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Kevin P. Knudson

Publications and source records attributed to Kevin P. Knudson.

17 recordsLinked to original sources
arXiv2024

Discrete Morse theory for open complexes

We develop a discrete Morse theory for open simplicial complexes $K=X\setminus T$ where $X$ is a simplicial complex and $T$ a subcomplex of $X$. A discrete Morse function $f$ on $K$ gives rise to a discrete Morse function on the order complex $S_K$ of $K$, and the topology change determined by $f$ on $K$ can be understood by analyzing the topology change determined by the discrete Morse function on $S_K$. This topology change is given by a structure theorem on the level subcomplexes of $S_K$. Finally, we show that the Borel-Moore homology of $K$, a homology theory for locally compact spaces, is isomorphic to the homology induced by a gradient vector field on $K$ and deduce corresponding weak Morse inequalities. The gradient vector field on $K$ provides a novel alternative to compute Borel-Moore homology.

math.AT↗
arXiv2020

Approximate triangulations of Grassmann manifolds

We define the notion of an approximate triangulation for a manifold $M$ embedded in euclidean space. The basic idea is to build a nested family of simplicial complexes whose vertices lie in $M$ and use persistent homology to find a complex in the family whose homology agrees with that of $M$. Our key examples are various Grassmann manifolds $G_k({\mathbb R}^n)$.

math.AT↗
arXiv2007

A refinement of multi-dimensional persistence

We study the multi-dimensional persistence of Carlsson and Zomorodian and obtain a finer classification based upon the higher tor-modules of a persistence module. We propose a variety structure on the set of isomorphism classes of these modules, and present several examples. We also provide a geometric interpretation for the higher tor-modules of homology modules of multi-filtered simplicial complexes.

math.AT↗
arXiv2005

Relative completions and $K_2$ of curves

We compute the completion of the special linear group over the coordinate ring of a curve over a number field $k$ relative to its representation in $\slnk$, and relate this to the study of $K_2$ of the curve.

math.KT↗
arXiv2004

On the kernel of the Gassner representation

We study the Gassner representation of the pure braid group $P_n$ by considering its restriction to a free subgroup $F$. The kernel of the restriction is shown to lie in the subgroup $[Γ^3 F,Γ^2 F]$, sharpening a result of Lipschutz.

math.GT↗
arXiv2004

Invariant chains and the homology of quotient spaces

For a finite group G and a finite G-CW-complex X, we construct groups H_\bullet(G,X) as the homology groups of the G-invariants of the cellular chain complex C_\bullet(X). These groups are related to the homology of the quotient space X/G via a norm map, and therefore provide a mechanism for calculating H_\bullet(X/G). We compute several examples and provide a new proof of ``Smith theory": if G=Z/p and X is a mod p homology sphere on which G acts, then the subcomplex X^G is empty or a mod $p$ homology sphere. We also get a new proof of the Conner conjecture: If G=Z/p acts on a Z-acyclic space X, then X/G is Z-acyclic.

math.AT↗
arXiv2003

Homology of linear groups via cycles in $BG\times X$

Let G be an algebraic group and let X be a smooth integral scheme over a field k. In this paper we construct homology-type groups $H_i(X,G)$ by considering cycles in the simplicial scheme $BG\times X (an idea suggested by Andrei Suslin). We discuss the basic properties of these groups and construct a spectral sequence, beginning with the groups $H_i(Δ^j,G)$, which converges to the etale cohomology of the simplicial group BG. These groups are therefore connected with the study of Friedlander's generalized isomorphism conjecture. We also compute some examples, focusing in particular on the case X=Spec(k). In the case where k is the real numbers, there is a connection between the groups $H_i$ and the Z/2-equivariant cohomology of the classifying space of the discrete group $G(\mathbb R)$.

math.KT↗
arXiv2003

The homology of invariant group chains

If $Q$ is a group acting as a group of automorphisms of another group $G$ (with finite orbits), denote by $C_*(G)^Q$ the subcomplex of $Q$-invariant chains in the bar complex $C_*(G)$. In this paper, we study the homology of the complex $C_*(G)^Q$ and compute several examples.

math.AT↗
arXiv2001

Relative completions and the cohomology of linear groups over local rings

We study the completion of a group relative to a Zariski dense representation in a reductive algebraic group over a field $k$. The characteristic zero case was worked out previously by R. Hain; we extend his results to arbitrary characteristic. The primary application is to the study of the cohomology of groups such as $SL_n(k[[T]])$.

math.KT↗
arXiv1998

On the K-theory of elliptic curves

Let A be the coordinate ring of an affine elliptic curve (over an infinite field k) of the form X-{p}, where X is projective and p is a closed point on X. Denote by F the function field of X. We show that the image of H_*(GL_2(A),Z) in H_*(GL_2(F),Z) coincides with the image of H_*(GL_2(k),Z). As a consequence, we obtain numerous results about the K-theory of A and X. For example, if k is a number field, we show that r_2(K_2> (A) x Q)=0, where r_m denotes the m-th level of the rank filtration.

math.KT↗
arXiv1998

Amalgamated free products, unstable homotopy invariance, and the homology of SL_2(Z[t])

We show that if R is an integral domain with many units, then the inclusion E_2(R) --> E_2(R[t]) induces an isomorphism in integral homology. This is a consequence of the existence of an amalgamated free product decomposition for E_2(R[t]). We also use this decomposition to study the homology of E_2(Z[t]). We show that H_i(E_2(Z[t]),Z) contains a countable rank free summand for each i>0 and that this summand maps nontrivially into H_i(SL_2(Z[t]),Z); hence, the latter is not finitely generated. This improves on a result of Grunewald, et.al., which states that SL_2(Z[t]) has free quotients of countable rank (and hence, H_1(SL_2(Z[t]) is not finitely generated).

math.KT↗
arXiv1998

Integral homology of PGL_2 over elliptic curves

Let E be an elliptic curve defined via a Weierstrass equation F(x,y)=0 over an infinite field k. Denote by A the coordinate ring of E. In this note we compute the integral homology of PGL_2(A). We obtain a rigidity result as a corollary.

math.KT↗
arXiv1998

Low dimensional homology of linear groups over Hensel local rings

We prove that if R is a Hensel local ring with infinite residue field k, the natural map H_i(GL(n,R),Z/p) ---> H_i(GL(n,k),Z/p) is an isomorphism for i <=3, p distinct from char(k). This implies rigidity for H_i(GL_n), i <=3, which in turn implies the Friedlander-Milnor conjecture in positive characteristic in degrees <=3.

math.KT↗
arXiv1998

Congruence subgroups and twisted cohomology of SL_n(F[t])

Let F be a field of characteristic zero and let V be an irreducible representation of SL_n(F). In this paper, we compute the first cohomology of SL_n(F[t]) with coefficients in V. It agrees with H^1(SL_n(F),V) if V is not the adjoint representation, while if V = Ad, the two groups differ by an F-vector space X. We show that if n=2, X is infinite dimensional, while if n>2, dim X = 1. We also study the abelianization of the kernel of the map SL_n(F[t])-->SL_n(F) given by setting t=0, where now F is any field. We conjecture that this abelianization is the adjoint representation sl_n(F) if n>2 and F is finite, and prove this in the case n=3, F=F_2, F_3.

math.KT↗
arXiv1998

Relative completions of linear groups over Z[t] and Z[t,t^{-1}]

We compute the completion of the groups SL_n(Z[t]) and SL_n(Z[t,t^{-1}]) relative to the obvious homomorphisms to SL_n(Q); this is a generalization of the classical Malcev completion. We also make partial computations of the rational second cohomology of these groups.

math.GR↗