Lefschetz

By Marlin Eller

Click! and you're awake. It happens that fast. One moment you are blissfully unaware and your ears are picking up the early morning sounds of a garbage truck, then you realize that you had been sleeping and are no longer. You haven't moved. Nothing has changed except that now you know that you're lying there with your eyes closed not yet alert but no longer unconscious. There is no way back now, you are on a one way street to awareness, awareness of self and surroundings

I realized that I was awake and pulled an exposed arm in under the blanket. It must be early. I don't usually wake up this early. I hadn't opened my eyes yet but through the lids I could tell it was dawn, just before sunrise, too bright for night but not yet day. No sound of rain, no drizzle off the roof. A car rolls by and from the whir of the tires on the pavement I know it's dry. It must not have rained all night. I remember it had just stopped raining when I had bicycled home from the library last night. I had been working late on a differential geometry problem: Prove that a smooth map from a compact manifold to itself has at most a finite number of Lefshetz fixed points. I knew in my heart why it was true but hadn't yet worked out the formal proof when they flickered the lights to announce closing time. The problem continued to play in my mind as I peddled down the dark damp back streets to home. But today would be dry. It was getting brighter and there was more street noise as people were trundling off to work. The air was crisp and cool and not even very humid, like the air in the desert. It was going to be a nice clear day, not a common occurrence in a Seattle winter. She stirred slightly and snuggled up a little closer and I could feel the warmth of her bottom pressing into my thigh.

Suddenly there was something wrong. What was it? Something did not fit. I had gone to bed alone last night. Someone had gotten into my bed! I had biked home, locked it in the basement, had a quick bite to eat and had come upstairs ... Wait!

The car noises were coming form the wrong direction. I was on the ground floor. Flexing my back I felt the mattress. It wasn't mine. I'd crawled into someone else's bed. The question was who. A quick review of my female acquaintances left none with whom I was sufficiently familiar for this. I tried to remember a party. I must have gotten really bombed. I'd heard of that sort of thing happening but had never experienced it directly. Where was the party? If I could remember whose house it was at I'd be able to deduce who this is.

There had been no party. I don't go to parties and I don't get drunk. I tried to make up a party, I tried to invent one but nothing came. I remembered the library clearly, and coming home but after that there was nothing. I'd gone to sleep and now I'm awake. It was time to open the eyes. The scene that greeted them was just what I expected, I was in bed with a woman, her back turned toward me. It was not my room and it was no room I recognized. The architecture was wrong. I probably wasn't even in Seattle. The air was wrong. She was a plump lady though not fat wearing a white night gown. She had shoulder length mousy brown hair with some streaks of gray. I guessed that she was in her late forties or there abouts. I had a very good feeling about her though I could not place the reason. Then I saw my hand.

I saw the back of my hand. I saw the skin draped around the tendons that control my fingers. I saw the triangular pattern of hair follicles that sparsely populated the side where the little finger attaches. I saw the arteries bulging with blood bearing another load of oxygen out to the fingertips. I saw the fine wrinkles across the folds of skin and the splotches of age. I saw the hand of an old man.

Suddenly I was aware of my body as it lay there. I was old. I was 50 years old. My toes felt 50 years old. I inhaled with lungs that had been bellowing in and out for 5 decades. My heart had been pumping for half a century. My eyes... well, it was early yet, maybe they'd focus a little more clearly as I woke up.

She let out a satisfied half asleep, "Ummm..." as she stretched and then snuggled up closer to me. The heaviness of my body was matched by the realizations I was coming to. She was my wife. She had to be. I knew it, the way you know something in a dream, where with no evidence to back it and no context to frame it you recognize someone or something and just know what it is.

"What time is it?" she asked.

I glanced over at the clock, "Six thirty two."

"You'd better get up or you'll be late for your class."

Late for my class. So I was still in school. I didn't know where I was, how I got there or even how old I was, but I was still in school. What a joke! I didn't even know what class I was taking. What if we have an exam today? I don't even remember what I've been studying, unless it's Lefschetz fixed points on compact manifolds. No, it couldn't be that. I was taking that when I was in grad school some 25 years ago. I must have gotten further than that in all this time. I can't go to class like this. "I think I'll just stay home and cut it today."

She rolled onto her back, yawned and rubbed her eyes and I got to look at her face for the first time. There were small, very small wrinkles at the corners of her mouth and eyes and pleasantly shaped features. She was a handsome woman and had a face with character. "You can't do that, your students are counting on you."

"Yeah, I know." So, I'm a teacher. A goddamn teacher! After all those years in grad school, I made it. You see, a Lefschetz fixed point is one where the eigenvalues of the Jacobian are all +/- 1. That definition had really pissed me off when I had looked it up because I couldn't remember what eigenvalues were so I had to look that up too. I wonder what I'm teaching? and where? I knew I should be panicking but for some reason my instincts told me to play it cool and not let on how little I knew or how confused I was. It was probably math or physics of computer science (Oh God, what if it's boys PE.!) I decided to see if I could direct the conversation toward answering some of my questions so I nonchalantly commented, "Yeah, they're counting on me, I guess. By the way, where is it I teach?"

She turned to look at me and held my head with both her hands as she stared deeply into my eyes. Hers were dark brown, very deep with a rich sparkle. She was a beautiful woman. Her face betrayed a calm buoyant spirit with a love and concern for people. I could easily fall in love with such a woman. I probably had at one time. I wondered when I had met her. Was it during grad school? Just after? Then she grinned, tousled my hair, and laughed as she said, "You teach at the University of Arizona at Phoenix. This semester you've got two calculus classes and one section of advanced algebra." then she leaned over and kissed me hard on the lips. It was a powerful kiss, not a light teasing sensual kiss, but one of deep friendship and love. I felt myself getting an erection. Then she rolled away and sat up to get out of bed. "And now, my little absent minded professor, why don't you see if you can dress yourself while I go get us some breakfast."

She slid on her slippers and headed toward the doorway. I was in awe. I was in shock. I was living in Arizona! I didn't know where I was or how I got there. I'd apparently spaced out some 25 years of my life. I'd turned into a zombie one night after working too hard on my differential geometry and was just now waking up. 25 years evaporated. I had to find out her name. "Honey, before you go, tell me something..."

She stopped in the doorway and turned to look at me. "What?"

"What's your ... um ..." I couldn't ask. "Tell me, have they been good years?"

She just looked at me and smiled, a knowing smile, an affirmative smile, an I'm-not-gonna-tell-you smile. "You'll be late." Then she turned to go saying, "Don't forget that Dorothy will be down this weekend."

"Dorothy?" I called after her as she disappeared down the hallway.

"Yeah, you remember, our daughter, off at Berkeley." Her voice came from a clatter of pans off to the right. That must be where the kitchen is.

"Oh yeah," Dorothy. In college. Dorothy. Dorothy Eller. I wonder what's her middle name. I wonder what she's majoring in? I majored in math. Yeah, heavy stuff that math. You start working on a problem and you sort of forget what's going on nearby and focus in on what you're doing. It's not a bad thing to study if you don't let it get out of hand. 25 years and a daughter too. Maybe it's time I took some time off and spent some of it with my family. Kind of get to know them better. You see, a compact manifold is sort of like being small so that if you put an infinite number of points in one, they can't just space themselves apart so they have to cluster somewhere, But Lefschetz fixed points can't cluster because although they twist up the space around them the don't stretch it so that in some small neighborhood around each one...

"Your egg is done. Come and get it."

Yeah, I'd better get out of bed. I sat up and looked around me. I was in my room in Seattle. My differential geometry book was there on the table where I'd left it when I’d come back from the library. I was back in my young body and felt very old. The whole scenario had been too real. I don't want to risk it. I've got to slow down or I'll go crazy. Then it occurred to me that I'd solved the problem. You see, you can show that there is an open neighborhood of each Lefschetz fixed point that contains no others...