# Topology: simple connectedness and fundamental groups

• quasar987
In summary, a pointed topological space is simply connected if and only if its fundamental group is trivial. This is in accordance with the R² characterization that U is simply connected if it is path-connected and has no holes in it. The closed unit-disk with a point of its boundary as base point is an example of a simply connected space with a fundamental group isomorphic to Z. However, all loops in the disk are homotopic to the constant loop, making the fundamental groupoid of the space trivial. This raises a debate on whether "trivial" should mean "trivial up to isomorphism" or "trivial up to equivalence." In the case of groupoids, these two concepts are not equivalent.

#### quasar987

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I read on wiki* that a (pointed) topological space is simply connected iff its fundamental group is trivial. But I don't see how this in accordance with the R² caracterisation that U is simply connected iff it is path-connected and has no holes in it.

Take the closed unit-disk with a point of its boundary as base point for example. Then what are the elements of the fundamental group? There is the trivial loop, which is the identity element, and then there are the homotopy equivalence classes of the loops that circles the boundary S^1 once, twice, three times, etc. such that the fundamental group is isomorphic to Z, not {e}.

*http://en.wikipedia.org/wiki/Simply_connected#Formal_definition_and_equivalent_formulations

All loops in the disk are homotopic to the constant loop. Also, it is usually specified that simply connected spaces are path connected (this does not follow from having trivial fundamental group).

quasar987 said:
Take the closed unit-disk with a point of its boundary as base point for example. Then what are the elements of the fundamental group?

they are all trivially trivial.

There is the trivial loop, which is the identity element, and then there are the homotopy equivalence classes of the loops that circles the boundary S^1 once, twice, three times, etc. such that the fundamental group is isomorphic to Z, not {e}.

rubbish. those maps are all trivially homotopic to the constant map. The space is the whole disk, not S^1.

StatusX said:
All loops in the disk are homotopic to the constant loop. Also, it is usually specified that simply connected spaces are path connected (this does not follow from having trivial fundamental group).

But it does from having trivial fundamental groupoid, right?

matt grime said:
But it does from having trivial fundamental groupoid, right?

I don't know, it depends what you mean by "trivial". The fundamental groupoid of a simply connected space is certainly not the trivial groupoid (which I'm taking to be the single element groupoid), but it is about as trivial as a fundamental groupoid can be (ie, every pair of points has a unique homotopy class of paths between them).

Ah, a debate on whether we should mean "trivial up to isomorphism" or "trivial up to equivalence". :tongue:

Hurkyl said:
Ah, a debate on whether we should mean "trivial up to isomorphism" or "trivial up to equivalence". :tongue:

What's the difference?

The groupoids G and H are isomorphic iff there are functors

P : G --> H
Q : H --> G

such that PQ and QP are equal to the appropriate identity functors.

The groupoids G and H are equivalent iff there are functors

P : G --> H
Q : H --> G

such that PQ and QP are naturally isomorphic to the appropriate identity functors.

An equivalent way of stating it is that two groupoids are equivalent iff they become isomorphic after we identify isomorphic points.

In particular, a groupoid is equivalent to the groupoid you stated if and only if, for any ordered pair of points (A, B), there is exactly one arrow A --> B.

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