What is Algebraic topology: Definition and 56 Discussions
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.
Note: I have many questions and will keep posting new ones as they come up. To find the questions simply scroll down to look for bold segments. Feel free to contribute any other comments relevant to the questions or the topic itself.
Here it is...
Let p:E->B be continuous and surjective...
I'm totally stuck on these two.
The first is...
Let A be a subset of X; suppose r:X->A is a continuous map from X to A such that r(a)=a for each a e A. If a_0 e A, show that...
r* : Pi_1(X,a_0) -> Pi_1(A,a_0)
...is surjective.
Note: Pi_1 is the first homotopy group and r* is the...
Please read the following problem first:
Suppose n > 1 and let S^n be the n-sphere in R^{n+1}. Let e be the unit-coordinate vector (1,0,...,0) on S^n. Prove that the fundamental group pi_1(S^n;e) is the trivial group.
Okay, now my question is what does the notation "pi_1(S^n;e)" mean...
i know that geometric topology is a field that is connected to knot theory, i wonder what are the similarities between the two subjects, and in what subject in particular they overlap?