Fundamental groups and homotopy type

In summary, the conversation discusses the link between the fundamental groups of homeomorphic and homotopy equivalent spaces. It is shown that homeomorphisms induce isomorphisms between fundamental groups, and that the fundamental group functor also takes homotopy equivalences to isomorphisms. This highlights the importance of thinking in terms of maps rather than just objects in understanding these concepts.
  • #1
quasar987
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I know of the "result" that if two pointed spaces are homeomorphic, then the group homomorphism induced by such an homeomorphism if actually an isomorphism between the fundamental groups of these pointed spaces.

But is there a link between the fundamental groups of homotopy equivalent spaces?
 
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  • #2
Homotopy equivalences also induce isomorphisms. This is easy to show if they are homotopies relative to the basepoint, but not too much harder even if they're not.
 
  • #3
a functor is something that atkes objkects to objects, takes maps betwen pairs of objects to similar maps, takes identities to identities, and takes compositions to compositions. hence it also takes inverses to inverses.

i.e. if ∏ is a functor from spaces to groups such as the fundamental group, and if f:X-->Y is a homeomorphism, that means there is a map g:Y-->X such that fg = idY and gf = idX are the identities on Y and X respectively.

Hence, since ∏ is a functor from top spaces to groups, then ∏(f) and ∏(g) are homomorphisms from ∏(f):∏(X)-->∏(Y), and ∏(g):∏(Y)-->∏(X), such that
∏(f)o∏(g) = ∏(fog) = ∏(idY) = id(∏(Y)), and similarly the other way.

Hence ∏(f) and ∏(g) are inverse homomorphisms of the groups ∏(Y) AND ∏(X), so those groups are isomorphic.

Now the fundamental group is a functor, so itab tkes homeomorphisms to isomorphisms of groups. But also the fundamental group is by tis very definition constant on homotopy classes of maps, hence also takes homotopy equivalences to isomorphisms.

so all this is ":trivial" from the category theoretic point of view. i.e., learn to think in terms of maps, not just objects, and these questions will become automatic to you.
 
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What is a fundamental group?

A fundamental group is a mathematical concept that represents the topological structure of a space. It is a way to measure the number of holes or "loops" in a space, and can reveal important information about its shape and connectivity.

How is the fundamental group calculated?

The fundamental group is calculated using the tools of algebraic topology, specifically homotopy theory. It involves identifying the loops in a space and then determining how they can be continuously deformed into one another.

What is homotopy type?

Homotopy type is a concept in topology that relates to the idea of continuous deformations between spaces. Two spaces are said to have the same homotopy type if they can be continuously deformed into one another. This is an important idea in understanding the structure of spaces.

How is homotopy type related to the fundamental group?

The fundamental group is a tool for determining the homotopy type of a space. In fact, the fundamental group is a complete invariant of the homotopy type of a space, meaning that if two spaces have different fundamental groups, they cannot have the same homotopy type.

What are some real-world applications of fundamental groups and homotopy type?

Fundamental groups and homotopy type have many applications in the fields of physics, engineering, and computer science. They can be used to analyze the properties of materials, model physical systems, and solve problems in network design and optimization.

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