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I can't think of another way to get $CP^2$ from an embedded $RP^2$ in the 4-sphere, so I would guess you are right. But why would the knottedness of the $RP^2$ imply that the $CP^2$ is exotic? After …

Nice question; there's not much known in general. The signature obstruction you mention goes back to Winkelnkemper (Bull. Amer. Math. Soc. 79 (1973), 45–51) and is sufficient for simply connected $4n$ …

In the case that $x\cdot x \neq 0$, topological methods based on the G-signature show that the genus goes to infinity more or less quadratically in $n$. (I'll be more specific below.) This goes back t …

There are some results in this general direction, but on non-compact manifolds. See for instance Instanton approximation, periodic ASD connections, and mean dimension, by Shinichiroh Matsuo, Masaki T …

A standard construction would be to take a 3-manifold $Y$ given by 0-surgery on a knot $K$. If $K$ is the boundary of a slice disk $D \subset B^4$, then the complement, say $W$ of a neighborhood of $D …

Let's take your $p$ to be prime. Then $X$ has to be simply connected, even without all of the hypotheses. Here is the argument. From the map on $\pi_1(L) \to Z_p$ you get a map $L \to BZ_p$. This ma …

There are not that many explicit handlebody pictures of exotic open 4-manifolds, because they get awfully complex in short order. The ones that I know of are in work of Žarko Bižaca from the mid-90's …

Since you've asked for an opinion, my answer is also an opinion. I would say that the embedding in $R^5$ is not going to be helpful. You are seeking to take $M = \partial N_i$ and `improve' the $N_i$ …

In $H_2(CP^2)$, every class $nH$ where $H$ is a generator and n is odd is characteristic. However, if $n >3$, then such a class is not represented by a torus. It is not represented by a disjoint unio …

Q1: This is true in the topological category and unknown in the smooth setting. In the topological setting, the fundamental group of $S^4 - K_1 \# K_2$ is $G_1 *_\mathbb{Z} G_2$ where $G_i$ are the fu …

The first is in Akbulut-Kirby, Branched covers of surfaces in 4-manifolds. (Math. Ann. 252, 111-131 (1980). See the end of Section 3; it's basically explained via a single example (the square knot rib …

Yes, this can be done, but requires a little care with the fundamental group. First, let me tighten up your description; one is attaching the 2-handle to $S^1 \times B^3$ along a curve $\gamma$ in its …

The paper of Fickle (not the paper of Gordon to which you link) has an explicit construction for $\Sigma(3,5,19)$. It's behind a paywall but maybe you can get it from interlibrary loan or some kind so …

(This is really a comment on the answer relating to concordance order.) Since your p is even, then your 2-bridge knot is actually a link. So, while it makes sense to ask if it's a slice or ribbon l …

The boundary of W may be described as 0-framed surgery on both components of the link you drew. The link can be drawn in a more symmetric fashion, so that it is clear that there is an involution inter …