rough idea behind this equivalence is the observation that the amount of connectivity (in the ways described above) is that we can shrink, contract the space as much as long as we want (as long we do not pass through a “hole”–which is impossible if we are not thinking about our space as being embedded in a larget space) and not change this type of connectivity. For example, consider the punctured plane $\mathbb{C}-\{0\}$. The only hole, the only issue with connectivity is at the hole at the origin. Thus, we should be able to shrink/contract any part of the space not containing this hole and maintain the same amount of connectivity. For example, one can shrink the region $(\mathbb{C}-\{0\})-\mathbb{S}^1$, which contains no holes, to the boundary of the unit circle. You can easily see that, indeed, in terms of connectivity the unit circle and the punctured plane are the same.

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Homotopy is precisely the formal notion capturing this ability to shrink/contract our space through new points. Since we live in fundamentally arrow obsessed society, it makes sense that the first step in defining this notion of weak equivalence should have to do with maps. Roughly, two maps are “homotopic” if one can find a continuous way of deforming one into the other. This captures precisely the notion of shrinking, because this deformation can involve any amount of contracting as is legall (i.e. while keeping the map continuous) is allowable. We are then able to transfer this notion to an equivalence of spaces by stating that for a continuous map $f:X\to Y$ to be an equivalence it doesn’t have to have a continuous inverse $g:Y\to X$ but only a continuous inverse up to homotopy. This gives exactly the notion of equivalence we want–one that allows are spaces to be the same “up to contraction of good regions”.

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Homotopy and Homotopy Equivalence

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Let $X$ and $Y$ be topological spaces and $f,g:X\to Y$ two continuous maps. A homotopy from $f$ to $g$ is a continuous map $H:X\times[0,1]\to Y$ such that $H(x,0)=f(x)$ and $H(x,1)=g(x)$ for all $x\in X$. If there exists a homotopy from $f$ to $g$ we say that $f$ and $g$ are homotopic and denote this by $f\simeq g$.

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Intuitively we can think of a homotopy from $f$ to $g$ as being a time-index set of continuous maps $H_t:X\to Y$ such that $H_0=f$ and $H_1=g$–and that $H_t$ varies smoothly with $t$. We can think about this process as deforming $f$ into $g$ where we start with the initial map $f=H_0$ and end with the map $g=H_1$ and have the intermediate maps $H_t, t\in(0,1)$ in between. The classic picture associated to this way of thinking about a homotopy is the following:

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where, in our notation, $\gamma_0=f$, $\gamma_1=g$, and $H(t,s)=H(s,t)$.

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Let’s give an example of a homotopy between two maps $\mathbb{D}\to\mathbb{D}$ where $\mathbb{D}$ is the standard unit disk sitting inside of $\mathbb{C}$. In particular, we claim that the identity map $\text{id}:\mathbb{D}\to\mathbb{D}$ and the constant map $c_0:\mathbb{D}\to\mathbb{D}$ with $c_0:(x,y)\mapsto (0,0)$ for all $(x,y)\in\mathbb{D}$. Indeed, define $H((x,y),t)=t(x,y)$. Clearly then $H$ is a continuous map $\mathbb{D}\times[0,1]\to\mathbb{D}$,

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$H((x,y),1)=(x,y)=\text{id}(x,y)$

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and

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$H((x,y),0)=0(x,y)=(0,0)$

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for all $(x,y)\in\mathbb{D}$. Thus, $\text{id}\simeq c_0$.

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References:

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[1] Spanier, Edwin Henry. Algebraic Topology. New York: McGraw-Hill, 1966. Print.

[2] Hatcher, Allen. Algebraic Topology. Cambridge: Cambridge UP, 2002. Print.

[3] Bredon, Glen E. Topology and Geometry. New York: Springer-Verlag, 1993. Print.

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