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There have been a couple of posts and questions on MathOverflow about the proofs of the following two facts: Fact 1: if $X$ is a topological space, then $\pi_k(X,x)$ is abelian for $k\ge 2$. Fact 2: …

Following Henrik Rüping's suggestion: if you take $M = S^2\times S^2$, you cannot find any $N$ doing the trick. I'll work with homology with integer coefficients. By excision and Poincaré–Lefschetz du …

Aru Ray and Danny Ruberman wrote a paper (here the arxiv version) about Dehn's lemma in dimension 4. From the abstract: We investigate certain 4-dimensional analogues of the classical 3-dimensional D …

For a closed, connected topological $n$-manifold, H_n(X)≠0 (in fact, it's either Z or Z/2Z, depending on orientability) $H_n(X;\mathbb{Z}/2\mathbb{Z})= \mathbb{Z}/2\mathbb{Z}$ and $H_m(X)=0$ if $m>n$, …

I'll expand a bit the comments by Igor Rivin and myself above. The way I see it, there are two ways of constructing such a curve, and they both involved what I'd call "embedded connected sum". This i …

Yes, you can compute them, and $\pi_1(M) = \mathbb{Z}$, $\pi_2(M) = \mathbb{Z}$. The way I see it, there are three steps in the proof. For convenience, let me call $Y = U(2)\times U(2)$ the set of $N …

The closure of the braid $\sigma_\kappa$ is a connected sum of torus links $T(2,k_i)$ (which are closures of 2-braids). Since the Alexander polynomial is multiplicative with respect to connected sums, …

This post is a summary of the comments above. No, such a manifold doesn't exist. Donaldson proved in 1982 that if the intersection form of a simply connected, closed, orientable, smooth 4-manifold is …

Let me assume that both $M$ and $B$ are orientable. From the long exact sequence of the pair $(M,B)$, Poincaré–Lefschetz duality, and the universal coefficient theorem, for every $k$ we get an exact s …

For singularities of the form $\{z^d = f(x,y)\}$ (which, I believe, are called suspension singularities), the Milnor fibre $M$ has a nice topological description as the $d$-fold cover of $B^4$ branche …

No. If $\Sigma$ is any homology 4-sphere with non-trivial fundamental group, $\mathbb{CP}^2 \# \Sigma$ is a homology $\mathbb{CP}^2$ with non-trivial fundamental group. (Here $\#$ denotes connected su …

As mme noted in the comments, such examples cannot exist in odd dimensions, for Euler characteristic reasons. They can't exist in dimension 2 either, by classification. I claim that in all other dimen …

One classical example of such move is Conway mutation, which falls into the category of tangle replacement, as Qiaochu Yuan mentioned in his comment. There's a very famous pair of mutants, the Kinoshi …

Yes, $H_1(\partial X)$ can be torsion, in which case it has to be metabolic (with respect to the linking form). To see this, consider any rational homology 3-sphere $Y$ that bounds a rational homology …

This is written in Némethi's book Normal surface singularities, Example 5.1.17. (He actually talks about the more general case of complete intersections of Brieskorn-type.) He refers to Jankins and Ne …