Homology

“So,” Ellie began, “what do you know about n-simplices?”

“Nothing.” Maud said. “I’ve told you this already. I learned about space groups to understand crystal structures. That’s as far as I got in math.”

“Great!” Ellie turned away from Maud to face her crotch. “Let’s start.” She took an index finger and used it to lightly tap Maud’s left nipple. Ignoring the reaction, she innocently continued.

“The 0-simplex is the first one. The number corresponds to the local dimension. Being 0-dimensional, it’s just a point. We’ll label it ‘0’.”

She leaned over to suck on the nipple she tapped for a second, breaking the suck with a lick.

“Going up to the 1-simplex, you have to add another 0-simplex, so another point, in a way that’s ‘orthogonal’ or ‘perpendicular’ to the existing simplex if you want it to be nondegenerate. But everything’s orthogonal to a single point, so I can place the second point anywhere. Let’s just say, oh, here.”

Ellie did the same suck then lick with Maud’s right nipple.

“We’ll label this point ‘1’. Then we connect them with a 1-dimensional space, that is, a line. The direction is important. We’ll respect the obvious ordering on the vertex labels.“

She traced a line from point 0 to point 1 with her tongue, down the side of Maud’s left teat and up the right.

“Now we want to place the third point in a roughly canonical way. Let’s start with the midpoint of the existing simplex.”

She placed the point of her index finger between Maud’s teats.

“Then let’s draw a perpendicular line.“

She did so, tracing a line down her abdomen until she hit the base of Maud’s clit.

“Perfect. That we can label ‘2’. Then we fill in the two new 1-simplices.”

She licked a line from Maud’s left nipple up to the base of her clit, and then did the same from her right nipple. She smiled as she felt Maud’s shudders and saw the proof of her excitement.

“Then we fill in the 2-dimensional space between them.”

Ellie caressed the area of Maud’s crotch that formed by the triangle of her teats and clit, then planted a kiss in the center. She placed a finger on the spot she kissed.

“So that’s a 2-simplex, embedded in the 2-dimensional space of the surface of your sexy body. To go up again, we need to go perpendicular to it.”

She traced a line upwards into the air above the aforementioned triangle.

“We can label this ‘3’, and you can imagine the tetrahedron that gets filled in. That gets us up to a 3-simplex, embedded in the 3-dimensional space we’re so familiar with. But to go up again, you have to go ‘perpendicular’ from this 3-dimensional solid you’re in the middle of, which doesn’t make sense. So we have to go ‘up’ to 4-dimensional space to get the fifth vertex we need to make a 4-simplex. And so on, and so on. That way you have a whole family of basic shapes you can use to understand higher-dimensional geometries with. The n-simplices.“

She looked down at Maud.

“Sorry, I couldn’t figure out how to make that part sexy. But I hope that it was satisfying conceptually.“

“It was.” Maud said in her monotone. “I like this lesson. I think I learned something.”

Ellie smiled. “Yeah, I find that nipple play really helps with the retention. Unfortunately, it’s a very difficult methodology to scale up to the classroom level.”

Maud looked at her without any expression in a way that meant rolling her eyes. Ellie laughed.

“Okay, now on to singular homology groups.”

“Oh.” Maud realized. “That wasn’t it?”

“Yeah, sorry…” Ellie said apologetically. “It gets a lot worse from here.” A rare frown flashed across Maud’s face as Ellie continued. “Now, I’ve been using ‘n-simplex’ as shorthand for ‘continuous map of the n-simplex’, which is what I was drawing on your body all along. I mean, ‘drawing on’ is a pretty good intuition for the idea of a continuous map without getting technical. So here are two 2-simplices.”

She drew a clockwise triangle from a point on Maud’s abdomen, to her left nipple, to her right, then back down. Then she drew another clockwise triangle, going from the base of her clit to her right nipple, then left, then back up.

“Of course, the 2-simplices are the filled-in triangle! I only drew their oriented boundaries. Now, a singular 2-chain is any formal sum of 2-simplices, so these two triangles together is an example. But notice what happens when we consider their boundaries together! We have six edges between the two triangles, but this edge”—Ellie drew a line from Maud’s right nipple, across her teats to her left, then back—”is repeated, with the directions canceling out! So the boundary of this 2-chain is actually this.”

She drew a diamond, starting from that point on Maud’s abdomen, brushing over her left nipple, touching the base of her clit, then brushing over her right nipple on the way back.

“It’s a 1-chain! Four 1-simplices added together. But I need a 2-chain with no boundary.”

“No boundary?”

“Yeah!” Ellie barely paused. “So imagine the four sides of a tetrahedron, like the 3-simplex we considered earlier.” She re-drew its base, the clockwise triangle going from Maud’s right nipple, to left, to base of clit and back. “We have this 2-simplex at the base, then we add three more of them for the sides.” She drew three triangles, all with their apex hovering above Maud’s crotch, their bases going from left nipple to right, from right nipple to clit, then from clit to left nipple, respectively. “All twelve edges of these four triangles are canceled out by a counterpart going in the other direction! So these four triangles form a singular 2-chain with no boundary. A singular 2-cycle.”

“Okay.” Maud said.

“But then imagine that we were drawing these on the surface of a tetrahedral solid. Say, a rock! You like those.”

“Rocks are very rarely tetrahedral.” Maud said.

“It could really be any solid shape! We’re just ‘drawing’ a 3-simplex made out of rock.” Ellie rolled her eyes. “The point is, if we consider the rock as a topological space, the rock itself can be thought of as a 3-simplex, thus, a singular 3-chain, whose boundary is the 2-chain we were just talking about. So this singular 2-cycle is also a singular 2-boundary.”

“You’re using the word ‘singular’ a lot.” Maud said. “So all cycles are boundaries?”

“No, that’s the thing! Like, what if we imagined that 2-cycle as on the surface of a tetrahedral geode?”

“A tetrahedral geode is even less likely than a tetrahedral rock.” Maud commented.

Ellie ignored her. “Then we can’t consider the ‘inside’ to be a single 3-simplex, because there’s a hole in it!“

“Cavity.” Maud corrected.

“Fine, cavity.” Ellie smiled. “So the surface of a geode is a 2-cycle that’s not a 2-boundary. So it represents a non-trivial homology class. An element of H_2, the second homology group.”

Maud didn’t say anything in response.

“You know, a hole!” Ellie laughed. “Actually, you’re right, ‘cavity’ is a better word for it. So you can think of H_2 as ‘counting’ three-dimensional cavities in a space, since each cavity can have a 2-cycle drawn around it that’s not a 2-boundary.”

“That was a lot of work to talk about counting cavities.” Maud said.

“Well, this was just the simplest possible example! Everything embedded in Euclidean 3-space.” Ellie shrugged. “But what I really wanted to talk about was cohomology groups.”

“There’s more?” Maud’s voice was actually inflected with dread. Ellie felt really bad, but could not stop herself now, due to the sunk cost fallacy.

“Yeah! I mean, you can add n-chains together, which means they form a group, C_n. So if you pick a coefficient group G, you can take the dual group Hom(C_n,G), calling the homomorphisms between them cochains, and the group of cochains C^n := Hom(C_n,G). This is an application of the contravariant functor Hom(-,G), which means it applies to the boundary maps \delta_*: C_* \rightarrow C_{*-1} as well, reversing their direction to form coboundary maps \delta^*: C^* \rightarrow C^{*+1}. We can analogously define dual versions of cycles and boundaries as well, that is, n-cochains in C^n are considered cocycles if they get mapped to zero in C^{n+1}, and coboundaries if they come from something in C^{n-1}. This way, by quotienting coboundaries from cocycles you get cohomology classes…“

Maud had clearly stopped listening long ago.

“But imagine!” Ellie put a hand on Maud’s upper foreleg, forcing her to make eye contact. “Imagine, say, the fifth cohomology group of a six-dimensional oriented, closed manifold, M? Something governed by not merely the embeddings of a five-dimensional space into a sixth-dimensional one, but also by a layer of algebraic abstraction on top of that?”

“No.” Maud said. “I don’t want to.”

“I don’t, either! That’s a terrible thing to imagine!”

“Or talk to your marefriend about.”

“I agree,” Ellie said apologetically. “And I am very sorry. But what I really want to talk about is something called Poincaré duality.“ Ellie emphasized the uvular “r” to the point of parody.

“You don’t have to say it like that. I don’t think even Prench ponies say it that way.”

“I like being terrible, though.” Ellie grinned. “But here’s what’s wonderful. Poincaré duality says that H^5(M) is actually isomorphic to H_1(M)! Isn’t that amazing? I mean, H_1(M) is just the homology classes that count two-dimensional holes. That’s pretty simple. They’re just rings.”

Ellie formed a circle with her thumb and middle finger. She then took the circle and inserted the tip of Maud’s nearly-flaccid clit into it, beginning to rub her up and down. She felt Maud’s spirits lift underneath her.

“Can you imagine it? You start with something so arcane and difficult to explain even the definition of, living in a dimension beyond human intuition. There’s no beauty there in that realm. No appeal. Just cold mechanics. But then…”

Ellie bent down to take the head of Maud’s clit into her mouth, running her tongue up and down her as she began to stiffen. She pulled back, observing the results of her work, and smiled.

“Then it’s transformed into something elegant. Something that can sing to the visual cortex. Geometric intuition, built out of what was needed for our minds to understand the physical space around us. The incarnation of the abstract. Word made flesh.”

She turned to look down at Maud.

“What do you think?” she asked.

“I think that I don’t want you to stop.” Maud said. “I also think that was very pretentious.”

Ellie laughed, and kissed her.

“You’re Poincaré duality, you know,” Ellie said, not pronouncing it terribly this time. She looked back down at her hand, and Maud’s stiffening clit. “You take something and you recontextualize it, because that’s what isomorphism is. The mechanics are the same, but it’s the names and the intuition that change. The feeling.“ Ellie sighed happily. “Something that I would otherwise see as having nothing to do with beauty, from a realm not my own, is suddenly made familiar. Understandable. Appealing. Beautiful. Things I never would have imagined I’d ever see the beauty in.” She drank in the sight for a few moments before turning around to look Maud in the eyes, smiling warmly. “But H_1(M) has nothing on the smell of your cum drying on my face.”

Maud’s face reddened. Ellie grinned lecherously.

“I don’t like you.” Maud said.

“Something in my hand says otherwise,” Ellie sang, terribly and very off-key. “But don’t worry. I’m done.“ She bent over to kiss Maud on the lips. “Don’t worry. No more math.” She planted more soft, reassuring kisses on her muzzle and cheek, down to her neck. “Thank you for suffering through this for me.”

“It wasn’t that bad.” Maud lied. Ellie smiled, and gazed into her eyes for a moment before kissing her again.

“You really are so sweet, lying for me that way.” Maud did not object. “But don’t worry. No more math. Really.” Another reassuring kiss. “Just going to have sex with you now.”

Maud smiled.