This semester I taught an undergraduate course on topology (continuity, compactness, connectedness, and basic homotopy theory) and on the last day of class I decided to give a brief introduction to the theory of higher homotopy groups. For motivation, consider the classical Brouwer fixed point theorem:

**Theorem:** Every continuous function has a fixed point, where is the closed unit ball in the plane.

*Proof:*

Suppose has no fixed point point, meaning and are distinct for every . Define a function (where is the boundary circle of ) as follows. Given there is a unique line in the plane containing both and , so there is a unique line segment containing whose endpoints consist of and a point on . Define to be the endpoint on . Explicit calculations (using the continuity of ) show that is continuous, and moreover if then . A continuous function from a topological space to a subset which restricts to the identity on is called a *retraction*; we have shown that if there is a continuous function with no fixed points then there is is a retraction .

Let us use algebraic topology to prove that there is no such retraction. Let denote the inclusion map, so that is the identity. Passing to the induced homomorphism on fundamental groups, this shows that is the identity and hence is surjective. But is the trivial group since is contractible and , so could not possibly be surjective, a contradiction. QED

One might wonder if the argument above works for the closed unit ball . Indeed, the first part of the argument works in higher dimensions almost verbatim, and one gets that any continuous function gives rise to a retraction onto the boundary sphere. But the second part of the argument fails: the fundamental group of is trivial for , so there is no contradiction. The solution is to replace the fundamental group with the higher homotopy group ; whereas is the group of homotopy classes of continuous maps , is the group of homotopy classes of continuous maps (of course, all spaces, maps, and homotopies must have base points).

In the proof of the Brouwer fixed point theorem above, we only needed three properties of the fundamental group:

- Every continuous map induces a group homomorphism .
- If are homotopic continuous maps then .
- is not the trivial group.

The first two of these properties generalize to higher homotopy groups with almost identical proofs. The counterpart of the third property, namely that is not the trivial group, is considerably more difficult. One typically computes using covering space theory, but there is no counterpart of covering space theory for higher homotopy groups. (Well, such a theory does exist in a manner of speaking, but it is much more complicated than covering space theory.)

To actually compute one needs some rather powerful tools in algebraic topology, such as the Freudenthal suspension theorem or the Hurewicz isomorphism. The difficulty of this computation is still a bit mysterious to me, and was the subject of one of my recent MathOverflow questions. Even the more modest goal of proving that is non-trivial is quite a bit more challenging for than for . Nevertheless, I came up with an argument in the case based on vector calculus which is suitable for undergraduates; I don’t think I’ve seen this exact argument written down anywhere else, so I thought I would write it up here. It is adapted from a more standard argument involving Stokes’ theorem on manifolds which works in any dimension (but which requires a semester’s worth of manifold theory to understand).

I will prove the following statement:

**Main Theorem:** There is no continuous retraction .

This alone is enough to prove the Brouwer fixed point theorem for without having to worry about higher homotopy groups, but in fact it implies that is nontrivial. Pick a base point and consider the identity map . This determines a class , so if is the trivial group then there is a base point preserving homotopy between and then constant map given by . This homotopy is a continuous map which satisfies:

- for all

Given such a homotopy, define by if and . It is not hard to check that is a retraction, contradicting the main theorem.

To prove the main theorem we need a technical lemma:

**Lemma:** If there is a continuous retraction then there is a smooth retraction.

The proof of this lemma uses some slightly complicated analysis, but ultimately it is fairly standard; see the final chapter of Gamelin and Greene’s “Introduction to Topology”, for example. The only other non-trivial input required to prove the main theorem is the following classical result from vector calculus:

**Divergence Theorem:** Let be a compact subset of whose boundary is a piecewise smooth surface , let denote the outward unit normal field on , and let be a smooth vector field on . Then:

Here (“divergence”) is the differential operator .

*Proof of Main Theorem:*

By the previous lemma it suffices to show that there is no smooth retraction from to , so suppose is such a retraction and denote its component functions by . Thus may be viewed as a smooth vector field on ; since for we have , , and for every .

Consider the smooth vector field where is the gradient operator. We will compute the integral of over in two different ways and get two different answers, giving a contradiction. Both computations will use the divergence theorem:

The first computation uses a bit of vector calculus. By the product rule for the divergence of a function multiplied by a vector field, we have:

The second term on the right-hand side vanishes by the product rule for the divergence of the cross product of two vector fields:

Here we used the fact that the curl of the gradient of any smooth function is the zero vector.

According to the standard “triple product” formula from vector algebra, the first term is the determinant of the Jacobian matrix associated to whose rows consist of , , and . I claim that this determinant is zero. Since takes values in we have that ; differentiating both sides of this equation with respect to gives , or equivalently . Similarly and , so the vectors , , and are all orthogonal to the same nonzero vector and hence there is a nontrivial dependence relation between them. But , , and are the columns of , so it follows that . We conclude:

by the divergence theorem. (The various identities used in this argument all appear in the Wikipedia page on Vector Calculus Identities, with the notation and .)

Now let us compute the same integral using the fact that on . Using , , and we calculate that and hence . By the divergence theorem we get:

This is a contradiction. QED

## Gender and the Mathematical Community

February 11, 2013

pwsiegel General Audience commentary 12 Comments

I still haven’t posted all that much in this blog, and essentially nothing research-related. I’ve been writing a bit offline, and I’ll probably adapt some of what I’ve been thinking about into blog form fairly soon. In this post I’d like to address some issues related to sexism and gender bias in the mathematical (and perhaps broader scientific) community. I think about these issues rather often, but I’m writing about them now because of recent posts in The Accidental Mathematician (Izabella Laba’s blog) and mathbabe (Cathy O’Neil’s blog).

The thrust of Izabella Laba’s post (entitled “Gender Bias 101 for Mathematicians”) is that gender bias in the mathematical community is not limited to a few grouchy old codgers, but rather that it is a systematic cultural and psychological phenomenon which afflicts everybody. There are two potentially controversial assertions implicit in this statement:

The first assertion is pretty hard to argue with, though I’m sure some people still try. Every math department with which I have been affiliated is *massively* male dominated, and there is ample evidence that hiring practices, salaries, journals, etc. are stacked in favor of men. I’m not going to try to document or justify this in any detail because I don’t have the facts available at my fingertips and because the issue has been argued to my satisfaction elsewhere (e.g. in the Accidental Mathematician).

The second assertion might be more surprising to some, and it’s the one I want to discuss here. Izabella Laba’s post quotes a recent study in which faculty from research oriented universities were presented with applications for a lab manager position with randomly assigned male or female names. The study found that a given application with a male name at the top was consistently rated more highly than the same application with a female name. Interestingly enough, the pattern was independent of the gender of the faculty evaluator: female professors were just as biased as male professors. Cathy O’Neil contributes another study which shows that 15-year-old girls outperform 15-year-old boys in science exams in some countries but not others (not in the United States), indicating that gender gaps in science are cultural rather than biological.

Both of these studies are quite compelling, and I’m sure there are others which point to the same conlusion. My intention is to participate in this discussion subjectively rather than objectively. In short, I am going to use the rest of this post to analyze my own gender-oriented biases. Something feels a bit self-indulgent about this exercise, but I think it will be healthy for me even if it isn’t useful for anyone else.

I will begin by admitting outright that I am biased against women. I consider myself to be a pretty progressive guy – perhaps even more progressive than most – and I think that most people who know me would say that overall I do a good job of treating women with the same respect with which I treat men. But this is not because I don’t have biases, it’s because I work very hard to identify them and eliminate them or at least minimize their impact on my behavior. I am unqualified to generalize my own psychological observations to everyone else, but I suspect that it is neurologically almost impossible for a person socialized in 20th or 21st century American society to avoid gender biases: we are bombarded with overt and covert messages about gender constantly and starting at a very young age. Given what I have been learning lately about how insignificant our conscious thought processes are in comparison to our subconscious psychological machinery, these messages must take their toll.

What forms do gender biases take? There are many answers with varying applicability to me. Here is a non-comprehensive unordered list that I have assembled from reading things online, talking to people, and making my own observations.

Intelligence and Competence Bias:This is simply the assumption that women are less intelligent or less competent than men. I have heard numerous stories in which Andrew launches into a lengthy explanation to Barbara about a subject in which Barbara is more of an expert than Andrew. Here is a particularly cringe-inducing example of this. I tend not to offer unsolicited explanations to men or women very often, and when providing solicited explanations I usually make an effort to identify my audience’s background, so I don’t think I am terribly guilty of this particular behavior. Instead, I notice this bias in myself when I am seeking an expert on a particular subject and I am presented with a male option and a female option. Sometimes I catch myself behaving or thinking according to the assumption that the male expert is more knowledgable or more adept than the female expert even if I have no particular reason to make such a judgement. I have to force myself to think deliberately about what I know and don’t know when making these sorts of comparisons.Experience Bias:Lately I have started noticing a disturbing pattern in my judgements about a person’s age, experience, education level, etc.: my estimates are consistently lower than reality for women and higher than reality for men. I have heard many stories from women in which they are demoted from faculty member to graduate student or from graduate student to undergraduate by a male interlocutor, and I am embarrassed to admit that I have done this before. I have also heard stories in which a female graduate student or faculty member has been assumed to be a secretary or staffperson; I don’t necessarily consider this to be a “demotion,” but I doubt I would appreciate it if it happend to me. These days I try to avoid guessing somebody’s position or experience level at all, and if I do make a guess it’s generally “faculty” regardless of gender (in a university setting). Still, it requires conscious effort on my part.Common Ground Bias:This is the assumption that, all things being equal, I will have more in common with a male than a female. This bias is fairly understandable – there are, after all, real biological and social differences between men and women – but I think it has unfortunate consequences in an academic setting. Few of my mathematical conversations with my peers begin, “Hi, my name is Paul. Would you like to have a conversation about elliptic cohomology?” Instead, they typically begin with the typical introductory social graces and lead into mathematical territory after a basic rapport has been established. This rapport is more difficult to establish with a person with whom I assume I will have a harder time identifying with before the conversation even begins, and consequently I am more likely to engage in mathematical conversations with my male colleagues than my female colleagues. I don’t know how socially isolated women in math departments feel, but I suspect that it’s more of a problem than I realize. My plan for reducing the impact of this bias is to simply be more bold and less awkward about engaging people in conversation, but this isn’t always easy.Sexual Biases:I am a heterosexual man who is attracted to intelligent and ambitious women, and the women that one finds employed in a math department often fit this description. Sexual attraction is firmly rooted in extremely powerful subconscious processes, and I am certain it affects my interactions with my female colleagues in ways that I don’t fully understand. If nothing else, it consumes some measure of my mental energy that is liberated when I’m interacting with men. It seems very hard to deal with the subconscious aspects of this bias, but I long ago adopted a mechanism which at least helps me manage the factors that are under my control. I decided early on in graduate school that I would categorically avoid romantically pursuing anyone in my own department. This allows me to sidestep the hazards associated with workplace romances in general, but mainly it helps me ensure that I treat all of my colleagues as professionally as possible. I don’t know how often the average woman in a math department is forced to deal with romantic overtures from her male colleagues, but given the highly skewed gender ratios I’m guessing it’s more than I imagine. I am also largely ignorant of the consequences of this behavior.I’m sure there are other biases worth mentioning, but this list feels like a good start. One interesting supplementary observation about biases in general is that thinking about them leads to an unfortunate feedback loop: worrying about biases against women affects my behavior toward women. I think this effect is fairly minimal in comparison to the consequences of ignoring my biases and failing to monitor my behavior at all, but it’s there all the same.

My final remark about this subject is that there are many other bias issues which are also largely ignored by the mathematical and scientific community. I have encountered some discussion of racial bias in science, but I have heard almost no discussion about biases related to sexual orientation. If anyone reading this is aware of any studies or references about these issues, I would be interested in seeing them. Also, in this post I have focused on the effects of bias on my interactions with my colleagues, but the way my biases manifest themselves in my teaching is a whole other subject which I might take up in the future.