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Perhaps the fundamental algebraic difference between a compact manifold with boundary and one without boundary is whether it satisfies Poincare duality or Lefschetz duality. Indeed, in algebraic surge …

Let $X$ be a connected CW complex. One can ask to what extent $H_\ast(X)$ determines $\pi_1(X)$. For example, it determines its abelianization, because the Hurewicz Theorem implies that $H_1(X)$ is is …

I think you're describing Spanier. Everyone I know who has seriously studied from Spanier swears by it- it's an absolute classic. The approach is exactly as you describe- algebraic topology for grown …

I quote Andrew Ranicki's answer here. The linking form appears in Example 12.44 of my recent book "Algebraic and geometric surgery" (Oxford University Press, 2002), and also in Chapter 3 of my earl …

The most convincing example I have found of "two basepoints being better than one" is the incorrect statement of the main result of the following paper: Garoufalidis, Stavros, and Andrew Kricker. "A …

My favourite application of the stable homotopy of spheres is the Rokhlin theorem that the signature of a compact smooth spin 4-manifold is divisible by 16. Rokhlin proved this as a corollary of πS3 t …

Your problem is that $T^n$ is not in general a co-Moore space. Therefore Eckmann-Hilton duality breaks down, as the dual spaces no longer form a spectrum, and there would be no (co)homology theory dua …

This question is inspired by an answer of Tim Porter. Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his …

I'm not an expert and this might be wrong, but I think that Cerf theory should be impossible for orbifolds, and therefore all that comes from it, e.g. Kirby Calculus. Could somebody who knows please c …

This question has already been answered, but there's a tiny piece of intuition which I'd like to make explicit: If you're thinking about a manifold in the PL world, surgery might look a bit arbitrary …

My recommendation would be the book of Freedman and Quinn, Topology of 4-manifolds. It's hands-on, very very good, and suitable I think for a reader of your background. Indeed, I would strongly recom …

The field of L-theory, the algebraic K-theory of quadratic forms, lies at the intersection of algebraic topology and of number theory. My impression is that it is an underpopulated discipline partiall …

Regarding your first question, in 1953 Moise proved the (manifold) Hauptvermutung for 3-manifolds (Ann. of Math. 58, pp. 458-480). One way to state his result is that every homeomorphism (diffeomorphi …

In my mind, algebraic topology is comprised of two components: Chain complex information, which is intrinsic information concerning how your object may be built up out of simple "lego blocks". Charac …

MR0809959 (87f:57016) Rourke, Colin . A new proof that $\Omega_3$ is zero. J. London Math. Soc. (2) 31 (1985), no. 2, 373--376. Edit: To summarize: Rourke's proof is short and elementary. Other proofs …