Systolic category

The systole (or systolic category) is a numerical invariant of a closed manifold M, introduced by Mikhail Katz and Yuli Rudyak in 2006, by analogy with the Lusternik–Schnirelmann category. The invariant is defined in terms of the systoles of M and its covers, as the largest number of systoles in a product yielding a curvature-free lower bound for the total volume of M. The invariant is intimately related to the Lusternik-Schnirelmann category. Thus, in dimensions 2 and 3, the two invariants coincide. In dimension 4, the systolic category is known to be a lower bound for the Lusternik–Schnirelmann category.

Bibliography
  • Dranishnikov, A.; Rudyak, Y. (2009) Stable systolic category of manifolds and the cup-length. Journal of Fixed Point Theory and Applications 6, no. 1, 165–177.
  • Katz, M.; Rudyak, Y. (2008) Bounding volume by systoles of 3-manifolds. Journal of the London Mathematical Society 78, no 2, 407–417.
  • Dranishnikov, A.; Katz, M.; Rudyak, Y. (2011) Cohomological dimension, self-linking, and systolic geometry. Israel Journal of Mathematics 184, no 1, 437–453. See arXiv:0807.5040.
  • Brunnbauer, M. (2008) On manifolds satisfying stable systolic inequalities. Mathematische Annalen 342, no. 4, 951–968.
  • Katz, M.; Rudyak, Y. (2006) Lusternik–Schnirelmann category and systolic category of low dimensional manifolds. Communications on Pure and Applied Mathematics 59, no. 10, 1433–1456.
gollark: As you can see, this was compiled with HTTP/2.
gollark: ```configure arguments: --prefix=/etc/nginx --conf-path=/etc/nginx/nginx.conf --sbin-path=/usr/bin/nginx --pid-path=/run/nginx.pid --lock-path=/run/lock/nginx.lock --user=http --group=http --http-log-path=/var/log/nginx/access.log --error-log-path=stderr --http-client-body-temp-path=/var/lib/nginx/client-body --http-proxy-temp-path=/var/lib/nginx/proxy --http-fastcgi-temp-path=/var/lib/nginx/fastcgi --http-scgi-temp-path=/var/lib/nginx/scgi --http-uwsgi-temp-path=/var/lib/nginx/uwsgi --with-cc-opt='-march=x86-64 -mtune=generic -O2 -pipe -fno-plt -D_FORTIFY_SOURCE=2' --with-ld-opt=-Wl,-O1,--sort-common,--as-needed,-z,relro,-z,now --with-compat --with-debug --with-file-aio --with-http_addition_module --with-http_auth_request_module --with-http_dav_module --with-http_degradation_module --with-http_flv_module --with-http_geoip_module --with-http_gunzip_module --with-http_gzip_static_module --with-http_mp4_module --with-http_realip_module --with-http_secure_link_module --with-http_slice_module --with-http_ssl_module --with-http_stub_status_module --with-http_sub_module --with-http_v2_module --with-mail --with-mail_ssl_module --with-pcre-jit --with-stream --with-stream_geoip_module --with-stream_realip_module --with-stream_ssl_module --with-stream_ssl_preread_module --with-threads```
gollark: Arch.
gollark: I believe my distro does that already.
gollark: Nginx is the thing doing the HTTP/2ing and is configured for it.
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