What are the caps in the proof of Poincare Conjecture and does the insertion of caps into initial manifold preserve homeomorphism? Quote from Wikipedia article "Poincare Conjecture":
"He wanted to cut the manifold at the singularities and paste in
caps (Question), and then run the Ricci flow again...
In essence, Perelman showed that all the strands that form can be
cut and capped (Question)..."
To my notice, the above "Question" is: the caps do not belong to the
original manifold; thus, there is no direct (one to one) correspondence between
the original manifold and the final sphere. In conclusion, this formulation
violates the Poincare Conjecture:
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
Should somebody explain this point in Wikipedia? The public deserves
to read a good article on Wikipedia.
 A: First of all, Wikipedia article is not meant to be a detailed technical explanation of how the proof of PC goes, only to give a casual reader with limited math background a general idea.
Here is what "caps" in Perelman's proof are. Suppose that $M$ is a simply-connected closed (compact with empty boundary) smooth 3-dimensional manifold and $S\subset M$ a smoothly embedded 2-dimensional sphere. Since $M$ is simply-connected, $S$ separates $M$ in two components, so removing it from $M$ yields two noncompact manifolds $N_1, N_2$. Both $N_1, N_2$ admit natural compactifications as smooth compact manifolds $\bar{N}_i, i=1,2$, with spherical boundaries. Attach (smoothly) 3-dimensional balls $B_1, B_2$ to $\bar{N}_1$, $\bar{N}_2$ along the boundary spheres. This results in closed 3-dimensional manifolds $M_1, M_2$. The "cups" are these 3-dimensional balls $B_1, B_2$. This process of decomposition of $M$ is called "connected sum decomposition":
$$
M= M_1\# M_2.
$$
It is vaguely similar to the process of decomposition of a natural number as a product of two smaller natural numbers, say
$$
12=6\cdot 2,
$$
or, more appropriately,
$$
1=1\cdot 1. 
$$
Repeating this process with $M_1, M_2$ inductively finitely many times, one obtains a connected sum decomposition
$$
M= L_1\# L_2 \# ... \#L_n.
$$
The way Perelman's proof works is that after finitely many decompositions, each $L_i$ is shown to be diffeomorphic to the usual 3-dimensional sphere $S^3$ (and this is the heart of the proof). Therefore,
$$
M= S^3\# ... \# S^3. 
$$
It is a standard (and easy) fact of geometric topology that if $K$ is any (smooth) connected manifold of dimension $m$ then
$$
K\# S^m=K,
$$
where $K$ means "diffeomorphic." In analogy with the product decomposition of natural numbers, $S^m$ plays the role of the unit: $$k\cdot 1=1$$
for all natural numbers $k$. (In fact, the analogy is deeper: The connected sum operation is commutative and associative. Moreover, every compact oriented connected 3-dimensional manifold $M$ admits a "prime decomposition" as a connected sum of manifolds which are not decomposable any further, and these "prime factors" are unique.)
In the setting of Perelman's proof, one concludes that
$$
M= S^3\# ... \# S^3= S^3, 
$$
thereby proving the PC.
