What is the lattice structure of manganese?
Ah, good old Mn. This is one of the uglier ones in terms of phases. 
$\alpha$-Mn, the room temperature phase, is CBCC (A12 family), fairly unusual. Near 1000K, Mn transitions to $\beta$-Mn, with a simple cubic (A13 family) crystal structure. $\gamma$-Mn is FCC, face-centered cubic (A1), and $\delta$-Mn is BCC, body-centered cubic (A2).

One excellent reference is A.T. Dinsdale, SGTE Data for Pure Elements, CALPHAD 15(4) 317-425 (1991). There is a readily available PDF of an updated version of that paper that can be found (Dinsdale). This paper gives the accepted Gibbs free energy functions for the elements in their various stable, and some unstable, forms.

Editing to add additional information on the $\alpha$- and $\beta$-Mn crystal phases. Both of these phases are pretty weird, and each is the first observed crystal with their structure (hence they are the 'prototype' in crystallography terms). 

$\alpha$-Mn was described by A.J. Bradley and J. Thewlis in 'The Crystal Structure of $\alpha$-Manganese', Proc. Royal Society 115, 456-471 (1927). The unit cell is based on body-centered cubic, but contains 58 atoms representing 4 distinct positions. It can be thought of as a bcc Bravais lattice with a 29 atom basis. Mn is the only element that exhibits this crystal structure.

$\beta$-Mn was described by G.D. Preston in Phil. Mag. 5(33) 1207-1212 (1928). The unit cell is simple cubic, containing 20 atoms in 2 groups. Preston's unit cell drawing is significantly worse than that of Bradley and Thewlis, although representations of the unit cell can be easily found on the web these days. Again, Mn is the only element that exhibits this crystal structure.

The $\delta$-Mn and $\gamma$-Mn are normal, boring bcc and fcc crystals, so no need to point to references for them.

I would provide links to the papers if I could - my institute does not have direct access to them.

As yet further info on the crystal structure references.  The A12, A13, ... notation is the Strukturebericht notation used by crystallographers. Unfortunately, there are a number of other notations used as well, including Schonenflies and Hermann-Mauguin. The Strukturebericht is pretty basic, mostly a catalog of crystal types, while the others convey the actual symmetries of the crystal structure. Some of this confusion of notations seems to date from the early days of crystallography when it was done in many places and languages. There are seven crystal systems, 14 Bravais lattices, 32 crystallographic point groups, and 230 space groups to be described to some level by the various notations. For this answer I stuck to the Strukturebericht because that is the notation used in the Dinsdale reference on Gibbs free energies of the elemental phases.