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.

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  1. P

    Algebraic topology for dummies?

    I am looking for the most basic but rigorous to some extent book on Algebraic topology out there.
  2. JasonRox

    Unique Partition of Evenly Covered Sets in Algebraic Topology

    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...
  3. JasonRox

    Surjectivity of Induced Homomorphism in Algebraic Topology

    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...
  4. B

    Proving Triviality of pi_1(S^n;e) in Algebraic Topology

    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...
  5. MathematicalPhysicist

    Geometric Topology Vs. Algebraic Topology.

    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?
  6. L

    Topology and algebraic topology?

    What are the main differences in approach between standard? topology and algebraic topology?
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