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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 …

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 …

Infinite dimensional constructions, such as spaces of diffeomorphisms, spectra, spaces of paths, and spaces of connections, appear all over topology. I rather like them, because they sometimes help me …

First, recall the slogan: Small constructions are good for making calculations, but large constructions are good for proving theorems. K-theory is certainly a large construction. In general, K-theory …

Given two morphisms between chain complexes $f_\bullet,g_\bullet\colon\,C_\bullet\longrightarrow D_\bullet$, a chain homotopy between them is a sequence of maps $\psi_n\colon\,C_n\longrightarrow D_{n …

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 …

Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the condi …

Le and Murakami (HERE and HERE) discovered several previously unknown relations between multiple zeta values through the study of quantum invariants of knots. Further relations were later discovered t …

Roughly speaking, algebraic topology works by reducing questions about topological objects such as manifolds and cell to questions about chain complexes. On the topological side, although in the PL …

T. Ohtsuki's Problems on invariants of knots and $3$--manifolds sounds to me like what you are looking for. Updates for problems in it, since it was published in 2002, are here. In my opinion, the bi …

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 …

This is an interesting question, and Greg's answer is excellent. Thinking about this was very nice! Regarding the Pontryagin-Thom construction, as opposed to the Pontryagin construction which other co …

Very much so. There are a number of small industries centred around studying equivalence classes of knot diagrams generated by a set of moves. The study of claspers. For example, $C_k$-moves are a sp …

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 …

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 …