https://www.youtube.com/watch?v=0H5_h-RB0T8
Gluing is a good method to construct new topological spaces from known ones. Here a rectangles is glued along the edges to form a torus.Often the fundamental

https://math.stackexchange.com/questions/4233879/disk-glued-to-a-torus-classifying-the-resulting-spaces-how-where-to-begin
Describe the homotopy type of the different spaces that can be obtained from gluing a disk to a torus. Obs: The gluing is only at the disk's border. On a first try, I was thinking of gluing the border of the disc to a single side of the torus (>), then to two sides (maybe continuous sides (>,>>) or inverting the orientation of the first gluing

https://mathcircle.berkeley.edu/sites/default/files/handouts/2018/topology_of_surfaces_2018_0.pdf
torus above. You decide to go for a walk. Trace your path. Be sure to exit some of the sides of the square and be careful about where you come back in! Do this several times. Draw some torus gluing diagrams of your own and practice some more. Example 3 (The Mobius Strip)¨. What happens if we start with a square and identify a pair of

https://mathworld.wolfram.com/Torus.html
An (ordinary) torus is a surface having genus one, and therefore possessing a single "hole" (left figure). The single-holed "ring" torus is known in older literature as an "anchor ring." It can be constructed from a rectangle by gluing both pairs of opposite edges together with no twists (right figure; Gardner 1971, pp. 15-17; Gray 1997, pp. 323-324). The usual torus embedded in three

https://en.wikipedia.org/wiki/Torus
A ring torus with aspect ratio 3, the ratio between the diameters of the larger (magenta) circle and the smaller (red) circle. In geometry, a torus ( pl.: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle.

https://www.quantamagazine.org/topology-101-how-mathematicians-study-holes-20210126/
A topologist is free to stretch and twist a shape. Even cutting and gluing are allowed, as long as the cut is precisely reglued. A sphere and a cube are distinct geometric objects, but to a topologist, they're indistinguishable. ... a set that allows addition and subtraction is called a group. So on the torus, for example, the one-dimensional

https://www.youtube.com/watch?v=YtPhIDibKxE
There are different ways of gluing solid tori along their boundaries, resulting in different manifolds (most common are S¹ × S² and S³). This video presents

http://www.geom.uiuc.edu/video/sos/materials/surfaces/gluing.html
B. f. Torus This object is easier to glue using the second page of templates (the wide gluing diagram). Students may recognize the shape as a donut, an innertube or a "torus" (not to be confused with the Zodiac sign "Taurus" which is spelled differently). This topology is often used in video games (go off one side come in the other). C. b

https://graphics.stanford.edu/courses/cs468-02-fall/notes/02.pdf
The torus is familiar to us as the surface of a bagel or a donut, as shown in Figure 2(a). We may describe a torus as a subspace of R3 geometrically. For example, a torus of revolution is created when we sweep a circle around ... one by gluing one end of a strip of paper to the other end with a single twist, as shown in the diagram (b). This

https://www.math.toronto.edu/drorbn/Students/Bettencourt/TorusKnotFibration/torusknot.html
Torus and Trefoil Knot. This preliminary visualization demonstrates how a square can be glued along its oposite edges to produce a torus. Use the sliders in the parameters panel to manipulate the visualization. Gluing along either the latitudinal or longitudinal will transform the square into a vertical or horizontal cylinder.

https://www.math.brown.edu/tbanchof/STG/ma8/papers/leckstein/Cosmo/torus.html
The Three-Torus. In the explanation of gluing, we looked at a square with edges glued.. If this square is glued only abstractly, it is called a flat two-dimensional torus. () But the attachments can also be made physically in 3-space out of a sheet of rubberThis produces a donut shape called a 2-torus.

https://math.stackexchange.com/questions/3914518/gluing-diagrams-for-tori
Gluing a 4k-gon by grouping the edges into groups of 4 like this produces a k-holed torus. Other gluings of 4k-gons also produce a k-holed torus, as long as they keep the surface orientable and ensure that all 4k vertices are glued together into a single point. This is a consequence of Euler's formula for CW complexes and the classification

https://discrete-notes.github.io/double-torus
We show how to build the torus by gluing faces of a polygon. The following picture is what is called a polygonal scheme of the double torus. It is an octagon whose edges have colors and orientations. There are exactly two edges for each color. The construction consists in the following operation: take two edges of the same color and glue them

https://www.open.edu/openlearn/science-maths-technology/mathematics-statistics/surfaces/content-section-4.5.2
4.5.2 n -fold toruses. We can use a similar technique to find the Euler characteristic of a 2-fold torus. If we cut the surface into two, as shown in Figure 95, and separate the pieces, we obtain two copies of a 1-fold torus with 1 hole, each with Euler characteristic −1. Figure 95. Consider any subdivision of the left-hand torus with 1 hole

https://math.stackexchange.com/questions/719055/manifolds-resulting-from-gluing-tori
To see this, let us first delete a meridian disk DM from T1 and then glue T1 ∖ DM and T2 by f(x, y) = (y, x). It results in a 3-ball, then gluing DM again produces S3. The other thing we can have is S2 × S1, which is realized when f is an identity. Note first that S2 × S1 is obtained by identifying two boundary components of S2 × [0, 1].

https://en.wikipedia.org/wiki/3-torus
The three-dimensional torus, or 3-torus, is defined as any topological space that is homeomorphic to the Cartesian product of three circles, In contrast, the usual torus is the Cartesian product of only two circles. The 3-torus is a three-dimensional compact manifold with no boundary. It can be obtained by "gluing" the three pairs of opposite

https://courses.cs.duke.edu/fall06/cps296.1/Lectures/sec-V-1.pdf
ine is obtained by gluing the two ends of an interval to the disk which is then shrunk to a point. For f(w) <a<f(z) the sublevel set is a capped torus. It has the homotopy type of a gure-8 obtained by gluing the two ends of another interval to the cylinder which is then shrunk to a circle. Finally, for f(z) <a we have the complete torus.

https://www.youtube.com/watch?v=0KwGE6hAtGc
Shows how you can get a torus from a square!

https://www.physicsforums.com/threads/is-this-the-correct-way-to-attach-moebius-strips-to-a-torus.897503/
A Torus with two Möbius Strips, also known as a double torus, is a topological shape that combines two Möbius strips with a torus. It has a doughnut-like appearance with two holes, and is a non-orientable surface, meaning it has no distinct inside or outside. 2. How is a Torus with two Möbius Strips different from a regular torus?

https://math.stackexchange.com/questions/214251/representation-of-s3-as-the-union-of-two-solid-tori
EDIT: this was worked out for me by the same team of gerbils that does Jordan Normal Form of matrices when I need that. Amazingly versatile. Take the coordinates as named $(w,x,y,z)$ to reduce subscripts. The sphere is $$ w^2 + x^2 + y^2 + z^2 = 1. $$ The common boundary is the Clifford Torus, $$ w^2 + x^2 = y^2 + z^2 = \frac{1}{2}.

https://www.researchgate.net/figure/Gluing-the-edges-of-a-hexagon-into-a-torus_fig5_324889547
The 3D surface thus appears as a torus with gaps where the sides of the tile meet. Next we present another surface, also a single Pied-de-poule tile, but with different tessellation type, a Klein

https://tomrocksmaths.com/2022/01/03/level-3-doughnuts-all-the-way-down/
Gluing two cross caps together gives you something that is equivalent to a Klein Bottle. So far we've looked at gluing a torus to a sphere, and a cross cap with another cross cap, but what happens when you glue a torus to a cross cap? Well, it turns out that attaching a handle to a cross cap is the same as attaching a Klein bottle to a cross cap.

https://www.youtube.com/watch?v=-be1ABH-ONM
This video demonstrate that gluing opposites sides of a hexagon is topologically equivalent to a torus.

https://math.stackexchange.com/questions/3255069/the-fundamental-group-of-the-gluing-of-two-genus-g-3-dimensional-handlebodie
Now, I want to see what happens when I glue two handlebodies along the boundary. This is where things get tricky for me. I want to use Seifert-Van-Kampen once again, but not entirely sure that I'm doing it correctly. ... But its fundamental group isn't the same a the fundamental group of a single solid torus (if I'm not mistaken). $\endgroup