In this section, Professor Haynes Miller describes the reading responses students complete for each paper they do not present.
Each participant submits a “reading response” to each paper he or she is not presenting, in advance of the talk, either on paper or by email. I reply by email to each reading response. Initially, students are often unsure of what is expected of them. I explain that I don’t want them to provide me with a sequence of statements and theorems; rather, I want to hear what they’re reminded of by this paper, what connections they see, or what surprised them. It’s an opportunity for them to engage me in a discussion about those connections and revelations.
Although they all produce deep and engaging responses, they are all so different! This is what makes responding to their writing so interesting for me as an instructor. Sometimes students pose questions in their responses. I usually cannot answer their specific questions, but I try to point them to resources and references that will help them. I almost always use their questions as springboards for writing about a related topic. Quite often we engage in a brief “back and forth” email conversation. I think it’s fantastic—It means students are really thinking and learning.
Below, Professor Haynes Miller and his student, Isabel Vogt, share their first reading response exchange.
Response to: Steenrod Operations (PDF - 1.7MB)
Dear Professor Miller,
Here is my response:
Before reading your notes I had seen the definition of the Steenrod squares given in Mosher-Tangora “Cohomology Operations and Applications in Homotopy Theory”. This approach makes more concerted use of the cell structure of B_(Z/2Z) = RP^\infty and E_(Z/2Z) = S^\infty and defines the Sq^i in terms of cup_i products on integral cochains.
I think that I understand the more geometric construction based upon the pi-adic construction on X that you present better than this construction, but I still have very little intuition for these operators geometrically based upon the definition. I was hoping you could help me understand this.
Supposed X is a smooth manifold. Then any cohomology class in H^q(X) corresponds under Poincare duality to a linear combination of codim q submanifolds. For u_Y a class in H^q(X) corresponding to a cidim q sub manifold Y, you can compute u_Y \cup u_Y as the cohomology class of the self-intersection of this submanifold — e.g. as the cohomology class of the zero locus of a section of the N_(Y/X) (or as the index q Stiefel-Whitney class of the normal bundle).
I’ve heard that Sq^i(u_Y) corresponds to the index i Stiefel-Whitney class of Y. This gives me more intuition for these cohomology operations.
Is there a way to understand this in terms of the construction you gave?
A more modern treatment of the cup-i approach is contained in the statement that the cochain functor (to DG algebras) lifts to a functor to E_\infty algebras. See Mandell, Cochain Multiplications, Am J Math 2002. This approach has its own virtues; for example, the cohomology of any co-commutative Hopf algebra comes equipped with Steenrod operations. (But they don’t satisfy Sq^0=1; instead, they are induced by the Frobenius in the dual Hopf algebra.)
I think a reference for the construction you describe is McCrory, Cobordism operations and singularities of maps, BAMS 82 (1976) 281ff. I guess it’s related to the statement, which I think we will hear from [Student Name], that SW classes correspond under the Thom isomorphism with Steenrod operations on the Thom class.
The complete correspondence between Professor Miller and Isabel is available in the table.