For me, one interesting thing about algebraic K-theory is L-theory (which I wish I understood better). This is in no way going to be coherent, but: Let's say you are interested in classifying manifolds. That's not going to be possible, because any finitely presented group is realizable as the fundamental group of some 4-manifold, and "most" finitely presented groups don't have solvable word problem. OK then, the next best thing is to try to classify manifolds within a fixed homotopy type. Surgery theory is a technique for doing this. Given a homotopy type, you construct a CW-complex X with that homotopy type, and your first question is whether X is homotopy equivalent to a manifold. Roughly speaking, this is determined by the normal bundle. So for X to be homotopy equivalent to a manifold, you want there to exist a suitably defined bundle map (f,b)\colon M \to X,which is normal bordant to a homotopy equivalence. This latter condition (for a bundle map to be normal bordant to a homotopy equivalence) is detected K-theoretically (I don't understand this well enough to attempt to try to explain it).
Note that homological techniques are sufficient for simply connected manifolds (Browder, Kervaire, Milnor, Novikov...), but are not useful for manifolds which are not simply connected. And this is where K-theory enters.
The main point seems to be that algebraic K-theory provides the machinery to work with quadratic forms over nonabelian group rings with involution (\mathbb{Z}[\pi_1(M)] in our case).
See also Walhausen's survery, which I don't yet understand.

For me, one interesting thing about algebraic K-theory is L-theory (which I wish I understood better). This is in no way going to be coherent, but: Let's say you are interested in classifying manifolds. That's not going to be possible, because any finitely presented group is realizable as the fundamental group of some 4-manifold, and "most" finitely presented groups don't have solvable word problem. OK then, the next best thing is to try to classify manifolds within a fixed homotopy type. Surgery theory is a technique for doing this. Given a homotopy type, you construct a CW-complex X with that homotopy type, and your first question is whether X is homotopy equivalent to a manifold. Roughly speaking, this is determined by the normal bundle. So for X to be homotopy equivalent to a manifold, you want there to exist a suitably defined bundle map $(f,b)\colon M \to X$,which is normal bordant to a homotopy equivalence. This latter condition (for a bundle map to be normal bordant to a homotopy equivalence) is detected K-theoretically (I don't understand this well enough to attempt to try to explain it).
Note that homological techniques are sufficient for simply connected manifolds (Browder, Kervaire, Milnor, Novikov...), but are not useful for manifolds which are not simply connected. And this is where K-theory enters.
The main point seems to be that algebraic K-theory provides the machinery to work with quadratic forms over nonabelian group rings with involution ($\mathbb{Z}[\pi_1(M)]$ in our case).
See also Walhausen's survery, which I don't yet understand.

For me, one interesting thing about algebraic K-theory is L-theory (which I wish I understood better). This is in no way going to be coherent, but: Let's say you are interested in classifying manifolds. That's not going to be possible, because any group is realizable as the fundamental group of some 4-manifold, and "most" groups don't have solvable word problem. OK then, the next best thing is to try to classify manifolds within a fixed homotopy type. Surgery theory is a technique for doing this. Given a homotopy type, you construct a CW-complex X with that homotopy type, and your first question is whether X is homotopy equivalent to a manifold. Roughly speaking, this is determined by the normal bundle. So for X to be homotopy equivalent to a manifold, you want there to exist a suitably defined bundle map (f,b)\colon M \to X,which is normal bordant to a homotopy equivalence. This latter condition (for a bundle map to be normal bordant to a homotopy equivalence) is detected K-theoretically (I don't understand this well enough to attempt to try to explain it).
Note that homological techniques are sufficient for simply connected manifolds (Browder, Kervaire, Milnor, Novikov...), but are not useful for manifolds which are not simply connected. And this is where K-theory enters.
The main point seems to be that algebraic K-theory provides the machinery to work with quadratic forms over nonabelian group rings with involution (\mathbb{Z}[\pi_1(M)] in our case).
See also Walhausen's survery, which I don't yet understand.

For me, one interesting thing about algebraic K-theory is L-theory (which I wish I understood better). This is in no way going to be coherent, but: Let's say you are interested in classifying manifolds. That's not going to be possible, because any finitely presented group is realizable as the fundamental group of some 4-manifold, and "most" finitely presented groups don't have solvable word problem. OK then, the next best thing is to try to classify manifolds within a fixed homotopy type. Surgery theory is a technique for doing this. Given a homotopy type, you construct a CW-complex X with that homotopy type, and your first question is whether X is homotopy equivalent to a manifold. Roughly speaking, this is determined by the normal bundle. So for X to be homotopy equivalent to a manifold, you want there to exist a suitably defined bundle map (f,b)\colon M \to X,which is normal bordant to a homotopy equivalence. This latter condition (for a bundle map to be normal bordant to a homotopy equivalence) is detected K-theoretically (I don't understand this well enough to attempt to try to explain it).
Note that homological techniques are sufficient for simply connected manifolds (Browder, Kervaire, Milnor, Novikov...), but are not useful for manifolds which are not simply connected. And this is where K-theory enters.
The main point seems to be that algebraic K-theory provides the machinery to work with quadratic forms over nonabelian group rings with involution (\mathbb{Z}[\pi_1(M)] in our case).
See also Walhausen's survery, which I don't yet understand.