Ontoness and Induced Maps on Fundamental Group.

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Discussion Overview

The discussion revolves around the induced maps on the fundamental group of a topological space under homeomorphisms, specifically questioning the surjectivity of the map from automorphisms to homomorphisms of the fundamental group. The scope includes theoretical aspects of topology and algebraic topology.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • The original poster (OP) questions whether the map g: Aut(X) → Hom(π₁(X), π₁(X)) is always surjective, noting that it is likely not injective due to the fundamental group functor being constant on homotopic maps.
  • One participant states that continuous maps induce homomorphisms of fundamental groups and that homeomorphisms induce isomorphisms.
  • The OP clarifies that they are asking if any permutation of the fundamental group can be induced by a self-homeomorphism of the space.
  • Another participant argues that the answer is no, providing an example involving the fundamental group of a wedge product of a circle and a circular strip, which cannot map generators via a self-homeomorphism.
  • A later reply acknowledges the example but corrects the homotopy type, stating that the fundamental group should be the free product Z * Z, while still asserting that swapping generators cannot be induced by a self-homeomorphism.
  • One participant suggests that in a specific case, it is possible to find images for a generating set of Z + Z using matrices, and discusses the mapping class group of the torus and its relation to automorphisms.
  • The last post asks for clarification on the generating matrices mentioned in the previous response.

Areas of Agreement / Disagreement

Participants express differing views on the surjectivity of the map g and whether specific permutations of the fundamental group can be induced by self-homeomorphisms. The discussion remains unresolved with multiple competing viewpoints.

Contextual Notes

Some limitations include the dependence on definitions of homeomorphisms and fundamental groups, as well as unresolved mathematical steps regarding the mapping class group and the specific examples provided.

Bacle
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Hi, everyone:

Given a top space X, and a homeo. h: X--->X , we get an induced map

(by functoriality ) h_*: Pi_1(X)---> Pi_1(X) . We can also write

the map as a map g: Aut(X) --->Hom(Pi_1(X),Pi_1(X))

Is the map g always surjective.? . Almost definitely not injective, since

the Fund. Group functor is constant on homotopic maps, but I have no idea

how to tell if it is injective.

I have no idea. Anyone know.?

Thanks
 
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If f is a continuous map between topological spaces, then it induces a homomorphism f* of their fundamental groups. If f has a continuous inverse, then (f^{-1})_*=(f_*)^{-1}. (This is because star distributes over function composition.) From that you get that homeomorphisms of topological spaces induce isomorphisms of fundamental groups.
 
The OP is asking whether given any permutation of the fundamental group there is a self-homeomorphism of the space that induces this permutation.
 
Oops, I did totally misread that. I'd say the answer is no. The fundamental group of the wedge product of a circle and a circular strip (circle cross an interval) is Z+Z, but we can't send a generator of one to a generator of the other via a self-homeomorphism.
 
Good example!
 
The example works but I botched the homotopy: the group should be the free product \mathbb{Z}*\mathbb{Z}...I had homology on the brain when I wrote that. But swapping generators is still an automorphism of the fundamental group that can't be induced by a self-homeomorphism of the space.
 
Thanks to both:

I think the answer is yes in this specific case: One just has to find images for
a generating set for Z(+)Z , which is finitely-generated ---3 matrices are enough.

And, re the mapping class group, thanks for the Rolfsen source. The idea is
nice:

Consider automorphisms h: T^2-->T^2 . By functoriality, we get a map

Pi_1(T^2)-->Pi_1(T^2) , or , in a more general way, we get a homomorphism :

f: Aut(T^2) --->Aut(Z(+)Z) ; a homeo. on the left and a homomorph. on the right.

Then we know Aut(Z(+)Z) ~ Gl(2,Z) .

So, ultimately, we have a homomorphism :

h: Aut(T^2)--->Gl(2,Z)

Since we "have proved" ontoness ( by finding images for the generating set),

we just mod out by the kernel , which is the set of maps that are isotopic to

the identity. I think that gives us the mapping class group for the torus.
 
What are your 3 generating matrices?
 

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